# On the number of nodal domains of homogeneous caloric polynomials

Matthew Badger  and  Cole Jeznach  Department of Mathematics, University of
Connecticut, Storrs, CT 06269-1009<EMAIL_ADDRESS>School of
Mathematics, University of Minnesota, Minneapolis, MN, 55455.
<EMAIL_ADDRESS>
(Date: January 13, 2024)

###### Abstract.

We investigate the minimum and maximum number of nodal domains across all
time-dependent homogeneous caloric polynomials of degree $d$ in
$\mathbb{R}^{n}\times\mathbb{R}$ (space $\times$ time), i.e., polynomial
solutions of the heat equation satisfying $\partial_{t}p\not\equiv 0$ and

$p(\lambda x,\lambda^{2}t)=\lambda^{d}p(x,t)\quad\text{for all
$x\in\mathbb{R}^{n}$, $t\in\mathbb{R}$, and $\lambda>0$.}$

When $n=1$, it is classically known that the number of nodal domains is
precisely $2\lceil d/2\rceil$. When $n=2$, we prove that the minimum number of
nodal domains is 2 if $d\not\equiv 0\pmod{4}$ and is 3 if $d\equiv 0\pmod{4}$.
When $n\geq 3$, we prove that the minimum number of nodal domains is $2$ for
all $d$. Finally, we show that the maximum number of nodal domains is
$\Theta(d^{n})$ as $d\rightarrow\infty$ and lies between
$\lfloor\frac{d}{n}\rfloor^{n}$ and $\binom{n+d}{n}$ for all $n$ and $d$. As
an application and motivation for counting nodal domains, we confirm existence
of the singular strata in Mourgoglou and Puliatti’s two-phase free boundary
regularity theorem for caloric measure.

###### Key words and phrases:

heat equation, caloric polynomial, nodal domain, free boundary regularity

###### 2020 Mathematics Subject Classification:

Primary: 35K05. Secondary: 26C05, 26C10, 35R35.

M. Badger was partially supported by NSF DMS grant 2154047. C. Jeznach was
partially supported by Simons Collaborations in MPS grant 563916 and NSF DMS
grant 2000288.

###### Contents

  1. 1 Introduction
    1. 1.1 Definitions and examples
    2. 1.2 Comparison with spherical harmonics and Grushin spherical harmonics
    3. 1.3 Free boundary regularity for caloric measure
    4. 1.4 Proof strategies and outline of the paper
  2. 2 Basic properties of homogeneous caloric polynomials
  3. 3 HCP in high spatial dimensions
  4. 4 HCP in $\mathbb{R}^{2+1}$, Part I: lower bounds
  5. 5 HCP in $\mathbb{R}^{2+1}$, Part II: constructions
    1. 5.1 Two nodal domains when $d\equiv 2\pmod{4}$
    2. 5.2 Two nodal domains when $d$ is odd
    3. 5.3 Three nodal domains when $d\equiv 0\pmod{4}$
  6. 6 Proof of Lemma 5.2

##  1\. Introduction

A general motivation for studying caloric polynomials comes from their
ubiquity in the theory of parabolic PDEs as finite order solutions of the heat
equation. After showing that tangent functions of solutions to a parabolic PDE
are homogeneous caloric polynomials (hereafter abbreviated _hcps_), one can
deduce strong unique continuation principles and estimate the dimension of
nodal and singular sets of solutions [Che98]. In a similar vein, it was
recently shown that zero sets of hcps appear as the supports of tangent
measures in non-variational free boundary problems for caloric measure [MP21].
Hcps also arise in geometric contexts such as understanding the dimension of
ancient caloric functions on manifolds with polynomial growth [CM21].

In this paper, with a goal of confirming existence of singular strata in the
aforementioned free boundary regularity problem for caloric measure (see
§1.3), we investigate basic topology of nodal domains of hcps in
$\mathbb{R}^{n+1}\equiv\mathbb{R}^{n}\times\mathbb{R}$ (space $\times$ time).
In particular, for each ambient dimension and for each parabolic degree (see
§1.1), we would like to determine the minimum and maximum number of possible
nodal domains. This type of question has been studied extensively for
spherical harmonics (see §1.2 for a brief survey). Also, up to scaling by a
constant, for each degree $d$, there is a unique hcp in $\mathbb{R}^{1+1}$ of
degree $d$ and the number of nodal domains is precisely $2\lceil d/2\rceil$
(see §2). Thus, the problem is to determine what happens for time-dependent
hcps in at least two space variables. We fully determine the minimum number of
domains (with detailed constructions) and establish asymptotic bounds for the
maximum number of domains.

######  Theorem 1.1 (minimum number of nodal domains).

When $n\geq 2$, the minimum number $m_{n,d}$ of nodal domains of time-
dependent homogeneous caloric polynomials in $\mathbb{R}^{n+1}$ of degree
$d\geq 2$ satisfies (see Figure 1.1)

(1.1) $m_{2,d}=\begin{cases}2,&\text{when }d\not\equiv 0\pmod{4},\\\
3,&\text{when }d\equiv 0\pmod{4}.\end{cases}$

When $n\geq 3$, we have $m_{n,d}=2$ for all $d\geq 2$.

Figure 1.1. Gallery of nodal sets of homogeneous caloric polynomials in
$\mathbb{R}^{2+1}$ achieving the minimum number $m_{2,d}$ of nodal domains.
For increased visibility, we show the intersection of the full nodal set with
a spherical annulus. Left: (5.7) with $d=4$, $\varepsilon=0.2$,
$\alpha=\pi/10$. Middle: (5.2) with $d=5$, $\varepsilon=0.3$, $\alpha=\pi/10$.
Right: (5.1) with $d=6$, $\varepsilon=0.05$.

We prove the case $n\geq 3$ of Theorem 1.1 in §3 by modifying a construction
from [BET17] for homogeneous harmonic polynomials (hereafter abbreviated
_hhps_). The case $n=2$ is established in §§4 and 5, partly by employing the
perturbation technique from [Lew77]. Notably, the proof that $m_{2,4k}=3$ is
the most difficult argument in the paper.

######  Theorem 1.2 (maximum number of nodal domains).

For all $n\geq 2$, the maximum number $M_{n,d}$ of nodal domains of time-
dependent homogeneous caloric polynomials in $\mathbb{R}^{n+1}$ of degree $d$
is $\Theta(d^{n})$ as $d\rightarrow\infty$. More precisely, for all $n\geq 2$
and $d\geq 2$,

(1.2) $\left\lfloor\frac{d}{n}\right\rfloor^{n}\leq
M_{n,d}\leq\binom{n+d}{n}.$

We prove Theorem 1.2 in §3. The lower bound on $M_{n,d}$ in (1.2) is based on
an elementary construction using products of hcps in $\mathbb{R}^{1+1}$. The
upper bound follows from an indirect application of Courant’s nodal domain
theorem, exploiting a connection between negative time-slices of caloric
polynomials in $\mathbb{R}^{1+1}$ and Hermite orthogonal polynomials.

######  Remark 1.3.

Classical theorems in algebraic geometry imply that the maximal number of
nodal domains of polynomials $p:\mathbb{R}^{n+1}\rightarrow\mathbb{R}$ of
degree $d$ is $\Theta(d^{n+1})$ as $d\rightarrow\infty$ and lies between
$\binom{d}{0}+\cdots+\binom{d}{n+1}$ and $(d+1)^{n}(d+2)$; see [Sta07,
Proposition 2.4] for the lower bound (from hyperplane arrangements) and
[Mil64, Theorem 3] for the upper bound. Thus, perhaps not unsurprisingly, the
behavior of hcps is distinct from the behavior of general polynomials. Finding
the exact value of $M_{n,d}$ appears to be a difficult problem. Except for
some low degrees ($d\leq 6$), the corresponding problem for spherical
harmonics in $\mathbb{R}^{3}$ is also open [Ley96].

We discuss proof strategies for the main theorems and outline the paper in
§1.4.

###  1.1. Definitions and examples

Let us decode the terminology in the statements of the main theorems. We study
the heat equation $(\partial_{t}-\Delta)u=0$ in
$\mathbb{R}^{n+1}=\\{(x,t):x\in\mathbb{R}^{n},t\in\mathbb{R}\\}$, where
$\Delta\equiv\Delta_{x}=\sum_{1}^{n}\partial_{x_{i}}\partial_{x_{i}}$. The
natural notion of homogeneity in this setting is anisotropic.

######  Definition 1.4.

A function $f(x,t)$ is parabolically homogeneous of degree $d\in\mathbb{R}$ if

(1.3) $\displaystyle f(\lambda x,\lambda^{2}t)$
$\displaystyle=\lambda^{d}f(x,t)\quad\text{for all $\lambda>0$ and $(x,t)$.}$

######  Definition 1.5.

A polynomial $p(x,t)$ is a homogeneous caloric polynomial (hcp) of degree $d$
if $p$ satisfies the heat equation and is parabolically homogeneous of degree
$d\in\mathbb{N}=\\{0,1,2,\dotsc\\}$. We say that $p$ is time-dependent if
$\partial_{t}p\not\equiv 0$.

######  Example 1.6.

For any exponent $k\in\mathbb{N}$ and multi-index $\alpha\in\mathbb{N}^{n}$,
the monomial $t^{k}x^{\alpha}$ is parabolically homogeneous of degree
$2k+\left\lvert\alpha\right\rvert=2k+\alpha_{1}+\cdots+\alpha_{n}$. In
particular, if $p(x,t)$ is parabolically homogeneous of degree
$d\in\mathbb{N}$, then each monomial in $p(x,t)$ has the form
$t^{k}x^{\alpha}$ where $2k+\left\lvert\alpha\right\rvert=d$. Parabolic and
algebraic homogeneity are distinct notions; e.g.,

(1.4) $p(x,t)=t^{2}+tx^{2}+x^{4}/12$

is an hcp of degree $4$ in $\mathbb{R}^{1+1}$, but $p$ is not _algebraically_
homogeneous. Nevertheless, the parabolic degree and the algebraic degree of an
hcp always coincide.

######  Remark 1.7.

If $p$ is a time-dependent hcp, then the degree of $p$ is at least 2.

######  Definition 1.8.

The nodal domains of a continuous function $u(x,t)$ on $\mathbb{R}^{n+1}$ are
the connected components of the set $\\{(x,t):u(x,t)\neq 0\\}$. We let
$\mathcal{N}(u)\in\mathbb{N}\cup\\{+\infty\\}$ denote the number of nodal
domains of $u$. The nodal set of $u$ is $\\{(x,t):u(x,t)=0\\}$.

######  Remark 1.9.

Let
$\mathbb{S}^{n}=\\{(x,t)\in\mathbb{R}^{n+1}:x_{1}^{2}+\cdots+x_{n}^{2}+t^{2}=1\\}$.
By parabolic homogeneity, if $u$ is an hcp of degree $d\geq 1$ in
$\mathbb{R}^{n+1}$, then $\mathcal{N}(u)$ is the number of connected
components of $\\{(x,t)\in\mathbb{S}^{n}:u(x,t)\neq 0\\}$ in $\mathbb{S}^{n}$.

######  Remark 1.10.

For any $n\geq 1$, the mean value property (e.g., see [Eva10, p. 50]) implies
that a non-constant solution $u:\mathbb{R}^{n+1}\rightarrow\mathbb{R}$ of the
heat equation takes positive and negative values in any neighborhood of a zero
of $u$. In particular, $m_{n,d}\geq 2$ for all $n\geq 1$ and $d\geq 2$.

######  Example 1.11.

For any $1\leq j\leq n$, the polynomial $p(x,t)=2t+x_{j}^{2}$ in
$\mathbb{R}^{n+1}$ is an hcp of degree $2$. Moreover, $p(x,t)$ has exactly two
nodal domains: $\\{t>x_{j}^{2}\\}$ and $\\{t<x_{j}^{2}\\}$. Thus, for all
$n\geq 1$, the minimum number of nodal domains of time-dependent hcps in
$\mathbb{R}^{n+1}$ of degree 2 is $m_{n,2}=2$.

######  Example 1.12.

Up to scaling by a constant multiple, for every $d\geq 1$, there exists a
unique hcp $p_{d}(x,t)$ of degree $d$ in $\mathbb{R}^{1+1}$ and
$m_{1,d}=M_{1,d}=\mathcal{N}(p_{d})=2\lceil d/2\rceil$. See §2 for the
details.

######  Example 1.13.

The polynomial

(1.5) $p(x,y,t)=150t(3x+y)+27x^{3}+267x^{2}y+144xy^{2}-64y^{3}$

is an hcp of degree 3 in $\mathbb{R}^{2+1}$ and $\mathcal{N}(p)=2$. This
example can be found by evaluating (5.2) with $d=3$, $\varepsilon=1$, and
$(\cos\alpha,\sin\alpha)=(3/5,4/5)$ and multiplying by a constant to obtain a
polynomial with integer coefficients. It can be checked that $\nabla
p(x,y,t)=0$ if and only if $(x,y,t)=(0,0,0)$.

######  Example 1.14.

The polynomial

(1.6) $\begin{split}p(x,y,t)=7500t^{2}&+150t(37x^{2}-7xy+13y^{2})\\\
&+192x^{4}+176x^{3}y+1623x^{2}y^{2}-351xy^{3}-108y^{4}\end{split}$

is an hcp of degree 4 in $\mathbb{R}^{2+1}$ and $\mathcal{N}(p)=3$. This
example can be found by evaluating (5.7) with $d=4$, $\varepsilon=1/2$, and
$(\cos\alpha,\sin\alpha)=(3/5,4/5)$ and multiplying by a constant to obtain a
polynomial with integer coefficients.

######  Example 1.15.

The polynomial

(1.7) $p(x,y,z,t)=12t^{2}+12tx^{2}+x^{4}+y^{4}-6y^{2}z^{2}+z^{4}$

is an hcp of degree 4 in $\mathbb{R}^{3+1}$ and $\mathcal{N}(p)=2$. See
Proposition 3.1. It can be checked that $\nabla p(x,y,z,t)=0$ if and only if
$(x,y,z,t)=(0,0,0,0)$.

###  1.2. Comparison with spherical harmonics and Grushin spherical harmonics

Steady-state solutions of the heat equation on $\mathbb{R}^{n+1}$ correspond
to harmonic functions on $\mathbb{R}^{n}$. Nodal geometry of homogeneous
harmonic polynomials $p$ in $\mathbb{R}^{n}$ (also called _solid harmonics_)
and of the so-called _spherical harmonics_ $p|_{\mathbb{S}^{n-1}}$ is well-
studied. See [EJN07, NS09, Log18a, Log18b] for a short sample, including
results for Laplace-Beltrami eigenfunctions on closed Riemannian manifolds
beyond the sphere.

Parallel to the quantities $m_{n,d}$ and $M_{n,d}$ defined in Theorems 1.1 and
1.2, we let $\tilde{m}_{n,d}$ and $\tilde{M}_{n,d}$ denote the minimum and
maximum number of nodal domains of hhps in $\mathbb{R}^{n}$ of degree $d$,
respectively. In the line ($n=1$), the only harmonic functions are affine and
$\tilde{m}_{1,1}=\tilde{M}_{1,1}=2$ trivially. In the plane ($n=2$), since
harmonic functions can be written as the real part of a complex-analytic
function, the nodal sets of hhp degree $d$ are rotations of
$\\{(x,y):\mathrm{Re}(x+iy)^{d}=0\\}$ and $\tilde{m}_{2,d}=\tilde{M}_{2,d}=2d$
for all $d\geq 1$. The situation becomes more interesting when $n\geq 3$. Lewy
[Lew77] proved that $\tilde{m}_{3,d}=2$ whenever $d\geq 1$ is odd and
$\tilde{m}_{3,d}=3$ whenever $d\geq 2$ is even. An explicit example of a
degree 3 hhp with exactly two nodal domains,

(1.8) $p(x,y,z)=x^{3}-3xy^{2}+z^{3}-(3/2)(x^{2}+y^{2})z,$

was found independently by Szulkin [Szu79]. Badger, Engelstein, and Toro
[BET17] gave a simple construction (utilizing the explicit description of hhps
in $\mathbb{R}^{2}$) that shows $\tilde{m}_{n,d}=2$ for all $n\geq 4$ and
$d\geq 1$, independent of the parity of $d$.

When $n\geq 3$, any hhp $p$ in $\mathbb{R}^{n}$ of degree $d$ satisfies the
equation

$-\Delta_{\mathbb{S}^{n-1}}p|_{\mathbb{S}^{n-1}}=d(d+n-2)p|_{\mathbb{S}^{n-1}},$

where $\Delta_{\mathbb{S}^{n-1}}$ denotes the Laplace-Beltrami operator on the
sphere. Courant’s nodal domain theorem asserts that when listed with
multiplicity, the $m$-th eigenfunction of the Laplace-Beltrami operator on a
closed $C^{1}$ Riemannian manifold has at most $m$ nodal domains [CH53,
CLMM20]. Since the dimension of the vector space of hhps of degree $d\geq 2$
in $\mathbb{R}^{n}$ is exactly $\binom{n+d-1}{n-1}-\binom{n+d-3}{n-1}$ (see
[ABR01, Proposition 5.8]), the maximal number of linearly independent hhps of
degree at most $d$ is exactly
$\binom{n+d-1}{n-1}+\binom{n+d-2}{n-1}=O(d^{n-1})$ as $d\rightarrow\infty$.
Thus, $\tilde{M}_{n,d}=O(d^{n-1})$ as $d\rightarrow\infty$. It is known that
the upper bound on $M_{n,d}$ provided by Courant’s theorem is not sharp. See
[Ley96] for further discussion and the (still to this day) state-of-the-art
bounds on $M_{2,d}$.

In [LTY15], Liu, Tian, and Yang study the minimum number $\tilde{m}^{G}_{2,d}$
of nodal domains of _Grushin spherical harmonics_ , i.e. parabolically
homogenenous polynomial solutions $p(x,y,t)$ of the operator
$L_{G}=\partial_{x}^{2}+\partial_{y}^{2}+(x^{2}+y^{2})\partial_{t}^{2}$ on
$\mathbb{R}^{2+1}$. In particular, they prove that $\tilde{m}^{G}_{2,d}=2$
when $d\equiv 0\pmod{4}$, whereas $\tilde{m}^{G}_{2,d}\geq 3$ when $d\equiv
0\pmod{4}$; moreover, they provide examples that show
$m^{G}_{2,4}=m^{G}_{2,8}=m^{G}_{2,12}=3$. The method of proof is the
perturbation technique of Lewy _op. cit._ Other than the fact that the
parabolic scaling is the natural scaling for solutions of the Grushin operator
and the heat operator, there does not seem to be any immediate connection
between Grushin spherical harmonics and hcps. To wit, when $p(x,y,t)$ is a
Grushin spherical harmonic, so is $p(x,y,-t)$, whereas this strong symmetry
property is not enjoyed by time-dependent solutions of the heat equation.
Thus, the main results in [LTY15] cannot be used to establish Theorem 1.1 or
vice-versa.

###  1.3. Free boundary regularity for caloric measure

The phrase _caloric measure_ refers to a family of probability measures
$\omega^{X,t}_{\Omega}$ that are supported on a subset of the boundary
$\partial\Omega$ of a space-time domain
$\Omega\subset\mathbb{R}^{n+1}=\mathbb{R}^{n}\times\mathbb{R}$ and indexed by
the points $(X,t)\in\Omega$. They arise in connection with the Dirichlet
problem for the heat equation. Stochastically, $\omega^{X,t}_{\Omega}(E)$ is
the probability that the trace $(B(t+s),t-s)_{s\geq 0}$ of a Brownian traveler
$B(s)$ starting at $B(t)=X$ and sent into the past first intersects
$\partial\Omega$ inside the set $E\subset\mathbb{R}^{n+1}$. For a consolidated
introduction to caloric measure, see [BG23, §3], and for extensive background,
see [Wat12] or [Doo01]. Recent progress on free boundary regularity for
caloric measure was made by Mourgoglou and Puliatti [MP21], propelling the
time-dependent theory for caloric measure closer to the better developed,
time-independent theory for harmonic and elliptic measure (see e.g. [KPT09,
AM19, BET22]). Among other results—and setting aside certain technical
assumptions related to the heat potential theory—their work leads to the
following description of the asymptotic shape of the free boundary in the two-
phase setting.

######  Theorem 1.16 (Mourgoglou-Puliatti).

Assume that $\Omega^{+}=\mathbb{R}^{n+1}\setminus\overline{\Omega^{-}}$ and
$\Omega^{-}=\mathbb{R}^{n+1}\setminus\overline{\Omega^{+}}$ are complementary
domains in $\mathbb{R}^{n+1}$ with a sufficiently regular (for heat potential
theory), common boundary
$\partial\Omega=\partial\Omega^{+}=\partial\Omega^{-}$. Let
$\omega^{\pm}=\omega^{X_{\pm},t_{0}}_{\Omega^{\pm}}$ be caloric measures for
$\Omega^{\pm}$ with poles at $(X_{\pm},t_{0})\in\Omega^{\pm}$ or poles at
infinity. If $\omega^{\pm}$ are doubling measures,
$\omega^{+}\ll\omega^{-}\ll\omega^{+}$, and the Radon-Nikodym derivatives
$d\omega^{-}/d\omega^{+}$ and $d\omega^{+}/d\omega^{-}$ are bounded continuous
functions on $\partial\Omega\cap\\{t\leq t_{0}\\}$, then
$\partial\Omega=\Gamma_{1}\cup\Gamma_{2}\cup\cdots\cup\Gamma_{d_{0}},$ where
geometric blow-ups (tangent sets)
$\Sigma=\lim_{i\rightarrow\infty}r_{i}^{-1}(\partial\Omega-x)$ of
$\partial\Omega$ at $x\in\Gamma_{d}$ along sequence of scales
$r_{i}\rightarrow 0$ are zero sets of homogeneous caloric polynomials $p$ of
degree $d$ such that $\\{p>0\\}$ and $\\{p<0\\}$ are connected. Cf. [MP21,
Theorems III, IV].

The main results in this paper validate Mourgoglou and Puliatti’s theory by
confirming existence of time-dependent hcps with two nodal domains.
Furthermore, we obtain a refined description of the free boundary in low
dimensions.

######  Theorem 1.17.

When $n=1$, $\partial\Omega=\Gamma_{1}$. When $n=2$,

(1.9) $\partial\Omega=\bigcup_{k\geq
0}\Gamma_{4k+1}\cup\Gamma_{4k+2}\cup\Gamma_{4k+3};$

for every $d\not\equiv 0\pmod{4}$, the stratum $\Gamma_{d}$ is nonempty for
some pair of domains satisying the free boundary condition. When $n=3$, the
stratum $\Gamma_{d}$ can be nonempty for every $d\geq 1$.

###### Proof.

When $n=1$, $\Gamma_{d}=\emptyset$ for all $d\geq 2$ by Example 1.12. When
$n=2$, $\Gamma_{d}=\emptyset$ for all $d=4k$ by Theorem 1.1. For the remaining
pairs of $n$ and $d$, the examples of hcps $p$ in $\mathbb{R}^{n+1}$ of degree
$d$ with $\mathcal{N}(p)=2$ constructed in the proof of Theorem 1.1 (see the
proofs of Proposition 3.1 and Theorems 5.4 and 5.5 for details) have smooth
zero sets outside any neighborhood of the origin. This fact is enough to
ensure that the domains $\Omega_{p}^{+}=\\{p>0\\}$ and
$\Omega_{p}^{-}=\\{p<0\\}$ associated to $p$ satisfy the background regularity
hypothesis in Theorem 1.16. If $\omega_{p}^{\pm}$ denote the caloric measures
on $\Omega_{p}^{\pm}$ with poles at infinity, then it is known that
$\omega_{p}^{+}=\omega_{p}^{-}$ and $d\omega_{p}^{-}/d\omega_{p}^{+}\equiv 1$
(see [MP21, §6]). Finally, by parabolic homogeneity, $\\{p=0\\}$ is the unique
blow-up of $\partial\Omega_{p}^{\pm}=\\{p=0\\}$ at the origin. Therefore,
$\Gamma_{d}$ is nonempty in $\partial\Omega^{\pm}_{p}$. ∎

###  1.4. Proof strategies and outline of the paper

In Section 2, we recall classical facts about hcps, including their connection
with Hermite polynomials. We also introduce a basis of hcps of degree $d$ in
$\mathbb{R}^{n+1}$, which we use in the constructions in Sections 3 and 5.

In Section 3, we first prove that $m_{n,d}=2$ for $n\geq 3$ and any $d\geq 1$
by modifying a construction in [BET17]. Next, we build time-dependent hcps in
$\mathbb{R}^{n+1}$ with a large number of nodal domains by taking products of
hcps in $\mathbb{R}^{1+1}$. Finally, after showing that any nodal domain of an
hcp necessarily intersects $\\{t=-1\\}$, we employ the proof of Courant’s
nodal domain theorem on negative time slices of hcps to establish the upper
bound on the number of nodal domains in Theorem 1.2. This leaves us to
determine the value of $m_{2,d}$ for $d\geq 1$.

In Section 4, we show that $m_{2,d}\geq 3$ whenever $d\geq 1$ satisfies
$d\equiv 0\pmod{4}$. The main argument leverages the fact that if $p$ is an
hcp of degree $d$ in $\mathbb{R}^{2+1}$, then the nodal set of
$p|_{\mathbb{S}^{2}}$ near the north and south poles is asymptotic to the zero
set of a homogeneous harmonic polynomial in two variables, which are easy to
describe and are perfectly understood.

In Section 5, we construct examples of time-dependent hcps of degree $d\geq 1$
in $\mathbb{R}^{2+1}$ with exactly $m_{2,d}$ nodal domains. The basic strategy
dates back to [Lew77]: starting with an hcp of degree $d$ in
$\mathbb{R}^{2+1}$ whose zero set we can explicitly describe, we perturb the
polynomial to produce a time-dependent hcp with the desired number of nodal
domains. This style of argument requires a careful analysis of how
perturbation affects the topology of nodal sets; see Lemma 5.2 for a precise
statement and Section 6 for the proof of the lemma.

##  2\. Basic properties of homogeneous caloric polynomials

Given an hcp $p(x,t)$ of degree $d$, we typically shall choose to write
$p(x,t)$ in the form

(2.1) $p(x,t)=t^{m}p_{m}(x)+t^{m-1}p_{m-1}(x)+\cdots+p_{0}(x)\quad\text{for
all $x\in\mathbb{R}^{n}$, $t\in\mathbb{R}$}.$

Each coefficient $p_{m-j}=p_{m-j}(x)$ is necessarily an algebraically
homogeneous polynomial of degree $d-2(m-j)$ (see Example 1.6). Moreover,
applying the heat operator to $p$ and collecting like powers of $t$, we obtain
the relations

(2.2) $0=\Delta p_{m},\quad mp_{m}=\Delta
p_{m-1},\quad\cdots,\quad(m-j)p_{m-j}=\Delta p_{m-j-1},\quad\cdots,\quad
p_{1}=\Delta p_{0}.$

As such, we refer to $p_{m}$ as the _harmonic coefficient_ of $p$ and obtain
that the other coefficients $p_{m-j}$ are polyharmonic:
$\Delta^{j+1}p_{m-j}=0$. In fact, the relations (2.2) and the requirement that
each coefficient $p_{i}(x)$ be homogeneous gives a characterization of
$p(x,t)$ being an hcp. Thus, we arrive at the following elementary method of
generating hcps: starting with any choice of $m\geq 0$ and hhp $p_{m}(x)$, use
(2.2) to inductively solve for homogeneous coefficients
$p_{m-1}(x),\dots,p_{0}(x)$; then $p(x,t)$ defined by (2.1) is an hcp.

Now, given any homogeneous polynomial $q(x)=bx^{d}$ with $x\in\mathbb{R}^{1}$,
the polynomial $r(x)=\frac{1}{(d+2)(d+1)}cx^{2}q(x)$ is the unique homogeneous
function such that $r^{\prime\prime}(x)=cq(x)$. Since the only hhps in
$\mathbb{R}^{1}$ are of the form $q(x)=b$ or $q(x)=bx$, it follows that, up to
scaling by a constant, there exists a unique hcp $p_{d}(x,t)$ in
$\mathbb{R}^{1+1}$ for each degree $d\geq 0$. We adopt the following
normalization for the hcps $p_{d}(x,t)$, emphasizing the time variable.

######  Definition 2.1.

For each $d\geq 0$, define $p_{d}:\mathbb{R}^{1+1}\rightarrow\mathbb{R}$ as
follows. When $d=2k$ is even,

$p_{d}(x,t):=t^{k}+\tfrac{k}{2!}t^{k-1}x^{2}+\tfrac{k(k-1)}{4!}t^{k-2}x^{4}+\cdots+\tfrac{k!}{(2k)!}x^{2k}=\sum_{j=0}^{k}\tfrac{k!}{(k-j)!(2j)!}t^{k-j}x^{2j}.$

When $d=2k+1$ is odd,

$p_{d}(x,t):=t^{k}x+\tfrac{k}{3!}t^{k-1}x^{3}+\tfrac{k(k-1)}{5!}t^{k-2}x^{5}+\cdots+\tfrac{k!}{(2k+1)!}x^{2k+1}=\sum_{j=0}^{k}\tfrac{k!}{(k-j)!(2j+1)!}t^{k-j}x^{2j+1}.$

######  Definition 2.2 (see [Sze75, §5.5]).

The _Hermite polynomials_ $H_{0}(x)=1$, $H_{1}(x)=2x$, $H_{2}(x)=4x^{2}-2$,
$H_{3}(x)=8x^{3}-12x$, $H_{4}(x)=16x^{4}-48x^{2}+12$,
$H_{5}(x)=32x^{5}-160x^{3}+120x$, _etc._ are the family of orthogonal
polynomials for the weighted space $L^{2}(\mathbb{R},e^{-x^{2}}dx)$ defined by
requiring that $\deg H_{d}=d$, the coefficient of $x^{d}$ in $H_{d}(x)$ is
positive, and

(2.3)
$\int_{\mathbb{R}}H_{d}(x)H_{d^{\prime}}(x)\,e^{-x^{2}}dx=\pi^{1/2}2^{d}d!\,\delta_{dd^{\prime}}\quad\text{for
all }d,d^{\prime}\in\mathbb{N}.$

######  Lemma 2.3 (see [Sze75, §5.5]).

Equivalently, for all $d\geq 0$,

(2.4) $H_{d}(x)=\sum_{j=0}^{\lfloor
d/2\rfloor}\frac{d!}{j!(d-2j)!}(-1)^{j}(2x)^{d-2j}.$

After establishing the connection between the “basic hcps” and the Hermite
polynomials (see e.g. [RW59], [Che98], [PKS21]), one can use facts about the
zeros of $H_{d}(x)$ and parabolic scaling to derive the following description
of $p_{d}(x,t)$ and its nodal set.

######  Theorem 2.4.

For all $d\geq 2$, the “basic hcp” $p_{d}(x,t)$ assumes the form

(2.5)
$p_{d}(x,t)=\left\\{\begin{array}[]{rl}(t+a_{d,1}x^{2})\cdots(t+a_{d,k}x^{2})&\text{when
$d=2k$ is even},\\\ x(t+a_{d,1}x^{2})\cdots(t+a_{d,k}x^{2})&\text{when
$d=2k+1$ is odd},\end{array}\right.$

for some distinct numbers $0<a_{d,1}<\dots<a_{d,k}$. Moreover, if we write

$\displaystyle p_{2k-1}(x,t)$
$\displaystyle=x(t+a_{1}x^{2})\cdots(t+a_{k-1}x^{2}),$ $\displaystyle
p_{2k+1}(x,t)$ $\displaystyle=x(t+c_{1}x^{2})\cdots(t+c_{k}x^{2}),$
$p_{2k}(x,t)=(t+b_{1}x^{2})\cdots(t+b_{k}x^{2}),$

with the $a_{i}$’s, $b_{i}$’s, and $c_{i}$’s each listed in increasing
order, then the coefficients associated with consecutive polynomials are
interlaced:

(2.6)
$\left\\{\begin{array}[]{l}b_{1}<a_{1}<b_{2}<a_{2}<\cdots<a_{k-1}<b_{k},\\\
\,c_{1}<b_{1}<c_{2}<b_{2}<\cdots<b_{k-1}<c_{k}<b_{k}.\end{array}\right.$

###### Proof.

Replacing $j$ by $k-j$ in the summations in Definition 2.1 yield that for all
$d\geq 0$,

(2.7) $p_{d}(x,t)=\sum_{j=0}^{\lfloor d/2\rfloor}\frac{\lfloor
d/2\rfloor!}{j!(d-2j)!}t^{j}x^{d-2j}.$

Comparing (2.4) and (2.7), we see that

(2.8) $p_{d}(x,-1)=\frac{\lfloor d/2\rfloor!}{d!}H_{d}(x/2).$

Because the Hermite polynomial $H_{d}$ is even, when $d$ is even, $H_{d}$ is
odd, when $d$ is odd, and orthogonal polynomials have a full number of
distinct real roots (see [Sze75, Theorem 3.3.1]), we can factor
$H_{d}(x)=(x^{2}-r_{1}^{2})\cdots(x-r_{k}^{2})$ for some
$0<r_{k}<\cdots<r_{1}$, when $d=2k$ is even, and
$H_{d}(x)=x(x^{2}-r_{1}^{2})\cdots(x^{2}-r_{k}^{2})$, when $d=2k+1$ is odd.
Hence

$p_{d}(x,-1)=\frac{\lfloor d/2\rfloor!}{d!}r_{1}^{2}\cdots
r_{k}^{2}x^{d-2\lfloor
d/2\rfloor}\left(\frac{x^{2}}{4r_{1}^{2}}-1\right)\cdots\left(\frac{x^{2}}{4r_{k}^{2}}-1\right).$

Together with parabolic homogeneity and the fact that $p_{d}(x,t)$ was
normalized to have leading term $t^{k}$ or $t^{k}x$, this yields (2.5) with
$a_{d,i}=1/(4r_{i}^{2})$. Thus, (2.6) follows from the interlacing of roots of
consecutive Hermite polynomials [Sze75, Theorem 3.3.2].∎

Figure 2.1. The nodal set of a degree $d$ hcp in $\mathbb{R}^{1+1}$ is a union
of $\lfloor d/2\rfloor$ nested, downward-opening parabolas with a common
turning point at the origin, and when $d$ is odd, an additional vertical line
(the $t$-axis). Thus, the number of nodal domains is precisely $2\lceil
d/2\rceil$. From left to right, we illustrate the cases $d=2$, …, $d=5$.
Inside the nodal set of $p_{d}p_{d+1}$, the “nodal parabolas” of consecutive
hcps $p_{d}$ and $p_{d+1}$ are intertwined: the “widest” parabola of $p_{d+1}$
sits above the “widest” parabola of $p_{d}$; the “widest” parabola of $p_{d}$
sits above the “second widest” parabola of $p_{d+1}$; _etc._

######  Corollary 2.5.

Any hcp $p(x,t)$ in $\mathbb{R}^{1+1}$ of degree $d\geq 1$ has exactly
$2\lceil d/2\rceil$ nodal domains. See Figure 2.1.

In contrast to the case $n=1$, given a homogeneous polynomial $q(x)$ with
$x\in\mathbb{R}^{n}$ for some $n\geq 2$, there is more than one way to produce
a homogeneous polynomial $r(x)$ such that $\Delta r(x)=cq(x)$. We can build an
explicit basis for the vector space $\mathfrak{H}_{d}(\mathbb{R}^{n+1})$ of
all hcps of degree $d$ in $\mathbb{R}^{n+1}$ (and the zero function) using
products of basic hcps in $\mathbb{R}^{1+1}$.

######  Definition 2.6.

Let $n\geq 2$. For each multi-index $\alpha=(\alpha_{1},\dots,\alpha_{n})$,
define

(2.9) $p_{\alpha}(x,t):=p_{\alpha_{1}}(x_{1},t)\cdots
p_{\alpha_{n}}(x_{n},t).$

######  Lemma 2.7.

For all $t<0$, the set $\\{p_{\alpha}(\cdot,t):\alpha\text{ is a multi-index
in }\mathbb{N}^{n}\\}$ is an orthogonal basis for the weighted space
$L^{2}(\mathbb{R}^{n},e^{-|x|^{2}/4t}\,dx)$.

###### Proof.

Let $t<0$ and let $\alpha$ and $\beta$ be multi-indices in $\mathbb{N}^{n}$
with $\alpha\neq\beta$. By Fubini’s theorem,

$\displaystyle\int_{\mathbb{R}^{n}}p_{\alpha}(x,t)p_{\beta}(x,t)e^{-\left\lvert
x\right\rvert^{2}/4\left\lvert t\right\rvert}\,dx$
$\displaystyle=\prod_{i=1}^{n}\int_{\mathbb{R}}p_{\alpha_{i}}(x_{i},t)p_{\beta_{i}}(x_{i},t)e^{-x_{i}^{2}/4\left\lvert
t\right\rvert}\,dx_{i}.$

Thus, the left hand side vanishes if and only if at least one of the terms on
the right hand side vanish. Since $\alpha\neq\beta$, there exists $1\leq i\leq
n$ such that $\alpha_{i}\neq\beta_{i}$. Write
$\gamma_{i}=(\alpha_{i}+\beta_{i})/2$. By a simple change of variables, with
$x_{i}=2|t|^{1/2}y$, parabolic homogeneity, (2.8), and (2.3),

$\begin{split}&|t|^{-1/2}\int_{\mathbb{R}}p_{\alpha_{i}}(x_{i},t)p_{\beta_{i}}(x_{i},t)e^{-x_{i}^{2}/4\left\lvert
t\right\rvert}\,dx_{i}=2\int_{\mathbb{R}}p_{\alpha_{i}}(2|t|^{1/2}y,t)p_{\beta_{i}}(2|t|^{1/2}y,t)\,e^{-y^{2}}dy\\\
&\quad=2|t|^{\gamma_{i}}\int_{R}p_{\alpha_{i}}(2y,-1)p_{\beta_{i}}(2y,-1)\,e^{-y^{2}}dy=2|t|^{\gamma_{i}}C(\alpha^{\prime},\beta^{\prime})\int_{\mathbb{R}}H_{\alpha_{i}}(y)H_{\beta_{i}}(y)\,e^{-y^{2}}dy=0.\end{split}$

By a similar argument, $\\{p_{\alpha}(\cdot,t):\alpha\text{ is a multi-index
in }\mathbb{N}^{n}\\}$ is a basis for
$L^{2}(\mathbb{R}^{n},e^{-|x|^{2}/4t}\,dx)$, because $\\{H_{d}(x):d\geq 0\\}$
is a basis for $L^{2}(\mathbb{R},e^{-x^{2}}dx)$. ∎

######  Lemma 2.8.

For all $n\geq 1$ and $d\geq 0$, the set
$\mathfrak{P}_{d}(\mathbb{R}^{n+1})=\\{p_{\alpha}:|\alpha|=d\\}$ is a basis
for $\mathfrak{H}_{d}(\mathbb{R}^{n+1})$ and
$\dim\mathfrak{H}_{d}(\mathbb{R}^{n+1})=\binom{n-1+d}{n-1}$.

###### Proof.

For a proof that $\dim\mathfrak{H}_{d}(\mathbb{R}^{n+1})=\binom{n-1+d}{n-1}$,
see [CM21, Lemma 2.2]. The fact that
$\\#\mathfrak{P}_{d}(\mathbb{R}^{n+1})=\binom{n-1+d}{n-1}$ is a standard
exercise; see e.g. [Eva10, p. 12]. Now, the set is
$\mathfrak{P}_{d}(\mathbb{R}^{n+1})$ is linearly independent, because
$\\{p_{\alpha}(\cdot,-1):|\alpha|=d\\}$ is linearly independent by Lemma 2.7.
Therefore, $\mathfrak{P}_{d}(\mathbb{R}^{n+1})$ is a basis for
$\mathfrak{H}_{d}(\mathbb{R}^{n+1})$. ∎

######  Example 2.9.

The hhp $x^{2}-y^{2}$ in $\mathbb{R}^{2}$ can be expressed as a linear
combination of the basic hcps $p_{1}(x,t)$ and $p_{1}(y,t)$:
$x^{2}-y^{2}=2(t+\frac{1}{2}x^{2})-2(t+\frac{1}{2}y^{2})$.

We will also need the following fact in the next section.

######  Lemma 2.10.

If $p$ is an hcp of degree $d$ in $\mathbb{R}^{n+1}$, then the function
$v(x):=p(x,-1)$ satisfies

(2.10) $\displaystyle-\mathrm{div}(e^{-\left\lvert x\right\rvert^{2}/4}\nabla
v(x))=(d/2)e^{-\left\lvert x\right\rvert^{2}/4}v(x).$

###### Proof.

Indeed, for $t<0$, we have
$p(x,t)=(-t)^{d/2}p((-t)^{-1/2}x,-1)=(-t)^{d/2}v((-t)^{1/2}x)$. Applying the
heat operator $\partial_{t}-\Delta$ and simplifying yields

$\displaystyle 0$
$\displaystyle=(-t)^{\frac{d}{2}-1}\left[-(d/2)v((-t)^{-1/2}x)+(1/2)\nabla
v((-t)^{-1/2}x)\cdot(-t)^{-1/2}x-\Delta v((-t)^{-1/2}x)\right].$

Thus, $(1/2)\nabla v(x)\cdot x-\Delta v(x)=(d/2)v(x)$. Therefore,

$-\mathrm{div}(e^{-|x|^{2}/4}\nabla v(x))=-e^{-|x|^{2}/4}\left[(1/2)\nabla
v\cdot x-\Delta v(x)\right]=(d/2)e^{-|x|^{2}/4}v(x).\qed$

##  3\. HCP in high spatial dimensions

In this section, we establish the case $n\geq 3$ of Theorem 1.1 and prove
Theorem 1.2.

######  Proposition 3.1 (Theorem 1.1 when $n\geq 3$).

If $\phi(x,y)$ is an hhp of degree $d\geq 1$ in $\mathbb{R}^{2}$ and
$\psi(z,t)$ is an hcp of degree $d$ in $\mathbb{R}^{1+1}$, then
$u(x,y,z,t):=\phi(x,y)+\psi(z,t)$ has exactly two nodal domains. In
particular, $m_{n,d}=2$ for all $n\geq 3$ and $d\geq 2$.

###### Proof.

We modify the proof of [BET17, Lemma 1.7], which gives a construction of hhps
in $\mathbb{R}^{n}$ with two nodal domains when $n\geq 4$. The essential
change is to show how to incorporate parabolic scaling. A trivial (but
important!) observation is that for any $(x,y)$ and $(z,t)$, the expressions
$|\phi(\lambda x,\lambda y)|=\lambda^{d}|\phi(x,y)|$ and $|\psi(\lambda
z,\lambda^{2}t)|=\lambda^{d}|\psi(z,t)|$ are (weakly) increasing as functions
of $\lambda\in[0,1]$. For any $(x,y)\neq(0,0)$, let “the line segment from
$(0,0)$ to $(x,y)$” have its usual meaning. For any $(z,t)\neq(0,0)$, let “the
line segment from $(0,0)$ to $(z,t)$” mean the curve described by $\\{(\lambda
z,\lambda^{2}t):\lambda\in[0,1]\\}$. Then $|\phi|$ and $|\psi|$ are increasing
along line segments started at the origin and $|\phi|$ and $|\psi|$ are
decreasing along line segments terminating at the origin. Also, the sets
$\\{\phi>0\\}$, $\\{\phi<0\\}$, $\\{\psi>0\\}$, and $\\{\psi<0\\}$ are each
nonempty. Keeping these preliminaries in mind, we will now show that
$\\{u>0\\}$ is path-connected. (Applying the same argument to $-u$ shows that
$\\{u<0\\}$ is path-connected.)

Suppose that $(x_{1},y_{1},z_{1},t_{1})$ and $(x_{2},y_{2},z_{2},t_{2})$ are
points at which $\phi(x_{i},y_{i})+\psi(z_{i},t_{i})>0$. Then for each
$i=1,2$, at least one of the terms $\phi(x_{i},y_{i})$ and $\psi(z_{i},t_{i})$
is positive. Because the argument that follows only involves “line segments”
with an endpoint at the origin and the monotonicity of $|\phi|$ and $|\psi|$
along those line segments, we may suppose without loss of generality that
$\psi(z_{1},t_{1})>0$ and $\psi(z_{2},t_{2})>0$. We emphasize that
$\phi(x_{1},y_{1})$ and $\phi(x_{2},y_{2})$ may have any sign (positive,
negative, zero). Choose any auxiliary point $(x_{+},y_{+})$ at which
$\phi(x_{+},y_{+})>0$. We can build a “piecewise linear” path in $\\{u>0\\}$
from $(x_{1},y_{1},z_{1},t_{1})$ to $(x_{2},y_{2},z_{2},t_{2})$ as follows.
The path is a concatenation of six “line segments” (see Figure 3.1):

Figure 3.1. Connecting $(v_{1},w_{1})=(x_{1},y_{1},z_{1},t_{1})$ to
$(v_{2},w_{2})=(x_{2},y_{2},z_{2},t_{2})$ in $\mathbb{R}^{3+1}\cap\\{u>0\\}$,
where $\phi(x,y)$ is a degree 4 hhp in $\mathbb{R}^{2}$ (left) and $\psi(z,t)$
is a degree 4 hcp in $\mathbb{R}^{1+1}$ (right).

  1. (1)

First follow the line segment from $(x_{1},y_{1},z_{1},t_{1})$ to
$(0,0,z_{1},t_{1})$.

  2. (2)

Second follow the line segment from $(0,0,z_{1},t_{1})$ to
$(x_{+},y_{+},z_{1},t_{1})$.

  3. (3)

Third follow the line segment from $(x_{+},y_{+},z_{1},z_{2})$ to
$(x_{+},y_{+},0,0)$.

  4. (4)

Fourth follow the line segment from $(x_{+},y_{+},0,0)$ to
$(x_{+},y_{+},z_{2},t_{2})$.

  5. (5)

Fifth follow the line segment from $(x_{+},y_{+},z_{2},t_{2})$ to
$(0,0,z_{2},t_{2})$.

  6. (6)

Sixth follow the line segment from $(0,0,z_{2},t_{2})$ to
$(x_{2},y_{2},z_{2},t_{2})$.

It remains to confirm that the six line segments lie in $\\{u>0\\}$. The first
segment lies in the positivity set, because $u(x_{1},y_{1},z_{1},t_{1})>0$ and
$\psi(z_{1},t_{1})>0$. (If $\phi(x_{1},y_{1})\leq 0$, then $u$ increases from
the initial point to the terminal point. If $\phi(x_{1},y_{1})\geq 0$, then
$u$ decreases from the initial point to the terminal point, but $u$ never
falls below $\psi(z_{1},t_{1})$.) Along the second segment, $\phi\geq 0$ and
$\psi>0$, so $u>0$. Along the third segment, $\phi>0$ and $\psi\geq 0$, so
$u>0$. After reversing the orientation, identical arguments show that the
fourth, fifth, and sixth segments lie in $\\{u>0\\}$, as well. ∎

We now move on to the lower and upper bounds on $M_{n,d}$ for any spatial
dimension.

######  Proposition 3.2.

For all $n\geq 2$ and $d\geq 2$, there exists a time-dependent hcp of degree
$d$ in $\mathbb{R}^{n+1}$ with at least $\lfloor d/n\rfloor^{n}$ nodal
domains.

###### Proof.

Let $c=\lfloor d/n\rfloor$. If $c\leq 1$, then the conclusion is trivial.
Thus, we may assume that $c\geq 2$. Let $p$ be an hcp in $\mathbb{R}^{1+1}$ of
degree $c$ and let $q$ be an hcp in $\mathbb{R}^{1+1}$ of degree
$b=d-(n-1)c\geq c$. A direct computation shows that
$u(x,t):=q(x_{n},t)\prod_{i=1}^{n-1}p(x_{i},t)$ is an hcp of degree $d$.
Moreover, $u$ is time-dependent, since $c\geq 2$ implies $p$ and $q$ are time-
dependent. By Corollary 2.5,

(3.1) $\mathcal{N}(u)\geq\mathcal{N}(p)^{n-1}\mathcal{N}(q)\geq c^{n-1}b\geq
c^{n}.$

We can verify the first inequality in (3.1) as follows. Suppose that $(x,t)$
and $(y,s)$ are points in $\mathbb{R}^{n+1}$ such that $u(x,t)\neq 0$,
$u(y,s)\neq 0$, and

  1. (a)

$(x_{i},t)$ and $(y_{i},s)$ belong to different nodal domains of $p$ for some
$1\leq i\leq n-1$, or

  2. (b)

$(x_{n},t)$ and $(y_{n},s)$ belong to different nodal domains of $q$.

If (a) holds, assign $j=i$ and $r=p$; otherwise, if (b) holds, assign $j=n$
and $r=q$. Let $\gamma:[0,1]\rightarrow\mathbb{R}^{n+1}$ be any curve such
that $\gamma(0)=(x,t)$ and $\gamma(1)=(y,s)$ and let $\pi$ be the orthogonal
projection onto the $e_{j}e_{n+1}$-plane. Then
$\pi\circ\gamma:[0,1]\rightarrow\mathbb{R}^{1+1}$ is a curve connecting
$(x_{j},t)$ to $(y_{j},s)$. Since $(x_{j},t)$ and $(y_{j},s)$ lie in different
nodal domains of $r$, it follows that $r(\pi(\gamma(c)))=0$ for some
$c\in(0,1)$. Thus, $u(\gamma(c))=0$, as well, by definition of $u$. Since
$\gamma$ was an arbitrary curve connecting $(x,t)$ to $(y,s)$, we conclude
that $(x,t)$ and $(y,s)$ lie in different nodal domains of $u$. Therefore,
$\mathcal{N}(u)\geq\mathcal{N}(p)^{n-1}\mathcal{N}(q)$. ∎

######  Example 3.3.

Let $u(x,y,t)=(2t+x^{2})(2t+y^{2})$. Then
$\mathcal{N}(u)=6>4=\mathcal{N}(2t+x^{2})\mathcal{N}(2t+y^{2})$. Thus, the
first inequality in (3.1) can be strict.

The next lemma will help us bound the number of nodal domains of an hcp from
above.

######  Lemma 3.4.

If $u$ is an hcp of degree $d\geq 1$ in $\mathbb{R}^{n+1}$, then every nodal
domain of $u$ intersects the hyperplane $\\{t=-1\\}$. Thus, $\mathcal{N}(u)$
is at most the number of connected components of
$\\{x\in\mathbb{R}^{n}:u(x,-1)\neq 0\\}$ in $\mathbb{R}^{n}$.

###### Proof.

By parabolic scaling, it suffices to prove that every nodal domain of $u$
intersects the lower half-space $\\{t<0\\}$. Suppose to get a contradiction
that there exists a nodal domain $U$ of $u$ such that $U\subset\\{t\geq 0\\}$.
In fact, observe that $U\subset\\{t>0\\}$, since $U$ is open. Replacing $u$ by
$-u$, if necessary, we may further assume that $U\subset\\{u>0\\}$. Write

$\mathbb{E}^{n}_{+}:=\\{(x,t)\in\mathbb{R}^{n+1}:x_{1}^{2}+\cdots+x_{n}^{2}+|t|=1\text{
and }t\geq 0\\}.$

Define an auxiliary hcp of degree $2d$ in $\mathbb{R}^{n+1}$ by
$f(x,t):=\sum_{i=1}^{n}p_{2d}(x_{i},t)$. By Theorem 2.4, there exist constants
$a_{1},\dots,a_{d}>0$ such that
$f(x,t)=\sum_{i=1}^{n}(t+a_{1}x_{i}^{2})\cdots(t+a_{d}x_{i}^{2}).$ Hence
$f(x,t)>0$ for all $(x,t)\in\mathbb{E}^{n}_{+}$. Thus, because $f$ is
continuous and $\mathbb{E}^{n}_{+}$ is compact, we can find $c>0$ such that
$f(x,t)>c$ for all $(x,t)\in\mathbb{E}^{n}_{+}$.

To proceed, choose $\varepsilon>0$ small enough so that
$v_{\varepsilon}:=u-\varepsilon f$ is positive at some point in $U$. Let $V$
be any connected component of $U\cap\\{v_{\varepsilon}>0\\}$. If $(x,t)\in V$
and $x_{1}^{2}+\cdots+x_{n}^{2}+|t|=\lambda^{2}$, then
$(x/\lambda,t/\lambda^{2})\in\mathbb{E}^{n}_{+}$ and we see that

$\displaystyle v_{\varepsilon}(x,t)$
$\displaystyle=\lambda^{d}u(x/\lambda,t/\lambda^{2})-\varepsilon\lambda^{2d}f(x/\lambda,t/\lambda^{2})\leq\lambda^{d}(\left\lVert
u\right\rVert_{L^{\infty}(\mathbb{E}^{n}_{+})}-\lambda^{d}\varepsilon c)<0$

whenever $\lambda\gg 1$ (depending only on $d$, $c$, $\varepsilon$, and
$\|u\|_{L^{\infty}(\mathbb{E}^{n}_{+})}$). This shows that $V$ is bounded.
Furthermore, since $\partial V\subset\partial
U\cup\\{v_{\varepsilon}=0\\}\subset\\{u=0\\}\cup\\{v_{\varepsilon}=0\\}$, we
have $v_{\varepsilon}\leq 0$ on $\partial V$. Therefore, $v_{\varepsilon}\leq
0$ throughout $V$ by the maximum principle for solutions of the heat equation.
This contradicts our assertion that $v_{\varepsilon}$ is positive at some
point of $V$. ∎

######  Example 3.5.

For all $k\geq 1$, the basic hcp $p_{2k}(x,t)$ of degree $2k$ in
$\mathbb{R}^{1+1}$ has $\mathcal{N}(p_{2k})=2k$, but
$\mathcal{N}(p_{2k}|\\{t=-1\\})=2k+1$. See Figure 2.1.

We are ready to present an analogue of Courant’s nodal domain theorem for
negative time-slices of hcps, which gives the upper bound on $M_{n,d}$. For
the original version of Courant’s theorem, see [CH53, Chapter VI, §6].

######  Theorem 3.6.

If $u$ is an hcp of degree $d$ in $\mathbb{R}^{n+1}$, then
$\mathcal{N}(u)\leq\binom{n+d}{n}$.

###### Proof.

Let $H$ denote the Hilbert space $L^{2}(\mathbb{R}^{n},e^{-x^{2}/4}dx)$. By
Lemma 2.7, as $\alpha$ ranges over all multi-indices in $\mathbb{N}^{n}$, the
polynomials $p_{\alpha}(x):=p_{\alpha}(x,-1)$ form an orthogonal basis for
$H$. Assign $q(x):=u(x,-1)$. By Lemma 3.4, $\mathcal{N}(u)\leq\mathcal{N}(q)$.
Thus, to establish the theorem, it suffices to prove that
$k:=\mathcal{N}(q)\leq\binom{n+d}{n}=\\#\\{p_{\alpha}(x):|\alpha|\leq d\\}$.
Enumerate the nodal domains of $q$ by
$V_{1},\dotsc,V_{k}\subset\mathbb{R}^{n}$. We suppose for the sake of
contradiction that $k>\binom{n+d}{n}$. Consider the homogeneous system of
$\binom{n+d}{n}$ linear equations $\sum_{i=1}^{k}c_{\alpha,i}a_{i}=0$, where

$c_{\alpha,i}:=\int_{\mathbb{R}^{n}}q(x)\chi_{V_{i}(x)}p_{\alpha}(x)\,e^{-x^{2}/4}dx\quad\text{for
all $|\alpha|\leq d$ and $1\leq i\leq k$}.$

Since $k>\binom{n+d}{n}$, we can find a non-zero solution vector
$(a_{1},\dots,a_{k})$. By definition of the linear system, the function $g$
defined by $g(x):=\sum_{i=1}^{k}a_{i}q(x)\chi_{V_{i}}(x)$ is orthogonal to
$p_{\alpha}$ in $H$ for all $|\alpha|\leq d$. On the one hand, by (2.10) and
integration by parts,

(3.2) $\begin{split}&\int_{\mathbb{R}^{n}}\left\lvert\nabla
g(x)\right\rvert^{2}e^{-\left\lvert
x\right\rvert^{2}/4}\;dx=\sum_{i=1}^{k}a_{i}^{2}\int_{V_{i}}\left\lvert\nabla
q(x)\right\rvert^{2}e^{-\left\lvert x\right\rvert^{2}/4}\;dx\\\
&\qquad=\sum_{i=1}^{k}a_{i}^{2}\int_{V_{i}}\frac{d}{2}q(x)^{2}e^{-\left\lvert
x\right\rvert^{2}/4}\;dx=\frac{d}{2}\int_{\mathbb{R}^{n}}g(x)^{2}e^{-\left\lvert
x\right\rvert^{2}/4}\;dx.\end{split}$

On the other hand, we can expand $g$ with respect to the orthogonal basis
$\\{p_{\alpha}\\}$ for $H$, say
$g(x)=\sum_{\left\lvert\alpha\right\rvert>d}b_{\alpha}p_{\alpha}(x)$ for some
coefficients $b_{\alpha}$. Now, by (2.10) and integration by parts

$\displaystyle\int_{\mathbb{R}^{n}}\nabla_{x}p_{\alpha}(x)\cdot\nabla_{x}p_{\beta}(x)e^{-\left\lvert
x\right\rvert^{2}/4}\,dx$
$\displaystyle=\int_{\mathbb{R}^{n}}\dfrac{\left\lvert\beta\right\rvert}{2}p_{\alpha}(x)p_{\beta}(x)e^{-\left\lvert
x\right\rvert^{2}/4}\,dx=0\quad\text{for all }\alpha\neq\beta.$

Thus, using (2.10) and integration by parts once more,

(3.3) $\begin{split}&\int_{\mathbb{R}^{n}}\left\lvert\nabla
g(x)\right\rvert^{2}e^{-\left\lvert
x\right\rvert^{2}/4}\;dx=\sum_{\left\lvert\alpha\right\rvert>d}b_{\alpha}^{2}\int_{\mathbb{R}^{n}}\left\lvert\nabla_{x}p_{\alpha}(x)\right\rvert^{2}e^{-\left\lvert
x\right\rvert^{2}/4}\;dx\\\
&\qquad=\sum_{\left\lvert\alpha\right\rvert>d}b_{\alpha}^{2}\int_{\mathbb{R}^{n}}\dfrac{\left\lvert\alpha\right\rvert}{2}p_{\alpha}(x)^{2}e^{-\left\lvert
x\right\rvert^{2}/4}\;dx\geq\dfrac{d+1}{2}\int_{\mathbb{R}^{n}}g(x)^{2}e^{-\left\lvert
x\right\rvert^{2}/4}\;dx.\end{split}$

Since $g\not\equiv 0$, (3.2) and (LABEL:eqn:eigp2) are incompatible.
Therefore, $k\leq\binom{n+d}{n}$. ∎

Note that Proposition 3.2 and Theorem 3.6 yield Theorem 1.2.

##  4\. HCP in $\mathbb{R}^{2+1}$, Part I: lower bounds

To complete the proof of Theorem 1.1, it remains to determine the minimum
possible number of nodal domains for hcps in $\mathbb{R}^{2+1}$, where the
story is more complicated than in $\mathbb{R}^{1+1}$ (see Corollary 2.5) and
in high enough spatial dimensions (see Proposition 3.1). In this section, we
aim to show that $m_{2,4k}\geq 3$ for all $k\geq 1$, i.e. the number of nodal
domains of an hcp in $\mathbb{R}^{2+1}$ of degree $d\geq 4$ with $d\equiv
0\pmod{4}$ is at least 3.

Towards our goal, we first lower bound the number of nodal domains of a
continuous function in $\mathbb{R}^{2}$ with “alternating nodal structure” at
the origin and at infinity. A _chamber of $u$ in $V$_ is a connected component
of $V\cap\\{u\neq 0\\}$ relative to $V$. A _positive chamber_ is a chamber on
which $u>0$; a _negative chamber_ is a chamber on which $u<0$.

######  Lemma 4.1.

Suppose that $u:\mathbb{R}^{2}\rightarrow\mathbb{R}$ is continuous, $u(0)=0$,
and $u$ has the following nodal structure near the origin and near infinity
for some integers $n^{+}_{\mathrm{in}},n^{+}_{\mathrm{out}}\geq 1$ with
$n^{+}_{\mathrm{in}}+n^{+}_{\mathrm{out}}\geq 3$:

  * •

There exists $\varepsilon>0$ (small) such that in $B_{\varepsilon}(0)$, the
chambers of $u$ are sectors based at the origin, $u$ alternates signs on
adjacent chambers, and $u$ is positive on $n^{+}_{\mathrm{in}}$ of the
chambers. (When $n^{+}_{\mathrm{in}}=1$, we allow either $0$ or $1$ negative
chambers.)

  * •

There exists $M>\varepsilon$ (large) such that in $B_{M}(0)^{c}$, the chambers
of $u$ are sectors extending off to infinity, $u$ alternates signs on adjacent
chambers, and $u$ is positive on $n^{+}_{\mathrm{out}}$ of the chambers. (When
$n^{+}_{\mathrm{out}}=1$, we allow either $0$ or $1$ negative chambers.)

The number of nodal domains of $u$ is at least
$n^{+}_{\mathrm{in}}+n^{+}_{\mathrm{out}}$.

Figure 4.1. Parameters $n^{+}_{\mathrm{in}}$ and $n^{+}_{\mathrm{out}}$ count
the number of positive chambers of $u$ near the origin and near infinity,
respectively. On the right, we illustrate the base case
$n^{+}_{\mathrm{in}}=2,n^{+}_{\mathrm{out}}=1$ (with zero negative chambers at
infinity).

###### Proof.

We adopt the phrase _chamber at the origin_ for chambers relative to
$B_{\varepsilon}(0)$ and the phrase _chamber at infinity_ for chambers
relative to $\mathbb{R}^{2}\setminus B_{M}(0)$. The proof is by induction on
$n^{+}_{\mathrm{in}}+n^{+}_{\mathrm{out}}$. For any fixed value of
$n^{+}_{\mathrm{in}}+n^{+}_{\mathrm{out}}$, it suffices to establish either
the case $n^{+}_{\mathrm{in}}\geq n^{+}_{\mathrm{out}}$ or the case
$n^{+}_{\mathrm{out}}\geq n^{+}_{\mathrm{in}}$, as each case follows from the
other case and inversion. By the alternating hypothesis, if
$n^{+}_{\mathrm{in}}\geq 2$, then $u$ also has $n^{+}_{\mathrm{in}}$ negative
chambers at the origin; if $n^{+}_{\mathrm{out}}\geq 2$, then $u$ also has
$n^{+}_{\mathrm{out}}$ negative chambers at infinity.

For the base case, suppose that $n^{+}_{\mathrm{in}}+n^{+}_{\mathrm{out}}=3$,
say $n^{+}_{\mathrm{in}}=2$ and $n^{+}_{\mathrm{out}}=1$. There are two
alternatives. _First alternative._ If the two positive chambers of $u$ at the
origin belong to different nodal domains, then $u$ has at least three nodal
domains, since $\\{u>0\\}$ has at least two connected components and
$\\{u<0\\}$ is nonempty. _Second alternative._ Suppose that the two positive
chambers of $u$ at the origin belong to the same nodal domain. Then we can
find a simple, closed curve $\gamma\subset\\{u>0\\}\cup\\{(0,0)\\}$ that
connects two points in distinct positive chambers of $B_{\varepsilon}(0)$; see
Figure 4.1. By the Jordan curve theorem, $\gamma$ disconnects the two negative
chambers at the origin. Hence $u$ has at least three nodal domains, since
$\\{u<0\\}$ has at least two connected components and $\\{u>0\\}$ is nonempty.
The base case holds.

For the induction step, suppose that there exists $m\in\mathbb{N}$ such that
the lemma holds whenever $3\leq n^{+}_{\mathrm{in}}+n^{+}_{\mathrm{out}}\leq
m$. Assume that $n^{+}_{\mathrm{in}}+n^{+}_{\mathrm{out}}=m+1$ and
$n^{+}_{\mathrm{in}}\geq n^{+}_{\mathrm{out}}$. We remark that this implies
$n^{+}_{\mathrm{in}}\geq\lceil\frac{1}{2}(m+1)\rceil\geq 2$. We must prove
that $u$ has at least $m+1$ nodal domains. There are two cases, one easy, one
harder.

_Easy case._ Suppose that no two chambers at the origin belong to the same
nodal domain of $u$. Then $\mathcal{N}(u)\geq 2n^{+}_{\mathrm{in}}\geq
2\lceil\frac{1}{2}(m+1)\rceil\geq m+1$ (since $n^{+}_{\mathrm{in}}\geq 2$).

_Harder case._ Suppose that two distinct chambers $C$ and $D$ at the origin
belong to the same nodal domain. Among all candidates, we choose $C$ and $D$
that have

(4.1) $\text{the least number of (positive and negative) chambers between
them}.$

Replacing $u$ by $-u$, if needed, we may assume without loss of generality
that $C$ and $D$ are positive chambers. By the alternating condition, the
minimum number of chambers strictly between $C$ and $D$ is at least one and is
odd. Suppose there are $2k+1$ such chambers and enumerate them and $C$ and $D$
in order:

(4.2) $C,\ N_{1},\ P_{1},\dots,\ P_{k},\ N_{k+1},\ D,$

where $N_{1},\dots,N_{k+1}$ are negative and $P_{1},\dots,P_{k}$ are positive.
(When $k=0$, the enumeration is $C,N_{1},D$.) Pause and note that the $2k+2$
chambers $C$, $N_{1}$, $P_{1}$, …, $P_{k}$, $N_{k+1}$ lie in $2k+2$ disjoint
nodal domains of $u$, otherwise we would violate (4.1). Also note that
$k\leq\frac{1}{2}n^{+}_{\mathrm{in}}-1$, when $n^{+}_{\mathrm{in}}$ is even,
and $k\leq\frac{1}{2}n^{+}_{\mathrm{in}}-\frac{3}{2}$ when
$n^{+}_{\mathrm{in}}$ is odd. (To see this, draw some examples for small
values of $n^{+}_{\mathrm{in}}$.) Either way, $n^{+}_{\mathrm{in}}\geq 2k+2$.

To proceed, let $\sigma\subset\\{u>0\\}\cup\\{(0,0)\\}$ be a simple closed
curve that connects a point in $C$ to a point in $D$, passes through the
origin in $\overline{C}$ and $\overline{D}$, and encloses the intermediate
chambers $N_{1},P_{1},\cdots,P_{k},N_{k+1}$ in the bounded component $\Omega$
of $\mathbb{R}^{2}\setminus\sigma$. Next, let
$\tilde{u}:\mathbb{R}^{2}\rightarrow\mathbb{R}$ be any continuous function
such that $\tilde{u}|_{\mathbb{R}^{2}\setminus\Omega}=u$ and $\tilde{u}>0$
throughout $\Omega$. Effectively, this collapses the chambers in (4.2) into a
single positive chamber. Since there are $k+2$ positive chambers in (4.2), it
follows that $n^{+}_{\mathrm{in}}(\tilde{u})=n^{+}_{\mathrm{in}}(u)-(k+1)$ and
$n^{+}_{\mathrm{out}}(\tilde{u})=n^{+}_{\mathrm{out}}(u)$. Suppose to get a
contradiction that
$n^{+}_{\mathrm{in}}(\tilde{u})+n^{+}_{\mathrm{out}}(\tilde{u})\leq 2$. Then

$k+2=(2k+2)-(k+1)+1\leq
n^{+}_{\mathrm{in}}(u)-(k+1)+n^{+}_{\mathrm{out}}(u)=n^{+}_{\mathrm{in}}(\tilde{u})+n^{+}_{\mathrm{out}}(\tilde{u})\leq
2.$

Hence $k=0$ and $4\leq
m+1=n^{+}_{\mathrm{in}}(u)+n^{+}_{\mathrm{out}}(u)=n^{+}_{\mathrm{in}}(\tilde{u})+n^{+}_{\mathrm{out}}(\tilde{u})+1\leq
3$, which is absurd. Therefore, $3\leq
n^{+}_{\mathrm{in}}(\tilde{u})+n^{+}_{\mathrm{out}}(\tilde{u})\leq
n^{+}_{\mathrm{in}}(u)+n^{+}_{\mathrm{out}}(u)-1\leq m$.

By the induction hypothesis, $\mathcal{N}(\tilde{u})\geq
n^{+}_{\mathrm{in}}(\tilde{u})+n^{+}_{\mathrm{out}}(\tilde{u})=n^{+}_{\mathrm{in}}(u)+n^{+}_{\mathrm{out}}(u)-(k+1)$.
Now, the nodal domains of $\tilde{u}$ are in one-to-one correspondence with
the nodal domains of $u$ that intersect $\mathbb{R}^{n}\setminus\Omega$.
Together with the additional $2k+1$ nodal domains of $u$ inside $\Omega$,
corresponding to the chambers $N_{1}$, $P_{1}$, …, $P_{k}$, $N_{k+1}$, we
conclude that in total $\mathcal{N}(u)\geq
n^{+}_{\mathrm{in}}(u)+n^{+}_{\mathrm{out}}(u)+k\geq
n^{+}_{\mathrm{in}}(u)+n^{+}_{\mathrm{out}}(u)=m+1$. This completes the
induction step. ∎

######  Lemma 4.2.

If $u=u_{d}+f$ near the origin in $\mathbb{R}^{2}$, where $u_{d}$ is an hhp in
$\mathbb{R}^{2}$ of degree $d\geq 1$, $f\in C^{1}$, and in polar coordinates,
$|f(r,\theta)|+\left\lvert\partial_{\theta}f(r,\theta)\right\rvert=o(r^{d})$
as $r\rightarrow 0$, then $u$ has $2d$ chambers with alternating signs in all
sufficiently small neighborhoods of the origin and $n^{+}_{\mathrm{in}}(u)=d$.

###### Proof.

Up to a rotation and renormalization, $u_{d}$ is given in polar coordinates by
$u_{d}(r,\theta)=r^{d}\sin(d\theta)$. Thus the nodal domains of $u_{d}$
consist of $2d$ sectors at the origin with opening $\pi/d$ with alternating
signs. If we consider the function
$g_{r}(\theta):\mathbb{S}^{1}\rightarrow\mathbb{R}$ defined by
$g_{r}(\theta)=r^{-d}u(r,\theta)$, then zero set of $g_{r}$ is the same as
that of $u$ intersected with $\partial B_{r}(0)$. Moreover, the assumptions on
$f$ near the origin give us that

(4.3) $\begin{split}g_{r}(\theta)=\sin(d\theta)+e_{1}(r,\theta),\\\
\partial_{\theta}g_{r}(\theta)=d\cos(d\theta)+e_{2}(r,\theta),\end{split}$

where $\left\lvert e_{i}(r,\theta)\right\rvert\rightarrow 0$ as $r\rightarrow
0$ for $i=1,2$. Since $\left\lvert\cos(d\theta)\right\rvert=1$ on whenever
$\sin(d\theta)=0$, it is straight-forward to check from (4.3) that the
following holds. For each $\varepsilon>0$, there is some $r_{0}>0$ small so
that for all $0<r<r_{0}$, $g_{r}(\theta)$ has exactly $2d$ zeros in
$\mathbb{S}^{1}$ (one in each of the sectors
$\\{\left\lvert\theta-j\pi/d\right\rvert<\varepsilon\\}$ for $0\leq j\leq
2d-1$), at which $g_{r}$ changes sign. If we order such zeros
$\theta_{0}(r),\theta_{1}(r),\dotsc,\theta_{2d-1}(r)$, then the
$\theta_{j}(r)$ are continuous in $r$ since $u$ is a continuous function, and
$\theta_{j}(r)\rightarrow j\pi/d$ as $r\downarrow 0$. Recalling that the nodal
set of $g_{r}$ coincides with $\mathcal{N}(u)\cap\partial B_{r}(0)$, one
obtains the desired conclusion. ∎

As an application of Lemmas 4.1 and 4.2, we obtain a lower bound on the number
of nodal domains of an hcp in $\mathbb{R}^{2+1}$ whose harmonic coefficient
has degree at least 2.

######  Proposition 4.3.

If $p(x,y,t)$ is an hcp in $\mathbb{R}^{2+1}$ of the form (2.1) and $\deg
p_{m}\geq 2$, then $p$ has at least $2\deg p_{m}$ nodal domains.

###### Proof.

Put $d=\deg p_{m}$. By Remark 1.9, it suffices to prove that the nodal set of
$p|_{\mathbb{S}^{2}}$ has at least $2d$ nodal domains. Since $d\geq 1$,
$p(N)=p(S)=0$, where $N=(0,0,1)$ and $S=(0,0,-1)$ are the north and south
poles of $\mathbb{S}^{2}$, respectively. Let $\pi_{N}$ denote the
stereographic projection from the north pole of $\mathbb{S}^{2}$ onto
$\widehat{\mathbb{R}}^{2}=\mathbb{R}^{2}\cup\\{\infty\\}$. Then $\pi_{N}(S)=0$
and $\pi_{N}(N)=\infty$. Thus, if we can show that the nodal domains of
$u:=p\circ\pi_{N}^{-1}$ near the origin and near infinity are homeomorphic to
$2d$ sectors with alternating signs, then
$n^{+}_{\mathrm{in}}(u)=n^{+}_{\mathrm{out}}(u)=d$ and
$\mathcal{N}(p)=\mathcal{N}(p|_{\mathbb{S}^{2}})=\mathcal{N}(u)\geq
n^{+}_{\mathrm{in}}(u)+n^{+}_{\mathrm{out}}(u)=2d\geq 4$ by Lemma 4.1.

The stereographic projection
$\pi_{N}:\mathbb{S}^{2}\setminus\\{N\\}\rightarrow\mathbb{R}^{2}$ is given by

$\displaystyle\pi_{N}(x,y,t)$
$\displaystyle=\left(\dfrac{x}{1-t},\dfrac{y}{1-t}\right),\quad\pi_{N}^{-1}(X,Y)=\left(\dfrac{2X}{R^{2}+1},\dfrac{2Y}{R^{2}+1},\dfrac{R^{2}-1}{R^{2}+1}\right),$

where $R=R(X,Y)=(X^{2}+Y^{2})^{1/2}$. Recalling (2.1), we have

(4.4)
$\begin{split}u(X,Y)&=\sum_{j=0}^{m}\left(\dfrac{R^{2}-1}{R^{2}+1}\right)^{j}p_{j}\left(\dfrac{2X}{R^{2}+1},\dfrac{2Y}{R^{2}+1}\right)\\\
&=\sum_{j=0}^{m}\left(\dfrac{R^{2}-1}{R^{2}+1}\right)^{j}\left(\dfrac{2}{R^{2}+1}\right)^{d+2(m-j)}p_{j}(X,Y)=:\sum_{j=0}^{m}a_{j}(R)p_{j}(X,Y),\end{split}$

where each term $p_{j}$ is algebraically homogeneous of order $d+2(m-j)$, the
lowest order term $p_{m}$ is an hhp of degree $d$ (see e.g. Figure 3.1 for the
case $d=4$), and the coefficients $a_{j}(R)$ are radial, bounded real-analytic
functions of $X$ and $Y$ with $\lim_{R\rightarrow
0}a_{j}(R)=(-1)^{j}2^{d+2(m-j)}$. In particular, $a_{m}$ does not vanish in
$\\{0<R\leq 1/2\\}$. It follows that $f:=u/a_{m}-p_{m}$ is real-analytic near
$0$ and in polar coordinates,
$|f(R,\theta)|+\left\lvert\partial_{\theta}f(R,\theta)\right\rvert=o(R^{d})$
as $R\rightarrow 0^{+}$, since each of the terms $p_{0}$, …, $p_{m-1}$ is
homogeneous with degree at least $d+2$. By Lemma 4.2, we conclude that
$u/a_{m}=p_{m}+f$ has $2d$ chambers at the origin with alternating signs.
Thus, since $a_{m}$ does not vanish near the origin, $u=a_{m}(u/a_{m})$ also
has $2d$ chambers at the origin with alternating signs and
$n^{+}_{\mathrm{in}}(u)=n^{+}_{\mathrm{in}}(u/a_{m})=d$.

Finally, let $v:=p\circ\pi_{S}^{-1}$, where $\pi_{S}$ denotes the
stereographic projection from the south pole of $\mathbb{S}^{2}$ onto
$\widehat{\mathbb{R}}^{2}$. Repeating the argument for $u$ shows that
$n^{+}_{\mathrm{in}}(v)=d$. Therefore, because $u$ and $v$ are related by
inversion, we have $n^{+}_{\mathrm{out}}(u)=n^{+}_{\mathrm{in}}(v)=d$.∎

Next, we present a lower bound on the number of nodal domains for a certain
class of hcps of degree $4k$.

######  Proposition 4.4.

If $p(x,y,t)$ is an an hcp in $\mathbb{R}^{2+1}$ of the form

(4.5) $\displaystyle p(x,y,t)$
$\displaystyle=t^{2k}+t^{2k-1}p_{2k-1}(x,y)+\cdots+p_{0}(x,y)$

for some $k\geq 1$, then $p$ has at least $3$ nodal domains.

Figure 4.2. Proof of Proposition 4.4. If the union $\tilde{\gamma}$ of the
path $\gamma$ and its reflection $R\circ\gamma$ lie in the positivity set of
$p|_{\mathbb{S}^{2}}$, then the negativity set of $p|_{\mathbb{S}^{2}}$ is
disconnected.

###### Proof.

By Remark 1.9, it suffices to prove that the positivity set of
$p|_{\mathbb{S}^{2}}$ is disconnected or the negativity set of
$p|_{\mathbb{S}^{2}}$ is disconnected. Thus, if $\\{p|_{\mathbb{S}^{2}}>0\\}$
is disconnected, we are done.

Suppose that the positivity set of $p|_{\mathbb{S}^{2}}$ is connected. By
(4.5), $p(0,0,1)=p(0,0,-1)=1$. Thus, the north pole $N=(0,0,1)$ and south pole
$S=(0,0,-1)$ belong to the positivity set of $p|_{\mathbb{S}^{2}}$. Hence
there exists a simple path
$\gamma:[0,1]\rightarrow\\{p|_{\mathbb{S}^{2}}>0\\}$ such that $\gamma(0)=N$
and $\gamma(1)=S$. Now, when $t$ is fixed, $p(x,y,t)$ is an even polynomial in
$x$ and $y$, since each coefficient $p_{i}(x,y)$ in (4.5) has even degree. In
particular, writing $R(x,y,t)=(-x,-y,t)$, we have $R\circ\gamma$ is a mirrored
path from $N$ to $S$ contained in $\\{p|_{\mathbb{S}^{2}}>0\\}$. Let
$\tilde{\gamma}$ be the union of the traces of $\gamma$ and its reflection
$R\circ\gamma$. See Figure 4.2. By the Jordan curve theorem, $\tilde{\gamma}$
separates $\mathbb{S}^{2}\setminus\tilde{\gamma}$ into two connected
components. Let $Q\in\mathbb{S}^{2}$ be any point such that $p(Q)<0$. Then
$p(R(Q))=p(Q)<0$, but $Q$ and $R(Q)$ lie in different connected components of
$\mathbb{S}^{2}\setminus\tilde{\gamma}$. Therefore, the negativity set of
$p|_{\mathbb{S}^{2}}$ is disconnected. ∎

Combining the previous two results, we see that $m_{2,d}\geq 3$ when $d\equiv
0\pmod{4}$.

######  Corollary 4.5 (A lower bound for $d\equiv 0\pmod{4}$).

If $d\equiv 0\pmod{4}$ and $d\geq 4$, then any time-dependent hcp $p$ of
degree $d$ in $\mathbb{R}^{2+1}$ has at least $3$ nodal domains. Hence
$m_{2,d}\geq 3$.

###### Proof.

Since $p$ has degree $d=4k\geq 4$, its leading $t$ term is either $t^{2k}$ or
of the form $t^{m}p_{m}(x,y)$ with $\deg p_{m}\geq 2$. In the first case,
$\mathcal{N}(p)\geq 3$ by Proposition 4.4. In the second case,
$\mathcal{N}(p)\geq 2\deg p_{m}\geq 4$ by Proposition 4.3. ∎

##  5\. HCP in $\mathbb{R}^{2+1}$, Part II: constructions

In this section, we construct examples of time-dependent hcps in
$\mathbb{R}^{2+1}$ of degree $d\geq 2$ with two nodal domains when
$d\not\equiv 0\pmod{4}$ and with three nodal domains when $d\equiv 0\pmod{4}$.
By Remark 1.9, counting the nodal domains of an hcp $p$ is equivalent to
counting the nodal domains of $p|_{\mathbb{S}^{2}}$. With this reduction, the
general strategy in constructing examples is the same in all cases (and is the
strategy introduced by [Lew77] and used by [EJN07], [LTY15], and [BH16] in
related contexts):

  1. (a)

Begin with an hcp $\phi_{1}$ of degree $d$ whose nodal set can be described
explicitly.

  2. (b)

Find another hcp $\phi_{2}$ of degree $d$ so that the nodal set of the
perturbation $u=\phi_{1}-\varepsilon\phi_{2}$ in $\mathbb{S}^{2}$ is either
one Jordan curve ($u$ has two nodal domains) or the nodal set of $u$ is the
union of two disjoint Jordan curves ($u$ has three nodal domains).

The key difficulty in this strategy is finding certain compatibility
conditions between $\phi_{1},\phi_{2}$. In general, the nodal set
$\\{\phi_{1}|_{\mathbb{S}^{2}}=0\\}$ is the union of a relatively open smooth
set, where $|\nabla\phi_{1}|\neq 0$, and isolated singular points, where
$|\nabla\phi_{1}|=0$. Understanding the picture near smooth portion of the
nodal set is straightforward; see e.g. [Lew77, Lemma 2]. However,
understanding how the nodal domains of $\phi_{1}$ change under perturbation
near a singular point is quite delicate, requiring knowledge both of the local
structure of $\phi_{1}$ and of the sign of $\phi_{2}$. This makes finding
$\phi_{1}$ and $\phi_{2}$ challenging. After reviewing the main perturbation
lemmas, we present our examples in order of increasing difficulty: $d\equiv
2\pmod{4}$ in §5.1, $d$ odd in §5.2, and $d\equiv 0\pmod{4}$ in §5.3.

Figure 5.1. Zero set of $\psi(x,y)=\mathrm{Im}((x+iy)^{6})$ (in black) and its
perturbation $\psi(x,y)-\varepsilon f(x,y)$ near the origin when $f$ is
$C^{1}$, $\varepsilon$ is small, and $f(0,0)>0$ (left, in green) or $f(0,0)<0$
(right, in red). The green lines lie in the positivity set for $\psi$, whereas
the red lines lie in the negativity set for $\psi$.

The first of two perturbation lemmas that we need, Lemma 5.1, is the
consequence of [Lew77, Lemma 4] recorded on the second paragraph on p. 1239 of
Lewy’s paper. In the original source, it is stated that $f$ should be real-
analytic, but inspecting the proof shows that it suffices to assume $f$ is
$C^{1}$.

######  Lemma 5.1.

Let $\psi(x,y)=\mathrm{Im}((x+iy)^{d})$ for some $d\geq 2$. If
$f:B_{r}(0)\rightarrow\mathbb{R}$ is $C^{1}$ and $f(0,0)>0$, then there exists
$\tau\in(0,r)$ and $\varepsilon_{0}>0$ such that for all
$\varepsilon\in(0,\varepsilon_{0})$, the nodal set of $\psi-\varepsilon f$ in
$B_{\tau}(0)$ consists of $d$ pairwise disjoint simple curves, with one curve
inside each of the connected components of $\\{\psi>0\\}$, and

$\lim_{\varepsilon\rightarrow 0}\mathrm{HD}\left(\\{\psi-\varepsilon
f=0\\}\cap B_{\tau}(0),\\{\psi=0\\}\cap B_{\tau}(0)\right)=0.$

The same conclusion holds when $f(0,0)<0$ except that then the nodal set of
the perturbation $\psi-\varepsilon f$ lies in $\\{\psi<0\\}$. See Figure 5.1.

We also need a variant of Lemma 5.1, in which $\psi(x,y)$ is replaced by a
function $G(x,y)$ whose nodal set is given locally by the union of $m$ graphs
with simple intersection at a common point. We emphasize that the following
lemma is inspired by [Lew77].

######  Lemma 5.2.

Suppose that $G:B_{r}(0)\subset\mathbb{R}^{2}\rightarrow\mathbb{R}$ takes the
form $G(x,y)=\prod_{i=1}^{m}g_{i}(x,y)$ for some $m\geq 2$, where
$g_{1},\dots,g_{m}:B_{r}(0)\rightarrow\mathbb{R}$ are real-analytic functions
satisfying

  * •

$g_{i}(0,0)=0$ and $\partial_{y}g_{i}(0,0)\neq 0$ for all $i$,

  * •

$\\{g_{i}=0\\}\cap\\{g_{j}=0\\}=\\{(0,0)\\}$ for all $i\neq j$.

If $F:B_{r}(0)\rightarrow\mathbb{R}$ is $C^{1}$ and $F(0,0)>0$, then there
exists $\tau\in(0,r)$ and $\varepsilon_{0}>0$ such that for all
$\varepsilon\in(0,\varepsilon_{0})$, the nodal set of $G-\varepsilon F$ in
$B_{\tau}(0)$ consists of $m$ pairwise disjoint simple curves, one inside each
of the $m$ connected components of $\\{G>0\\}$, and

$\lim_{\varepsilon\rightarrow 0}\mathrm{HD}\left(\\{G-\varepsilon F=0\\}\cap
B_{\tau}(0),\\{G=0\\}\cap B_{\tau}(0)\right)=0,$

where $\mathrm{HD}$ is the Hausdorff distance. The same conclusion holds when
$F(0,0)<0$ except that then the nodal set of the perturbation $G-\varepsilon
F$ lies in $\\{G<0\\}$. See Figure 5.2.

Figure 5.2. Zero set of
$G(x,y)=(x^{4}-y-y^{2})(x^{2}(x^{2}-1)+\frac{1}{2}y)(3x^{3}-y)$ and its
perturbation $G-\varepsilon F$: $\varepsilon=10^{-5}$, $F(x,y)=1$ (left),
$F(x,y)=-1$ (right).

######  Remark 5.3.

When $F(0,0)>0$, the negativity set of $G-\varepsilon F$ near the origin is
connected. When $F(0,0)<0$, the positivity set of $G-\varepsilon F$ near the
origin is connected.

We defer the proof of Lemma 5.2, which may be considered a (somewhat long)
exercise with the implicit function theorem, to §6. Note that by rotating
coordinates, Lemma 5.1 follows from Lemma 5.2.

###  5.1. Two nodal domains when $d\equiv 2\pmod{4}$

When $d=4k+2$ for some $k\geq 0$, the basic hcp
$p_{d}(x,t)=t^{2k+1}+c_{2k}t^{2k}x^{2}+\cdots+c_{0}x^{4k+2}$ in
$\mathbb{R}^{1+1}$ (see Definition 2.1) satisfies

$p_{d}(0,1)>0\quad\text{and}\quad p_{d}(0,-1)<0.$

This simple observation will let us build hcps in $\mathbb{R}^{2+1}$ of degree
$d$ with two nodal domains by essentially copying Lewy’s construction of odd
degree hhps in $\mathbb{R}^{3}$ with two nodal domains.

######  Theorem 5.4 (cf. [Lew77, Theorem 1]).

Assume $d=4k+2$ for some $k\geq 0$. Let $\psi(x,y)=\mathrm{Im}((x+iy)^{d})$
and let $p_{d}(x,t)$ be the basic hcp in $\mathbb{R}^{1+1}$. For all
sufficiently small $\varepsilon>0$,

(5.1) $u_{\varepsilon}(x,y,t)\coloneqq\psi(x,y)-\varepsilon p_{d}(x,t)$

is a time-dependent hcp in $\mathbb{R}^{2+1}$ of degree $d$ and
$u_{\varepsilon}$ has two nodal domains.

###### Proof.

It is clear that $u_{\varepsilon}$ is a time-dependent hcp of degree $d$
whenever $\varepsilon\neq 0$. To proceed, we argue that the nodal set of
$u_{\varepsilon}$ looks like the one in Figure 1.1 (right) when
$\varepsilon>0$ is small. Note that the nodal set of $\psi|_{\mathbb{S}^{2}}$
has only two singular points: the north and south pole.

We start with a description of the nodal set of
$u_{\varepsilon}|_{\mathbb{S}^{2}}$ near the north pole. Define
$v_{\varepsilon}(x,y):=\psi(x,y)-\varepsilon p_{d}(x,1)$ so that by parabolic
homogeneity,

$u_{\varepsilon}(x,y,t)=t^{d/2}u_{\varepsilon}(x/\sqrt{t},y/\sqrt{t},1)=t^{d/2}v_{\varepsilon}(x/\sqrt{t},y/\sqrt{t})\quad\text{for
all $t>0$}.$

Hence the nodal set of $u_{\varepsilon}|_{\mathbb{S}^{2}}$ in each time-slice
$\\{t=t_{0}\\}$ (with $0<t_{0}<1$) agrees with the zeros of $v_{\varepsilon}$
on the circle $\\{x^{2}+y^{2}=(1-t_{0}^{2})/t_{0}\\}$. Thus, the nodal set
$u_{\varepsilon}$ on a small spherical cap at the north pole is homeomorphic
to the nodal set of $v_{\varepsilon}$ in a small disk at the origin (see
Figure 5.1). Recall that $p_{d}(0,1)>0$. Therefore, we can use Lemma 5.1 to
conclude that when $\varepsilon>0$ is sufficiently small, the nodal set of
$u_{\varepsilon}$ in a small spherical cap at the north pole consists of $d$
“southward-opening U-shaped” curves lying in every other longitudinal sector
in $\mathbb{S}^{2}$ of angle $\vartheta:=\frac{\pi}{d}$, starting with
$\\{0<\theta<\vartheta\\}$ and ending with
$\\{(2d-2)\vartheta<\theta<(2d-1)\vartheta\\}$.

For all $t<0$, parabolic homogeneity instead yields

$u_{\varepsilon}(x,y,t)=|t|^{d/2}u_{\varepsilon}(x/\sqrt{|t|},y/\sqrt{|t|},-1)=|t|^{d/2}\big{(}\psi(x/\sqrt{|t|},y/\sqrt{|t|})-\varepsilon
p_{d}(x/\sqrt{|t|},-1)\big{)}.$

Since $p_{d}(0,-1)<0$, we can use Lemma 5.1 to conclude that when
$\varepsilon>0$ is sufficiently small, the nodal set of $u_{\varepsilon}$ in a
small spherical cap at the south pole consists of $d$ “northward-opening
U-shaped” curves lying in every other longitudinal sector in $\mathbb{S}^{2}$
of angle $\vartheta=\frac{\pi}{d}$, starting with
$\\{\vartheta<\theta<2\vartheta\\}$ and ending with
$\\{(2d-1)\vartheta<\theta<2d\vartheta\\}$.

Outside of the polar regions, i.e. in the complement of the union of fixed
spherical caps at the north and south pole, the nodal set of
$\psi|_{\mathbb{S}^{2}}(x,y)$ consists of $d$ disjoint smooth arcs, along
which $|\nabla\psi(x,y)|\geq c>0$ for some constant $c$ (depending on the size
of the caps). Hence the same is true for $u_{\varepsilon}|_{\mathbb{S}^{2}}$
when $\varepsilon$ is sufficiently small by the implicit function theorem.

In the end, since the chambers occupied at the north and south pole alternate,
we see that when $\varepsilon$ is sufficiently small, the nodal set of
$u_{\varepsilon}|_{\mathbb{S}^{2}}$ is a single closed, smooth, Jordan curve.
Therefore, $u_{\varepsilon}$ has two nodal domains.∎

###  5.2. Two nodal domains when $d$ is odd

See Figure 1.1 (middle) for an illustration of the example in Theorem 5.5 when
$d=5$.

######  Theorem 5.5.

Assume $d\geq 3$ is odd. For all sufficiently small $\varepsilon>0$ and
$\alpha>0$,

(5.2) $\displaystyle u_{\varepsilon,\alpha}(x,y,t):=yp_{d-1}(x,t)-\varepsilon
p_{d}(x\cos\alpha-y\sin\alpha,t)$

is a time-dependent hcp in $\mathbb{R}^{2+1}$ of degree $d$ and
$u_{\varepsilon,\alpha}$ has two nodal domains.

###### Proof.

Let $d=2k+1$ for some $k\geq 1$. By Theorem 2.4, we can write the basic hcps
$p_{d-1}$ and $p_{d}$ in $\mathbb{R}^{1+1}$ from Definition 2.1 as

$p_{d-1}(x,t)=(t+b_{1}x^{2})\cdots(t+b_{k}x^{2})\quad\text{and}\quad
p_{d}(x,t)=x(t+c_{1}x^{2})\cdots(t+c_{k}x^{2})$

for some numbers $0<c_{1}<b_{1}<c_{2}<b_{2}<\cdots<b_{k-1}<c_{k}<b_{k}$. Note
that the expression $p_{d}(x\cos\alpha-y\sin\alpha,t)$ is just the composition
of $p_{d}(x,t)$ with a rotation in the $x$ and $y$ coordinates and the
Laplacian is rotationally-invariant. Thus, $p_{d}(x\cos\alpha-y\sin\alpha,t)$
and $u_{\varepsilon,\alpha}(x,y,t)$ are time-dependent hcps in
$\mathbb{R}^{2+1}$ for all $\varepsilon$ and $\alpha$. To proceed, fix
$\varepsilon>0$ and $\alpha>0$ (small) and write $u=u_{\varepsilon,\alpha}$,
$p=p_{d-1}$, $q=p_{d}$, and
$q_{\alpha}(x,y,t)=p_{d}(x\cos\alpha-y\sin\alpha,t)$. Our goal is to show that
when $\varepsilon$ and $\alpha$ are small enough that
$\\{u=0\\}\cap\mathbb{S}^{2}$ is a Jordan curve, whence
$\mathcal{N}(u)=\mathcal{N}(u|_{\mathbb{S}^{2}})=2$.

Figure 5.3. Proof of Theorem 5.5 (1/2): Nodal set of $\overline{p}$ (top),
$\overline{q}$ (middle), and $\overline{u}$ (bottom) when $k=3$ and
$\varepsilon$ and $\alpha$ are sufficiently small.

Consider the standard spherical coordinates on $\mathbb{S}^{2}$ given by

(5.3)
$x=\cos\theta\cos\phi,\;y=\sin\theta\cos\phi,\;t=\sin\phi,\qquad-\pi<\theta\leq\pi,\;-\pi/2\leq\phi\leq\pi/2$

and write $\overline{p}$, $\overline{q}$, $\overline{q}_{\alpha}$, and
$\overline{u}$ for the functions corresponding to $yp_{d}(x,t)$, $q(x,t)$,
$q_{\alpha}(x,y,t)$, and $u_{\varepsilon,\alpha}(x,y,t)$ on $\mathbb{S}^{2}$
written in spherical coordinates. Hence

(5.4)
$\begin{split}\overline{p}(\theta,\phi)&=\sin\theta\cos\phi\prod_{i=1}^{k}\left(\sin\phi+b_{i}\cos^{2}\theta\cos^{2}\phi\right),\\\
\overline{q}(\theta,\phi)&=\cos\theta\cos\phi\prod_{i=1}^{k}\left(\sin\phi+c_{i}\cos^{2}\theta\cos^{2}\phi\right),\\\
\overline{q}_{\alpha}(\theta,\phi)&=\overline{q}(\theta+\alpha,\phi),\quad\overline{u}(\theta,\phi)=\overline{p}(\theta,\phi)-\varepsilon\overline{q}_{\alpha}(\theta,\phi).\end{split}$

As an aid for the reader, in Figure 5.3, we depict nodal domains of
$\overline{p}$, $\overline{q}$, and $\overline{u}$ in the $\theta\phi$-plane
when $k=3$. When $\alpha>0$ is small, the picture for $\overline{q}_{\alpha}$
is obtained by translating the picture for $\overline{q}$ slightly to the
left. This is crucial for the construction. Observe that the positivity and
negativity sets for $\overline{u}$ in the figure are connected. We must
explain why this is so.

The nodal set of $\overline{p}$ is comprised of the great circle
$\\{\theta=0\text{ or }\pi\\}$ (including the north and south poles
$\\{\phi=\pm\frac{\pi}{2}\\}$) and the graphs
$\\{(\theta,h_{i}(\theta)):-\pi<\theta\leq\pi\\}$ of the $k$ functions

(5.5)
$h_{i}(\theta):=\sin^{-1}\left(\frac{1-\sqrt{1+4b_{i}^{2}\cos^{4}\theta}}{2b_{i}\cos^{2}\theta}\right)\qquad(i=1,\dots,k).$

To find the formula for $h_{i}$, expand $\cos^{2}\phi=1-\sin^{2}\phi$ in the
equation $\sin\phi+b_{i}\cos^{2}\theta\cos^{2}\phi=0$ and use the quadratic
formula to solve for $\sin\phi$ (recalling the restriction that
$|\sin\phi|\leq 1$). The functions $h_{i}$ are $\pi$-periodic,
$h_{i}(\pm\pi/2)=0$, and $h_{i}(\theta)>h_{i+1}(\theta)$ for all $1\leq i\leq
k-1$ and $\theta\neq\pm\pi/2$, since $b_{1}<\cdots<b_{k}$. Moreover, from the
definition of $\overline{p}$, we observe that $\overline{p}$ takes opposite
signs in adjacent nodal domains as indicated in Figure 5.3.

To understand the nodal structure of $\overline{u}$ when $\varepsilon$ is
small using Lemma 5.2 and Remark 5.3, we must determine the signs of
$\overline{q}_{\alpha}$ at the singular points in the nodal set of
$\overline{p}$ (i.e. the points where two or more nodal lines of
$\overline{p}$ intersect). See Figure 5.4.

Figure 5.4. Proof of Theorem 5.5 (2/2): Nodal sets of $\overline{p}$ (top) and
$\overline{u}$ (bottom) near $\theta=0$ (left) and $\theta=\pi/2$ (right) when
$k=3$. The sign of $\overline{q}_{\alpha}$ at singular points in the nodal set
of $\overline{p}$ determines the local configuration of nodal domains of
$\overline{u}$ (see Lemma 5.2 and Remark 5.3).

The nodal set of $\overline{q}$ is the union of the great circle
$\\{\theta=\pm\pi/2\\}$ and the graphs
$\\{(\theta,g_{i}(\theta)):-\pi<\theta\leq\pi\\}$ of the $k$ functions

(5.6)
$g_{i}(\theta):=\sin^{-1}\left(\frac{1-\sqrt{1+4c_{i}^{2}\cos^{4}\theta}}{2c_{i}\cos^{2}\theta}\right)\qquad(i=1,\dots,k).$

Because of the interlacing property $c_{1}<b_{1}<\cdots<c_{k}<b_{k}$, the
order of the graphs of the functions $g_{i}$ and $h_{i}$ alternate: $g_{1}\geq
h_{1}\geq\cdots\geq g_{k}\geq h_{k}$ with strict inequality when
$\theta\neq\pm\pi/2$. In particular, along the lines $\theta=0$ and
$\theta=\pi$, the sign of the function $\overline{q}$ at the singular point in
the nodal set of $\overline{p}$ corresponding to $h_{i}$ is $(-1)^{i}$ when
$\theta=0$ and $(-1)^{i+1}$ when $\theta=\pi$. By continuity, the same
alternating sign pattern persists for $\overline{q}_{\alpha}$ if $\alpha$ is
sufficiently small. At the two remaining singular points $(-\pi/2,0)$ and
$(\pi/2,0)$ in the nodal set of $\overline{p}$, the function $\overline{q}$ is
zero.111Exceptionally, when $d=3$ and $k=1$, the nodal set for $\overline{p}$
is regular at $(\pm\pi/2,0)$. This is why we introduce the parameter $\alpha$.
Taking $\alpha>0$ (and small), we get
$\overline{q}_{\alpha}(-\pi/2,0)=\overline{q}((-\pi/2)+\alpha,0)>0$ and
$\overline{q}_{\alpha}(\pi/2,0)=\overline{q}((\pi/2)+\alpha)<0$.

By the perturbation lemma, the nodal lines for $\overline{p}$ transform into
the nodal lines for $\overline{u}$ as indicated in the figures provided that
$\varepsilon$ is sufficiently small (with $\alpha$ fixed before
$\varepsilon$). Outside of a neighborhood of the singular points, the nodal
lines for $\overline{u}$ are homeomorphic to the nodal lines for
$\overline{p}$ by the implicit function theorem; cf. proof of Theorem 5.4. (We
have purposely ignored the singularities of $\overline{p}$ on the lines
$\phi=\pm\pi/2$, because the nodal lines of $p|_{\mathbb{S}^{2}}$ are regular
at the north and south poles.) A careful stitching of the local structure of
$\\{\overline{u}=0\\}$ yields that $\\{u=0\\}\cap\mathbb{S}^{2}$ is a single
Jordan curve. ∎

###  5.3. Three nodal domains when $d\equiv 0\pmod{4}$

See Figure 1.1 (left) for an illustration of the example in Theorem 5.6 when
$d=4$.

######  Theorem 5.6.

Assume $d=4k$ for some $k\geq 1$. For all sufficiently small $\varepsilon>0$
and $\alpha>0$,

(5.7) $\displaystyle
u_{\varepsilon,\alpha}(x,y,t):=p_{2k}(x,t)p_{2k}(y,t)+\varepsilon
p_{2k+1}(x\cos\alpha-y\sin\alpha,t)p_{2k-1}(x\sin\alpha+y\cos\alpha,t)$

is a time-dependent hcp in $\mathbb{R}^{2+1}$ of degree $d$ and
$u_{\varepsilon,\alpha}$ has three nodal domains.

###### Proof.

The shell of the proof is the same as for proof of Theorem 5.5, but the
construction is more intricate. Assign $p(x,y,t):=p_{2k}(x,t)p_{2k}(y,t)$ and
$q(x,y,t):=p_{2k+1}(x,t)p_{2k-1}(y,t)$. By Theorem 2.4, we may express

$\displaystyle p(x,y,t)$
$\displaystyle=(t+b_{1}x^{2})\cdots(t+b_{k}x^{2})(t+b_{1}y^{2})\cdots(t+b_{k}y^{2}),$
$\displaystyle q(x,y,t)$
$\displaystyle=xy(t+c_{1}x^{2})\cdots(t+c_{k}x^{2})(t+a_{1}y^{2})\cdots(t+a_{k-1}y^{2}),$

where $a_{1},\dots,a_{k-1}$, $b_{1},\dots,b_{k}$, and $c_{1},\dots,c_{k}$ are
positive constants satisfying (2.6). Fix small parameters $\varepsilon>0$ and
$\alpha>0$ and assign
$q_{\alpha}(x,y,t):=q(x\cos\alpha-y\sin\alpha,x\sin\alpha+y\cos\alpha,t)$ and
$u:=u_{\varepsilon,\alpha}=p+\varepsilon q_{\alpha}$. Since the Laplacian is
rotationally invariant, we conclude that $q_{\alpha}$ and $u$ are time-
dependent hcps of degree $d$. Our goal is to prove that the nodal set of
$u|_{\mathbb{S}^{2}}$ is the union of two disjoint, closed Jordan curves,
which implies that $u$ has 3 nodal domains.

Figure 5.5. Proof of Theorem 5.6 (1/2): The nodal set of $\overline{p}$ (top),
$\overline{q}$ (middle), and $\overline{u}$ when $k=3$ and $\varepsilon$ and
$\alpha$ are sufficiently small. The two Jordan curves which form
$\\{\overline{u}=0\\}$ are depicted in blue and orange.

Once again, we write $\overline{p}$, $\overline{q}$, $\overline{q}_{\alpha}$,
and $\overline{u}$ to denote the functions $p$, $q$, $q_{\alpha}$, and $u$
expressed in the standard spherical coordinates (5.3). Thus,

(5.8)
$\begin{split}\overline{p}(\theta,\phi)&=\prod_{i=1}^{k}(\sin\phi+b_{i}\cos^{2}\theta\cos^{2}\phi)\prod_{i=1}^{k}(\sin\phi+b_{i}\sin^{2}\theta\cos^{2}\phi),\\\
\overline{q}(\theta,\phi)&=\cos\theta\sin\theta\cos^{2}\phi\prod_{i=1}^{k}(\sin\phi+c_{i}\cos^{2}\theta\cos^{2}\phi)\prod_{i=1}^{k-1}(\sin\phi+a_{i}\sin^{2}\theta\cos^{2}\phi).\end{split}$

See Figure 5.5 for an illustration of the nodal domains of $\overline{p}$,
$\overline{q}$, and $\overline{u}$ when $k=3$. The remainder of the proof is
devoted to showing that $\overline{u}$ has two disjoint, closed nodal curves,
bounding one positive component and two negative components.

The nodal set of $\overline{p}$ is the union of the $2k$ graphs of the
functions $h^{p}_{i}(\theta)$ defined by (5.5) and the functions
$g^{p}_{i}(\theta):=h^{p}_{i}(\theta-\pi/2)$ for all $1\leq i\leq k$. (Here we
used the elementary fact that the positive $y$-axis is the rotation of the
positive $x$-axis by $\pi/2$ radians.) Recall that the functions $h^{p}_{i}$
are $\pi$-periodic, $h^{p}_{i}(\pm\pi/2)=0$, and
$h^{p}_{i}(\theta)>h^{p}_{i+1}(\theta)$ for all $1\leq i\leq k-1$ and
$\theta\neq\pm\pi/2$. Similarly, the functions $g^{p}_{i}$ are $\pi$-periodic,
$g^{p}_{i}(0)=g^{p}_{i}(\pi)=0$, and $g^{p}_{i}(\theta)>g^{p}_{i+1}(\theta)$
for all $1\leq i\leq k-1$ and $\theta\neq 0,\pi$. The sign of $\overline{p}$
changes across adjacent nodal domains.

Figure 5.6. Proof of Theorem 5.6 (2/2): Sign pattern for
$\overline{q}_{\alpha}$ at singular points in the nodal set of $\overline{p}$
near $\theta=\pi/4$ when $k=3$.

As usual, we must next identify the singular points in the nodal set of
$\overline{p}$. The points $(-\pi/2,0)$ and $(\pi/2,0)$ common to the graph of
each $h^{p}_{i}$ and the points $(0,0)$ and $(\pi,0)$ common to the graph of
each $g^{p}_{i}$ are singular points.222When $d=4$ and $k=1$, the nodal set
for $\overline{p}$ is regular at $(-\pi/2,0)$, $(0,0)$, $(\pi/2,0)$, and
$(\pi,0)$. In addition, the nodal set of $\overline{p}$ has $4k^{2}$ singular
points where the graph of an $h^{p}_{i}$ intersects the graph of a
$g^{p}_{j}$. For each $1\leq i,j\leq k$, there is a unique angle
$\theta_{i,j}\in(0,\pi/2)$ so that the graphs of $h^{p}_{i}$ and $g^{p}_{j}$
intersect precisely at

(5.9)
$(\theta_{i,j},h_{i}^{p}(\theta_{i,j})),\quad(\theta_{i,j}-\pi,h_{i}^{p}(\theta_{i,j})),\quad(-\theta_{i,j},h_{i}^{p}(\theta_{i,j})),\quad(-\theta_{i,j}+\pi,h_{i}^{p}(\theta_{i,j})).$

The nodal set of $\overline{q}$ includes two great circles
$\\{(\theta,\phi)\;:\;\theta=-\pi/2,0,\pi/2,\pi\\}$ and the $2k-1$ graphs of
functions $h_{i}^{q}$ defined by

(5.10)
$h^{q}_{i}(\theta):=\sin^{-1}\left(\frac{1-\sqrt{1+4c_{i}^{2}\cos^{4}\theta}}{2c_{i}\cos^{2}\theta}\right)\qquad(i=1,\dots,k)$

and the functions $g^{q}_{i}$ defined by

(5.11)
$g_{i}^{q}(\theta)\coloneqq\sin^{-1}\left(\frac{1-\sqrt{1+4a_{i}^{2}\sin^{4}\theta}}{2a_{i}\sin^{2}\theta}\right)\qquad(i=1,\dots,k-1).$

The sign of $\overline{q}$ alternates on adjacent nodal domains. In view of
(2.6), we have

(5.12) $\displaystyle h_{1}^{q}$ $\displaystyle\geq h_{1}^{p}\geq
h_{2}^{q}\geq h_{2}^{p}\geq\cdots\geq h_{k}^{q}\geq h_{k}^{p},$ (5.13)
$\displaystyle g_{1}^{p}$ $\displaystyle\geq g_{1}^{q}\geq g_{2}^{p}\geq
g_{2}^{q}\geq\cdots\geq g_{k-1}^{q}\geq g_{k}^{p},$

with strict inequalities in (5.12) unless $\theta=\pm\pi/2$ and strict
inequalities in (5.13) unless $\theta=0,\pi$. From the strict inequalities,
periodicity, and evenness, one can deduce the following sign pattern for
$\overline{q}$ at the singular points in the nodal set of $\overline{p}$:

(5.14)
$\begin{split}\mathrm{sgn}(\overline{q}(\theta_{i,j},h_{i}^{p}(\theta_{i,j}))&=\mathrm{sgn}(\overline{q}(\theta_{i,j}-\pi,h_{i}^{p}(\theta_{i,j}))=(-1)^{i+j+1},\\\
\mathrm{sgn}(\overline{q}(-\theta_{i,j},h_{i}^{p}(\theta_{i,j}))&=\mathrm{sgn}(\overline{q}(-\theta_{i,j}+\pi,h_{i}^{p}(\theta_{i,j}))=(-1)^{i+j}.\end{split}$

See Figure 5.6. By continuity, (5.14) persists for $\overline{q}_{\alpha}$ if
$\alpha>0$ is small enough. In addition,

(5.15)
$\overline{q}_{\alpha}(0,0)=\overline{q}_{\alpha}(\pi,0)>0,\quad\overline{q}_{\alpha}(-\pi/2,0)=\overline{q}_{\alpha}(\pi/2,0)<0.$

Near the singular points of $\overline{p}$, we again apply Lemma 5.2 / Remark
5.3 (in a rotated coordinate system, as necessary) to see that locally the
nodal set of $\overline{u}$ is given by finitely many simple curves, which are
contained either in the positive or negative components of
$\\{\overline{p}\neq 0\\}$, if $\overline{q}_{\alpha}$ is negative or positive
there, respectively. Moreover, the curves approach the nodal set of
$\overline{p}$ when $\varepsilon\downarrow 0$. (Note that
$\overline{u}=\overline{p}+\varepsilon\overline{q}_{\alpha}=\overline{p}-\varepsilon(-\overline{q}_{\alpha})$.)
Careful piecing of all of this information together, one deduces that for
$\alpha>0$ and $\varepsilon>0$ small, the nodal domain of $\overline{u}$
consists of two disjoint simple curves that separate $\mathbb{R}^{2}$ into
three connected components (one positive connected component and two negative
connected components).

Indeed away from the singular points of $\overline{p}$, the nodal set of
$\overline{u}$ consists of smooth curves tending to the nodal set of
$\overline{p}$ as $\varepsilon\downarrow 0$. The key observation is then that
at the singular points corresponding to the $\theta_{i,j}$ and
$\theta_{i,j}-\pi$, connectivity is gained in the horizontal direction. That
is to say, at the points $(\theta_{i,j},h_{i}^{p}(\theta_{i,j}))$ and
$(\theta_{i+1,j+1},h_{i+1}^{p}(\theta_{i+1,j+1}))$, $\overline{q}_{\alpha}$
has the same sign, which is equal to the sign of $\overline{p}$ at
$(\theta,h_{i}^{p}(\theta_{i,j}))$ for
$\theta\in(\theta_{i,j}-\delta,\theta_{i,j}+\delta)\setminus\\{\theta_{i,j}\\}$
and $\delta>0$ sufficiently small. Thus for the singular points
$(\theta_{i,j},h_{i}^{p}(\theta_{i,j}))$, horizontally-adjacent chambers of
$\overline{p}$ become connected in the nodal domain of $\overline{u}$ by Lemma
5.2. On the other hand, connectivity is gained in the vertical direction at
the singular points corresponding to the $-\theta_{i,j},-\theta_{i,j}+\pi$ in
similar manner. Along with the fact that all positive chambers of
$\overline{p}$ meeting $(0,0)$ and $(0,\pi)$ become connected in the nodal
domain of $\overline{u}$, and all the negative chambers of $\overline{p}$
meeting $(\pm\pi/2,0)$ become connected in the nodal domains of
$\overline{q}$, one can conclude that the positivity set of $\overline{u}$ is
connected, and $\overline{u}$ has only $2$ negative chambers. See Figure 5.6.
∎

##  6\. Proof of Lemma 5.2

Towards the proof of Lemma 5.2, we start with an easy variation of [Lew77,
Lemma 3], which assumed that $g(x)=x^{k}$ for some integer $k\geq 2$ and $f$
is real-analytic.

######  Lemma 6.1.

Suppose that $f,g:[0,t]\rightarrow\mathbb{R}$ are Lipschitz functions such
that $f(0)=1$ and for some numbers $k>1$, $a>0$, and $C>0$,

(6.1) $|g(x)-ax^{k}|\leq Cx^{k+1}\quad\text{for all $x\in[0,1]$,}$ (6.2)
$|g^{\prime}(x)-akx^{k-1}|\leq Cx^{k}\quad\text{and}\quad|f^{\prime}(x)|\leq
C\quad\text{for a.e.\leavevmode\nobreak\ $x\in[0,1]$.}$

There exist $\varepsilon_{0}=\varepsilon_{0}(C,a,k)>0$ such that for all
$\varepsilon\in(0,\varepsilon_{0})$, the function
$F_{\varepsilon}:[0,1]\rightarrow\mathbb{R}$,

(6.3) $F_{\varepsilon}(x)=g(x)-\varepsilon f(x)\quad\text{for all
$x\in[0,1]$},$

has a unique root $x_{0}\in(0,\tau)$, where
$\tau=\min\\{1,\frac{1}{2}C^{-1}ak\\}$. Moreover,
$x_{0}\in(\varepsilon^{1/(k-\frac{1}{2})},\varepsilon^{1/(k+1)})$,
$F_{\varepsilon}(x)<0$ for all $0\leq x<x_{0}$, $F_{\varepsilon}(x)>0$ for all
$x_{0}<x\leq\tau$, and $F_{\varepsilon}$ is strictly increasing on
$[\varepsilon^{1/(k-\frac{1}{2})},\tau]$.

###### Proof.

Let $0<\varepsilon\leq\varepsilon_{0}<1$. Write $j=k-\frac{1}{2}$ and
$\ell=k+1$, so that $k-1<j<k<\ell$. For all $0\leq x\leq\varepsilon^{1/j}$, we
have

(6.4) $\begin{split}F_{\varepsilon}(x)&\leq
ax^{k}+Cx^{\ell}-\varepsilon(1-Cx)\\\ &\leq
a\varepsilon^{k/j}+C\varepsilon^{\ell/j}+\varepsilon(C\varepsilon^{1/j}-1)=(a\varepsilon^{1/(2j)}+C\varepsilon^{3/(2j)}+C\varepsilon^{1/j}-1)\varepsilon\leq-\frac{1}{2}\varepsilon\end{split}$

provided that $\varepsilon_{0}$ (hence $\varepsilon$) is sufficiently small
depending only on $C$, $k$, and $a$. Similarly, for all
$\varepsilon^{1/\ell}\leq x\leq\min\\{1,\frac{1}{2}C^{-1}a\\}$,

(6.5) $\begin{split}F_{\varepsilon}(x)\geq
ax^{k}-Cx^{k+1}-\varepsilon(1+C)&=(a-Cx)x^{k}-\varepsilon(1+C)\\\
&\geq\tfrac{1}{2}a\varepsilon^{k/\ell}-\varepsilon(1+C)\geq(\tfrac{1}{2}a\varepsilon^{-1/\ell}-1-C)\varepsilon\geq\varepsilon\end{split}$

provided that $\varepsilon_{0}$ is sufficiently small depending only on $C$,
$k$, and $a$. Since $F$ is continuous, it follows that $F$ has at least one
root $x_{0}$ in the interval $(\varepsilon^{1/j},\varepsilon^{1/\ell})$. Now,
for any $\varepsilon^{1/j}\leq x\leq\min\\{1,\frac{1}{2}C^{-1}ak\\}=:\tau$ at
which $F$ is differentiable,

(6.6) $\begin{split}F_{\varepsilon}^{\prime}(x)\geq
akx^{k-1}-Cx^{k}-C\varepsilon&=(ak-Cx)x^{k-1}-C\varepsilon\\\
&\geq\tfrac{1}{2}ak\varepsilon^{(k-1)/j}-C\varepsilon\geq(\tfrac{1}{2}ak\varepsilon^{-1/(2j)}-C)\varepsilon\geq\varepsilon\end{split}$

provided that $\varepsilon_{0}$ is sufficiently small depending only on $C$,
$k$, and $a$. It follows that $F$ is strictly increasing on
$[\varepsilon^{1/j},\tau]$. Choosing $\varepsilon_{0}$ to be sufficiently
small guarantees that $\varepsilon^{1/j}<x_{0}<\varepsilon^{1/\ell}\leq\tau$.
In conjunction with (6.4), this shows that $F_{\varepsilon}(x)<0$ for all
$0\leq x<x_{0}$ and $F_{\varepsilon}(x)>0$ for all $x_{0}<x\leq\tau$.
Therefore, $x_{0}$ is the unique root of $F_{\varepsilon}$ in $(0,\tau)$.∎

######  Remark 6.2.

The proof of Lemma 6.1 shows that in fact, $\varepsilon_{0}$ and $\tau$ can be
chosen to depend only on $C$, $k$ and $a_{0}>1$ provided that
$a\in[a_{0}^{-1},a_{0}]$.

Using Lemma 6.1, we prove Lemma 5.2.

###### Proof of Lemma 5.2.

It suffices to prove the result for $G-\varepsilon F$, since the nodal set of
$G+\varepsilon F$ is the same as $-G-\varepsilon F$, and then we may apply the
result for $-G$ in place of $G$. Since $\partial_{y}g_{i}(0,0)\neq 0$, the
(real-analytic) implicit function theorem implies that in a neighborhood of
the origin, the zero set of $g_{i}$ is given by the graph of real analytic
function of one variable, $h_{i}(x)$, so $g_{i}(x,h_{i}(x))=0$. Each $h_{i}$
satisfies $h_{i}(0)=0$ since $g_{i}$ vanishes at the origin. Let us continue
by proving the result for the nodal set of $G-\varepsilon F$ in $\\{x\geq
0\\}$, since similar reasoning applies to the nodal set in $\\{x\leq 0\\}$.

Note that since the $g_{i}$ only share a common zero at the origin, the
functions $h_{i}(x)$ are distinct real-analytic functions. In particular, for
$\tau$ chosen small enough, then the $h_{i}$ must be ordered, and thus we may
as well assume that $h_{1}(x)<h_{2}(x)<\cdots<h_{m}(x)$ for $0<x<\tau$. For
each $i$, we write $h_{i}(x)=\sum_{k=k_{i}}^{\infty}a_{i,k}x^{k}$, where
$a_{i,k_{i}}\neq 0$ is the first non-zero in the expansion of $h_{i}$. Of
course $k_{i}\geq 1$ since $h_{i}(0)=0$. It is straightforward to see from the
fact that the nodal set of $g_{i}$ is given by the graph of $h_{i}$, that the
Taylor series of $g_{i}$ takes the form

(6.7) $\displaystyle g_{i}(x,y)$
$\displaystyle=y\left(b_{i}^{*}+\sum_{\alpha=(\alpha_{1},\alpha_{2}),\left\lvert\alpha\right\rvert\geq
1}b_{\alpha}x^{\alpha_{1}}y^{\alpha_{2}}\right)+\sum_{k=k_{i}}^{\infty}b_{i,k}x^{k},$

where $b_{i}^{*}=\partial_{y}g_{i}(0,0)\neq 0$, and $b_{i,k_{i}}\neq 0$. In
particular, the first nonzero pure $x^{k}$ term in the expansion for $g_{i}$
has the same power as that of $h_{i}$ (and in fact,
$a_{i,k_{i}}=-b_{i,k_{i}}/b_{i}^{*}$).

We remark that since $\partial_{y}g(0,0)\neq 0$, $g_{i}$ changes sign about
its nodal set and $G$ changes sign about the graphs $h_{i}$. Note that for
$\varepsilon_{0}$ and $\tau$ small, the zero set of $G-\varepsilon F$ in
$B_{\tau}(0)$ is contained in the positivity set of $G$, since $F(0,0)>0$.
Hence we consider a chamber

$U_{i_{0}}\coloneqq
B_{\tau}(0)\cap\\{(x,y)\;:\;x>0,\;h_{i_{0}}(x)<y<h_{i_{0}+1}(x)\\}.$

(the case when one of these chambers is just $\\{y>h_{m}\\}$ or
$\\{y<h_{1}\\}$ is similar) and prove the conclusion of the lemma inside this
chamber. First, we need some estimates on $G$ and $\nabla G$, especially near
the graphs of $h_{i_{0}}$ and $h_{i_{0}+1}$, so define the region
$U_{i_{0}}^{\delta}$ for $\delta>0$ small by

$\displaystyle U_{i_{0}}^{\delta}\coloneqq\left\\{(x,y)\in
U_{i_{0}}\;:\;y=th_{i_{0}}(x)+(1-t)h_{i_{0}+1}(x)\text{ for some
}t\in[0,\delta)\cup(1-\delta,1]\right\\}.$

See Figure 6.1.

Figure 6.1. The region $U_{i_{0}}^{\delta}$ when the boundary curves
$h_{i_{0}}(x)$ and $h_{i_{0}+1}(x)$ are quadratic polynomials.

Note that for $\ell_{i_{0}}\coloneqq\min\\{k_{i_{0}},k_{i_{0}+1}\\}$ and
$(x,y)\in U_{i_{0}}$, we have $\left\lvert y\right\rvert\lesssim\left\lvert
x\right\rvert^{\ell_{i_{0}}}$, i.e. $\left\lvert y\right\rvert\leq
C\left\lvert x\right\rvert^{\ell_{i_{0}}}$ for some constant $C>0$. In
addition, from (6.7), we see that for $i\neq i_{0}$ and such $(x,y)$,

(6.8) $\displaystyle\left\lvert g_{i}(x,y)\right\rvert\lesssim\left\lvert
x\right\rvert^{\min\\{k_{i},\ell_{i_{0}}\\}}$

and moreover,

(6.9) $\left\lvert g_{i_{0}}(x,y)\right\rvert\lesssim\left\lvert
x\right\rvert^{\ell_{i_{0}}},\;\left\lvert
G(x,y)\right\rvert\lesssim\left\lvert
x\right\rvert^{\sum_{i=1}^{m}\min\\{k_{i},\ell_{i_{0}}\\}}.$

We can compute directly

$\displaystyle\partial_{y}G$
$\displaystyle=\sum_{i=1}^{m}\partial_{y}g_{i}\prod_{j\neq
i}g_{j},\quad\partial_{y}^{2}G=\sum_{i=1}^{m}\partial_{y}^{2}g_{i}\prod_{j\neq
i}g_{j}+\sum_{i=1}^{m}\sum_{j\neq
i}\partial_{y}g_{i}\partial_{y}g_{j}\prod_{k\neq i,j}g_{j}.$

Since $\nabla G$ and $\nabla^{2}G$ are bounded in a neighborhood of $(0,0)$,
it follows that for all $(x,y)\in U_{i_{0}}$,

(6.10)
$\displaystyle\left\lvert\partial_{y}G(x,y)\right\rvert\lesssim\left\lvert
x\right\rvert^{\sum_{i=1}^{m}\min\\{k_{i},\ell_{i_{0}}\\}-\ell_{i_{0}}}\quad\text{and}\quad\left\lvert\partial_{y}^{2}G(x,y)\right\rvert$
$\displaystyle\lesssim\left\lvert
x\right\rvert^{\sum_{i=1}^{m}\min\\{k_{i},\ell_{i_{0}}\\}-2\ell_{i_{0}}}$

by (6.8) and the fact that $\min\\{k_{i},\ell_{i_{0}}\\}\leq\ell_{i_{0}}$. Of
course, similar estimates hold for $\partial_{x}\partial_{y}G$ and
$\partial_{x}^{2}G$, so we obtain
$\left\lvert\nabla^{2}G(x,y)\right\rvert\lesssim\left\lvert
x\right\rvert^{\sum_{i=1}^{m}\min\\{k_{i},\ell_{i_{0}}\\}-2\ell_{i_{0}}}$ in
$U_{i_{0}}$.

Next we estimate $\left\lvert\nabla G(x,y)\right\rvert$ from below in
$U_{i_{0}}^{\delta}$. From our computation of $\partial_{y}G$,

$\displaystyle\partial_{y}G(x,h_{i_{0}}(x))$
$\displaystyle=\partial_{y}g_{i_{0}}(x,h_{i_{0}}(x))\prod_{i\neq
i_{0}}g_{i}(x,h_{i_{0}}(x)).$

Recalling that $\partial_{y}g_{i}(0,0)\neq 0$, then for $\tau$ chosen
sufficiently small, we have

(6.11)
$\displaystyle\left\lvert\partial_{y}G(x,h_{i_{0}}(x))\right\rvert\gtrsim\prod_{i\neq
i_{0}}\left\lvert g_{i}(x,h_{i_{0}}(x))\right\rvert.$

We may estimate from below for $i\neq i_{0}$,

(6.12) $\begin{split}\left\lvert
g_{i}(x,h_{i_{0}}(x))\right\rvert&=\left\lvert\int_{0}^{1}\dfrac{d}{ds}\left(g_{i}(x,sh_{i_{0}}(x)+(1-s)h_{i}(x))\right)\;ds\right\rvert\\\
&=\left\lvert\left(h_{i_{0}}(x)-h_{i}(x)\right)\int_{0}^{1}\partial_{y}g_{i}(x,sh_{i_{0}}(x)+(1-s)h_{i}(x))\;ds\right\rvert\\\
&\gtrsim\left\lvert h_{i_{0}}(x)-h_{i}(x)\right\rvert\gtrsim\left\lvert
x\right\rvert^{\min\\{k_{i},\ell_{i_{0}}\\}},\end{split}$

where in the second to last inequality, we again used that
$\partial_{y}g_{i}(0,0)\neq 0$ (and take $\tau$ small), and in the last, we
are simply using the definition of the $k_{i}$ and the expansions of the
$h_{i}$. In conjunction with (6.11), we see that

$\displaystyle\left\lvert\partial_{y}G(x,h_{i_{0}}(x))\right\rvert\gtrsim\left\lvert
x\right\rvert^{\sum_{i\neq i_{0}}\min\\{k_{i},\ell_{i_{0}}\\}},$

as long as $\tau$ is taken sufficiently small, and similarly for
$\left\lvert\partial_{y}G(x,h_{i_{0}+1}(x))\right\rvert$. Along with (6.10),
this gives by the mean value theorem that

(6.13) $\begin{split}\left\lvert\nabla
G(x,y)\right\rvert\geq\left\lvert\partial_{y}G(x,y)\right\rvert&\gtrsim\left\lvert
x\right\rvert^{\sum_{i=1}^{m}\min\\{k_{i},\ell_{i_{0}}\\}-\ell_{i_{0}}}=\left\lvert
x\right\rvert^{\sum_{i\neq i_{0}}^{m}\min\\{k_{i},\ell_{i_{0}}\\}}\end{split}$

when $(x,y)\in U_{i_{0}}^{\delta}$ provided that $\delta$ is chosen small
enough.

Set $\ell^{*}=\sum_{i=1}^{m}\min\\{k_{i},\ell_{i_{0}}\\}$ for convenience. By
(6.9), we have that, as long as $\lambda>0$ is chosen sufficiently small (but
not depending on $\varepsilon$), $G-\varepsilon F\neq 0$ in
$B_{(\lambda\varepsilon)^{1/\ell^{*}}}(0)$, since $F\gtrsim 1$ there while
$\left\lvert G\right\rvert\lesssim\lambda\varepsilon$. Now if $(x,y)\in
U_{i_{0}}^{\delta}$, and $(x,y)\not\in
B_{(\lambda\varepsilon)^{1/\ell^{*}}}(0)$, then by estimate (6.13),

$\displaystyle\left\lvert\partial_{y}G(x,y)\right\rvert\gtrsim\left\lvert
x\right\rvert^{\ell^{*}-\ell_{i_{0}}}\gtrsim(\lambda\varepsilon)^{1-\ell_{i_{0}}/\ell^{*}},$

since $(x,y)\in U_{i_{0}}$ implies $\left\lvert
y\right\rvert\lesssim\left\lvert x\right\rvert^{\ell_{i_{0}}}$ with
$\ell_{i_{0}}\geq 1$, so $\left\lvert
x\right\rvert\simeq\left\lvert(x,y)\right\rvert$ there. Since $F$ is $C^{1}$
in a neighborhood of the origin, then as long as $\tau$ and $\varepsilon_{0}$
are chosen sufficiently small, then $\varepsilon\left\lVert\nabla
F\right\rVert_{L^{\infty}(B_{r/2}(0))}\leq\left\lvert\partial_{y}G(x,y)\right\rvert/8$
for such points, and thus in $U_{i_{0}}^{\delta}\setminus
B_{(\lambda\varepsilon)^{1/\ell^{*}}}(0)$,

(6.14) $\displaystyle\left\lvert\partial_{y}(G-\varepsilon
F)\right\rvert\geq\left\lvert\partial_{y}G\right\rvert-\varepsilon\left\lvert\nabla
F\right\rvert\geq(7/8)\left\lvert\partial_{y}G\right\rvert>0.$

In particular, the implicit function theorem implies that locally near any
zero $(x,y)$ of $G-\varepsilon F$ in $U_{i_{0}}^{\delta}$, the set
$\\{G-\varepsilon F=0\\}$ is the graph of a $C^{1}$ function over the
$x$-axis.

For $(x,y)\in U_{i_{0}}\setminus U_{i_{0}}^{\delta/2}$, we apply Lemma 6.1. In
particular, for $t\in[\delta/2,1-\delta/2]$, write $y_{t}(x)\coloneqq
th_{i_{0}}(x)+(1-t)h_{i_{0}+1}(x)$,

$\displaystyle g_{t}(x)\coloneqq G(x,y_{t}(x))\quad\text{and}\quad
f_{t}(x)\coloneqq F(x,y_{t}(x)).$

Since $h_{i_{0}}$, $h_{i_{0}+1}$, and $G$ are real-analytic in a neighborhood
of the origin, so is $g_{t}$, and similarly, $f_{t}$ is Lipschitz since $F$ is
(with Lipschitz constant independent of $t$). The fact that
$G(0,0)=h_{i_{0}}(0)=h_{i_{0}+1}(0)=0$ and $\nabla G(0,0)=0$ forces the Taylor
expansion of $g_{t}$ at the origin to have leading term $a(t)x^{k(t)}$ for
some $k(t)\geq 2$ and $a(t)>0$, since $g_{t}$ is a non-trivial, non-negative
analytic function that is positive in $\\{x>0\\}$. (Recall that $U_{i_{0}}$ is
a positive chamber for $G$.) In fact, $k(t)\equiv\ell^{*}$ for all such $t$,
as can be seen from the estimates (6.9) and an estimate similar to (6.12),
from which one deduces that $\left\lvert
G(x,y_{t}(x))\right\rvert\simeq_{t}x^{\ell^{*}}$ for all $x$ sufficiently
small, which forces $k(t)=\ell^{*}$. Thus, Lemma 6.1 gives for each
$t\in[\delta/2,1-\delta/2]$, some $\varepsilon_{0}(t)$ and $\tau(t)$ so that
the function $g_{t}-\varepsilon f_{t}$ has exactly one root in $(0,\tau(t))$
for all $\varepsilon<\varepsilon_{0}(t)$. Since $t$ lives in a compact
interval and $a(t)$ is continuous and strictly positive, then it attains its
positive minimum and finite maximum in $[\delta/2,1-\delta/2]$. In particular,
$\varepsilon(t)$ and $\tau(t)$ can be chosen to depend only on the maximum and
minimum of $a(t)$. By Remark 6.2, we can choose $\varepsilon_{0}$ and $\tau$
small independently of $t$ so that satisfies the conclusion of Lemma 6.1 for
all $t\in[\delta/2,1-\delta/2]$. Of course, $\varepsilon_{0}$ and $\tau$ will
depend on $\delta$, but this is a small, fixed quantity depending on $F$ and
$G$.

We are in a position to conclude the proof. Our work so far shows that at any
zero $z\in\mathcal{N}(G-\varepsilon F)\cap U_{i_{0}}$, there is a neighborhood
$V_{z}$ of $z$ such that $\mathcal{N}(G-\varepsilon F)\cap V_{z}$ coincides
with a curve passing through $z$: in $U_{i_{0}}^{\delta}$ this is by the
implicit function theorem and in $U_{i_{0}}\setminus U_{i_{0}}^{\delta/2}$
from the fact that the (unique) zeros of $G-\varepsilon F$ along the
trajectories $x\rightarrow(x,y_{t}(x))\in U_{i_{0}}$ indexed by
$t\in[\delta/2,1-\delta/2]$ form a continuous curve in $t$. Let $\gamma$
denote some connected component of $\mathcal{N}(G-\varepsilon
F)\cap\overline{U_{i_{0}}}$. If $\gamma$ meets $U_{i_{0}}^{\delta}$, define

$\displaystyle x_{\mathrm{min}}=\inf\\{x>0\;:\;(x,y)\in
U_{i_{0}}^{\delta}\cap\gamma\text{ for some }y\\},$

which exists since $G-\varepsilon F\neq 0$ in
$B_{(\lambda\varepsilon)^{1/\ell^{*}}}(0)$. Choose a minimizing sequence
$x_{k}\rightarrow x_{\mathrm{min}}$ with corresponding points $y_{k}$ such
that $(x_{k},y_{k})\in U_{i_{0}}^{\delta}\cap\gamma$, and note that we may
assume (up to a subsequence) that
$(x_{k},y_{k})\rightarrow(x_{\mathrm{min}},y_{\mathrm{min}})$ for some
$y_{\mathrm{min}}$. Note that $(x_{\min},y_{\min})$ is a zero of
$G-\varepsilon F$, and moreover, $y_{\min}=y_{t}(x_{\min})$ for either
$t=\delta$ or $t=1-\delta$. Indeed $G-\varepsilon F$ is nonzero on the graphs
of $h_{i_{0}}$ and $h_{i_{0}+1}$, and (6.14) says that if
$(x_{\min},y_{\min})\in\mathrm{int}(U_{i_{0}}^{\delta})$, then locally near
this point, $\mathcal{N}(G-\varepsilon F)$ coincides with the graph of a
$C^{1}$ function over the $x$-axis, which would contradict the definition of
$x_{\min}$. By the remarks above, there is a neighborhood $V$ of
$(x_{\min},y_{\min})$ in which $\mathcal{N}{(G-\varepsilon F)}$ coincides with
a curve passing through $(x_{\min},y_{\min})$, which therefore must coincide
with $\gamma\cap V$. Hence
$(x_{\min},y_{\min})\in\gamma\,\cap(U_{i_{0}}\setminus U_{i_{0}}^{\delta/2})$.
We have proved that every connected component of $\mathcal{N}(G-\varepsilon
F)\cap\overline{U_{i_{0}}}$ meets $U_{i_{0}}\setminus U_{i_{0}}^{\delta/2}$.

Recall that $\mathcal{N}(G-\varepsilon F)\cap(U_{i_{0}}\setminus
U_{i_{0}}^{\delta/2})$ has exactly one connected component, given by the curve
of zeros of $G-\varepsilon F$ along the trajectories
$x\rightarrow(x,y_{t}(x))$. Since any connected component of
$\mathcal{N}(G-\varepsilon F)\cap\overline{U_{i_{0}}}$ meets this region, it
follows that $\mathcal{N}(G-\varepsilon F)\cap\overline{U_{i_{0}}}$ has
exactly one connected component and we know that this component is a simple
curve. We leave it to the reader to verify that the curve tends to $\\{G=0\\}$
near the origin as $\varepsilon\downarrow 0$. ∎

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