Let $\lambda_{j}\in C^{0}(I)$, $j=1,\ldots,d$, be a continuous
parameterization of the roots of $P_{a}$ on $I$. Then the distributional
derivative of each $\lambda_{j}$ in $I$ is a measurable function
$\lambda_{j}^{\prime}\in L^{q}(I)$ for every $q\in[1,p)$. In particular, each
$\lambda_{j}\in W^{1,q}(I)$ for every $q\in[1,p)$.

  2. (2)

This regularity of the roots is uniform. Let
$\\{P_{a_{\nu}}:\nu\in\mathcal{N}\\}$,

$P_{a_{\nu}(t)}(Z)=Z^{d}+\sum_{j=1}^{d}a_{\nu,j}(t)Z^{d-j}\in
C^{k}(\overline{I})[Z],~{}\nu\in\mathcal{N},$

be a family of curves of polynomials, indexed by $\nu$ in some set
$\mathcal{N}$, so that the set of coefficients
$\\{a_{\nu,j}:\nu\in\mathcal{N},j=1,\ldots,d\\}$ is bounded in
$C^{k}(\overline{I})$. Then the set

$\qquad\\{\lambda_{\nu}\in C^{0}(I):P_{a_{\nu}}(\lambda_{\nu})=0\text{ on
$I$},~{}\nu\in\mathcal{N}\\}$

is bounded in $W^{1,q}(I)$ for every $q\in[1,p)$.

For instance, for $d=3$ the proof given in [83, Part 3] provides $k(3)=6$ and
$p(3)=6/5$, which is worse than the optimal $k=3$, $p=3/2$, but still better
than Spagnolo’s result which gives the absolute continuity (i.e. $p=1$) for
$k=25$.

The proof of the above theorem is based on formulas for the roots of the
universal monic polynomial $P_{a}(Z)=Z^{d}+\sum_{j=1}^{d}a_{j}Z^{d-j}$ in
terms of its coefficients $a=(a_{1},\ldots,a_{d})\in\mathbb{C}^{d}$. They are
obtained as follows. By Hironaka’s resolution of singularities [49], there is
a tower of smooth principalizations

(37)
$\displaystyle\mathbb{C}^{d}\stackrel{{\scriptstyle\sigma_{2}}}{{\longleftarrow}}M_{2}\stackrel{{\scriptstyle\sigma_{3,2}}}{{\longleftarrow}}M_{3}\stackrel{{\scriptstyle\sigma_{4,3}}}{{\longleftarrow}}\cdots\stackrel{{\scriptstyle\sigma_{d,d-1}}}{{\longleftarrow}}M_{d}$

which successively principalize the generalized discriminant ideals
$\mathscr{D}_{m}$ in $\mathbb{C}[a]$, $m=2,\ldots,d$, defined in [83, Section
1.1]. These ideals give the stratification of the space of polynomials by root
multiplicities. Then, locally on $M_{d}$, the roots of the pulled back
polynomial $P_{\sigma_{d}^{*}(a)}$ are given by linear combinations with
rational coefficients

(38)
$\displaystyle\sum_{m=1}^{d}A_{m}\,(\varphi_{m}\circ\sigma_{d,m}),\quad\text{
where }$
$\displaystyle\varphi_{m}=f_{m}^{\alpha_{m}}\psi_{m}(y_{{m},1}^{1/q_{m}},\ldots,y_{{m},r_{m}}^{1/q_{m}},y_{{m},r_{m}+1},\ldots,y_{{m},d}).$

Here $\sigma_{m}=\sigma_{2}\circ\sigma_{3,2}\circ\cdots\circ\sigma_{m,m-1}$,
$\sigma_{d,m}=\sigma_{m+1,m}\circ\cdots\circ\sigma_{d,d-1}$,
$f_{m}\in\mathscr{D}_{m}$ is a local generator of
$\sigma_{m}^{*}(\mathscr{D}_{m})$, $\psi_{m}$ is a convergent power series,
$q_{m}\in\mathbb{N}_{\geq 1}$, $\alpha_{m}\in\frac{1}{q_{m}}\mathbb{N}_{\geq
1}$, and $y_{{m},i}$ is a local system of coordinates so that $f_{m}^{-1}(0)$
is given by $y_{{m},1}y_{{m},2}\cdots y_{{m},r_{m}}=0$. These formulas reduce
the problem to radicals and then one may use Theorem 4.1.

The theorem is first proven for, sufficiently generic, real analytic curves of
polynomials. Let $a(t)$ be such a curve, for genericity it suffices that the
discriminant of $P_{a(t)}$ is not identically equal to zero. Note that $a(t)$
can be uniquely lifted over the sequence of blowings-up (37) by the universal
property of blowing-ups. The main difficulty is to get uniform bounds for the
norms independent of the choice of the curve.

If $a(t)$ is arbitrary, then by the Weierstrass approximation theorem it can
be approximated by a sequence of polynomial curves $(a_{\nu})\subseteq
C^{\omega}(\overline{I},\mathbb{C}^{d})$, such that

$a_{\nu}\to a\quad\text{ in }C^{k}(\overline{I},\mathbb{C}^{d})\quad\text{ as
}\nu\to\infty.$

Then, thanks to the uniformity of the bounds, one concludes by the
Arzelá–Ascoli theorem, or alternatively by the Rellich–Kondrachov compactness
theorem, by passing to a subsequence.

###  4.4. Multiparameter case

Though Theorems 4.1, 4.2, 4.9 are one dimensional, they can be fairly easily
extended to the multiparameter case by a standard sectioning argument. We
recall the statement for the analogue of Theorem 4.2. For Theorem 4.1 and
Theorem 4.9, see [37, Theorem 2.3] and [83, Multiparameter Theorem],
respectively.

######  Theorem 4.10 ([84, Theorem 2]).

Let $U\subseteq\mathbb{R}^{n}$ be open and let

(39) $P_{a}(x)(Z)=Z^{d}+\sum_{j=1}^{d}a_{j}(x)Z^{d-j},\quad x\in U,$

be a monic polynomial with complex valued coefficients $a_{j}\in
C^{d-1,1}(U)$, $j=1,\ldots,d$. Let $\lambda\in C^{0}(V)$ be a root of $P_{a}$
on a relatively compact open subset $V\Subset U$. Then $\lambda$ belongs to
the Sobolev space $W^{1,p}(V)$ for every $1\leq p<d/(d-1)$. The distributional
gradient $\nabla\lambda$ satisfies

(40) $\|\nabla\lambda\|_{L^{p}(V)}\leq C(d,n,p,\mathcal{K})\max_{1\leq j\leq
d}\|a_{j}\|^{1/j}_{C^{d-1,1}(\overline{W})},$

where $\mathcal{K}$ is any finite cover of $\overline{V}$ by open boxes
$\prod_{i=1}^{n}(\alpha_{i},\beta_{i})$ contained in $U$ and
$W=\bigcup\mathcal{K}$; the constant $C(d,n,p,\mathcal{K})$ depends only on
$d$, $n$, $p$, and the cover $\mathcal{K}$.

There is a simpler version of the above theorem, avoiding covers by open boxes
in the statement, valid for bounded Lipschitz domains, see [85, Theorem A.1].

####  4.4.1. Bounded variation of the roots

There is nevertheless an important difference between the one parameter and
the multiparameter case. Recall that by a theorem of Kato [52, II.5.2] a
continuous family of complex monic polynomials depending on one real parameter
always admits a continuous choice of roots. This is no longer true if the
dimension of the parameter space is at least two. In that case, monodromy may
prevent the existence of continuous roots. For instance, the radical
$\mathbb{R}^{2}=\mathbb{C}\ni(x+iy)\mapsto(x+iy)^{1/d}$, $d>1$, does not admit
continuous parameterizations near the origin.

Functions of bounded variation ($BV$) are integrable functions whose
distributional derivative is a vector valued finite Radon measure (cf. Section
7.4). They form an algebra of discontinuous functions. Due to their ability to
deal with discontinuities, they are widely used in the applied sciences, see
e.g. [53]. It is shown by Parusiński and Rainer in [85] that the roots of a
polynomial (39) with coefficients in a differentiability class of sufficiently
high order can be represented by functions of bounded variation.

######  Theorem 4.11 ([85, Theorem 1.1]).

For all integers $d,n\geq 2$ there exists an integer
$k=k(d,n)\geq\max\\{d,n\\}$ such that the following holds. Let
$\Omega\subseteq\mathbb{R}^{n}$ be a bounded Lipschitz domain and let (39) be
a monic polynomial of degree $d$ with complex valued coefficients
$a=(a_{1},\ldots,a_{d})\in C^{k-1,1}(\overline{\Omega},\mathbb{C}^{d})$.

Then the roots of $P_{a}$ admit a parameterization
$\lambda=(\lambda_{1},\ldots,\lambda_{d})$ by special functions of bounded
variation ($SBV$) on $\Omega$ such that

(41) $\|\lambda\|_{BV(\Omega)}\leq
C(d,n,\Omega)\,\max\big{\\{}1,\|a\|_{L^{\infty}(\Omega)}\big{\\}}\max\big{\\{}1,\|a\|_{C^{k-1,1}(\overline{\Omega})}\big{\\}}.$

There is a finite collection of $C^{k-1}$ hypersurfaces $E_{j}$ in $\Omega$
such that $\lambda$ is continuous in the complement of $E:=\bigcup_{j}E_{j}$.
Any hypersurface $E_{j}$ is closed in an open subset of $\Omega$ but possibly
not in $\Omega$ itself. All discontinuities of $\lambda$ are jump
discontinuities.

The main difficulty of the problem is to make a good choice of the
discontinuity set $E$ of the roots. On the complement of $E$, the roots are of
optimal Sobolev class $W^{1,p}$, for all $1\leq p<\frac{d}{d-1}$, by [84], see
Theorem 4.10. In general, it is not possible to choose the discontinuity set
$E$ of finite codimension one Hausdorff measure $\mathcal{H}^{n-1}$, [85,
Example 2.4]. Thus, in order to have bounded variation it is crucial that the
jump height of a selection of a root is integrable (with respect to
$\mathcal{H}^{n-1}$) along its discontinuity set, see also the end of Section
6.1.

The proof is based on the radical case, see [85, Theorem 1.4], which is
significantly simpler, and on the formulas (38) for the roots of the universal
polynomial $P_{a}$, $a\in\mathbb{C}^{d}$, obtained in [83]. The result of
Ghisi and Gobbino [37] is used in several places. Interestingly, the method of
[83] seems to be better suited for the control of the discontinuities and
integrability along them than a more elementary method of [84].

##  5\. Lifting maps over invariants of group representations

###  5.1. A reformulation of the regularity problem for hyperbolic polynomials

The regularity problem for the roots of hyperbolic polynomials can be seen as
a special lifting problem: let the symmetric group $\operatorname{S}_{d}$ act
on $\mathbb{R}^{d}$ by permuting the coordinates. The algebra of invariant
polynomials $\mathbb{R}[\mathbb{R}^{d}]^{\operatorname{S}_{d}}$ is generated
by the elementary symmetric functions

$\sigma_{i}=\sum_{j_{1}<\cdots<j_{i}}x_{j_{1}}\cdots x_{j_{i}},\quad 1\leq
i\leq d.$

Each point in the image of the map
$\sigma=(\sigma_{1},\ldots,\sigma_{d}):\mathbb{R}^{d}\to\mathbb{R}^{d}$
represents, in view of Vieta’s formulas, a monic hyperbolic polynomial $P_{a}$
of degree $d$, in a unique way. The fiber of $\sigma$ over this point is the
orbit
$\operatorname{S}_{d}(\lambda_{1},\ldots,\lambda_{d})=\\{(\lambda_{\tau(1)},\ldots,\lambda_{\tau(d)}):\tau\in\operatorname{S}_{d}\\}$,
where $\lambda_{1},\ldots,\lambda_{d}$ are the roots of $P_{a}$ (with
multiplicities). Thus $\sigma(\mathbb{R}^{d})=\operatorname{Hyp}(d)$.

In this picture, a family of monic hyperbolic polynomials $P_{a}(x)$, $x\in
U\subseteq\mathbb{R}^{n}$, is a map $p:U\to\mathbb{R}^{d}$ with image
contained in $\sigma(\mathbb{R}^{d})$ and a system of roots for the family
$P_{a}$ is a lifting $\overline{p}:U\to\mathbb{R}^{d}$ of $p$ over $\sigma$,
i.e., $\sigma\circ\overline{p}=p$. The regularity problem can be rephrased as
follows: Given that $p$ has some regularity, how regular can a lifting of $p$
be chosen?

$\textstyle{\mathbb{R}^{d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{p}}$$\scriptstyle{p}$$\textstyle{\sigma(\mathbb{R}^{d})}$

This setup can be generalized considerably and we will see that Bronshtein’s
theorem 3.2 and many other results on the perturbation theory of hyperbolic
polynomials hold in greater generality.

We shall also discuss a generalization of the nonhyperpolic case along the
same lines in Section 5.5 and subsequent sections.

###  5.2. Orthogonal representations of compact Lie groups

Let $G$ be a compact Lie group and let $\rho:G\rightarrow\operatorname{O}(V)$
be an orthogonal representation in a real finite dimensional Euclidean vector
space $V$ with inner product $\langle\cdot\mid\cdot\rangle$. We will often
just write $G\circlearrowleft V$. In particular, $G$ may be a finite group. By
a classical theorem of Hilbert and Nagata, the algebra $\mathbb{R}[V]^{G}$ of
invariant polynomials on $V$ is finitely generated. So let
$\\{\sigma_{i}\\}_{i=1}^{n}$ be a system of homogeneous generators of
$\mathbb{R}[V]^{G}$ and call it a system of _basic invariants_.

A system of basic invariants $\\{\sigma_{i}\\}_{i=1}^{n}$ is called _minimal_
if there is no polynomial relation of the form
$\sigma_{i}=P(\sigma_{1},\ldots,\widehat{\sigma_{i}},\ldots,\sigma_{n})$, or
equivalently, $\\{\sigma_{i}\\}_{i=1}^{n}$ induces a basis of the real vector
space $\mathbb{R}[V]^{G}_{+}/(\mathbb{R}[V]^{G}_{+})^{2}$, where
$\mathbb{R}[V]^{G}_{+}=\\{f\in\mathbb{R}[V]^{G}:f(0)=0\\}$; cf. [31, Section
3.6]. The elements in a minimal system of basic invariants may not be unique,
but their number and their degrees $d_{i}:=\deg\sigma_{i}$ are unique. Let us
set

$d:=\max_{1\leq i\leq n}d_{i}$

and consider the _orbit map_
$\sigma=(\sigma_{1},\ldots,\sigma_{n}):V\rightarrow\mathbb{R}^{n}$. The image
$\sigma(V)$ is a semialgebraic set in the categorical quotient
$V/\\!\\!/G:=\\{y\in\mathbb{R}^{n}:P(y)=0~{}\mbox{for
all}~{}P\in\mathscr{I}\\}$, where $\mathscr{I}$ is the ideal of relations
between $\sigma_{1},\ldots,\sigma_{n}$. Since $G$ is compact, $\sigma$ is
proper and separates orbits of $G$, and it thus induces a homeomorphism
between the orbit space $V/G$ and $\sigma(V)$.

Note that, by the differentiable slice theorem, $V/G$ is a local model for the
orbit space of differentiable $G$-manifolds.

The given data, i.e., the representation $G\circlearrowleft V$, a fixed system
of basic invariants $\\{\sigma_{i}\\}_{i=1}^{n}$ with degrees
$d_{i}=\deg\sigma_{i}$, and the corresponding orbit map $\sigma$, will be
abbreviated by the triple $(G\circlearrowleft V,d,\sigma)$.

In this setting, we may consider the following lifting problem: let
$f:U\to\mathbb{R}^{n}$, $U\subseteq\mathbb{R}^{m}$, be a map with some
regularity ($C^{\omega}$, $C^{\infty}$, $C^{p}$, etc.) and image contained in
$\sigma(V)$; for brevity we will write $f\in\mathscr{C}(f,\sigma(V))$, where
$\mathscr{C}$ is the respective regularity class. How regular can a lifting
$\overline{f}:U\to V$ of $f$ be?

$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{f}}$$\scriptstyle{f}$$\textstyle{\sigma(V)}$

Note that any two systems of basic invariants differ by a polynomial
diffeomorphism. Thus this question is independent of the choice of
$\\{\sigma_{i}\\}_{i=1}^{n}$ and hence of $\sigma$.

####  5.2.1. Differentiable lifting

The results known for this lifting problem generalize the results for
hyperbolic polynomials. The role of the polynomial degree (as an order of
differentiablity of the coefficients needed to guarantee certain regularity
properties of the roots) is here played by $d$, the maximal degree of a
minimal system of basic invariants.

######  Theorem 5.1 ([56, Proposition 3.1 and Theorem 4.4]).

Let $(G\circlearrowleft V,d,\sigma)$ be a real finite dimensional orthogonal
representation of a compact Lie group. Let $I\subseteq\mathbb{R}$ be an open
interval. Then:

  1. (1)

Each continuous curve $c:I\to\sigma(V)$ has a continuous lifting
$\overline{c}:I\to V$.

  2. (2)

Each $C^{d}$ curve $c:I\to\sigma(V)$ has a differentiable lifting
$\overline{c}:I\to V$.

In Parusiński and Rainer [82], the proof of Bronshtein’s theorem presented in
Section 3.2 was generalized and led to the following essentially optimal
results.

The next theorem is the analogue of Bronshtein’s theorem 3.2.

######  Theorem 5.2 ([82, Theorem 1]).

Let $(G\circlearrowleft V,d,\sigma)$ be a real finite dimensional orthogonal
representation of a compact Lie group. Let $I\subseteq\mathbb{R}$ be an open
interval. Each curve $c\in C^{d-1,1}(I,\sigma(V))$ has a lifting
$\overline{c}\in C^{0,1}(I,V)$ such that, for any pair of relatively compact
open intervals $I_{0}\Subset I_{1}\Subset I$,

(42) $\operatorname{Lip}_{I_{0}}(\overline{c})\leq C\,\max_{1\leq i\leq
n}\|c_{i}\|_{C^{d-1,1}(\overline{I}_{1})}^{1/d_{i}}.$

The constant $C>0$ depends only on $I_{0}$, $I_{1}$, and the isomorphism
classes of the representation $G\circlearrowleft V$ and respective minimal
systems of basic invariants.

In analogy to Theorem 3.9, we have

######  Theorem 5.3 ([82, Theorem 2]).

Let $(G\circlearrowleft V,d,\sigma)$ be a real finite dimensional orthogonal
representation of a compact Lie group. Let $I\subseteq\mathbb{R}$ be an open
interval. Each curve $c\in C^{d}(I,\sigma(V))$ has a lifting $\overline{c}\in
C^{1}(I,V)$.

For finite groups and polar representations, a precursor of this theorem was
obtained by [57, 59] (by reduction to the case
$\operatorname{S}_{d}\circlearrowleft\mathbb{R}^{d}$ and thus to Bronshtein’s
theorem).

An orthogonal representation $G\circlearrowleft V$ is called _polar_ , if
there exists a linear subspace $\Sigma\subseteq V$, called a _section_ or a
_Cartan subspace_ , which meets each orbit orthogonally; cf. [28], [29]. The
trace of the $G$-action on $\Sigma$ is the action of the _generalized Weyl
group_ $W(\Sigma)=N_{G}(\Sigma)/Z_{G}(\Sigma)$ on $\Sigma$, where
$N_{G}(\Sigma):=\\{g\in G:g\Sigma=\Sigma\\}$ and $Z_{G}(\Sigma):=\\{g\in
G:gs=s\text{ for all }s\in\Sigma\\}$. This group is finite, and it is a
reflection group if $G$ is connected. The algebras $\mathbb{R}[V]^{G}$ and
$\mathbb{R}[\Sigma]^{W(\Sigma)}$ are isomorphic via restriction, by a
generalization of Chevalley’s restriction theorem, due to [29] and
independently [110], and thus the orbit spaces $V/G$ and $\Sigma/W(\Sigma)$
are isomorphic. Consequently, the lifting problem can be reduced to a finite
group action.

For finite group actions, Theorem 5.2 and Theorem 5.3 can be sharpened.

######  Theorem 5.4 ([82, Corollary 1 and 3]).

Let $(G\circlearrowleft V,d,\sigma)$ be a real finite dimensional orthogonal
representation of a finite group. Let $I\subseteq\mathbb{R}$ be an open
interval. Then:

  1. (1)

Any continuous lifting $\overline{c}$ of $c\in C^{d-1,1}(I,\sigma(V))$ is
locally Lipschitz and satisfies (42).

  2. (2)

Any differentiable lifting $\overline{c}$ of $c\in C^{d}(I,\sigma(V))$ is
$C^{1}$.

  3. (3)

Each curve $c\in C^{2d}(I,\sigma(V))$ has a $C^{1}$ lifting whose second order
derivatives exist everywhere.

If $G$ is a compact Lie group of positive dimension, then a Lipschitz (or a
$C^{1}$) lifting $\overline{c}$ of $c$ can be distorted to a continuous
lifting that is not locally Lipschitz simply by taking $g\overline{c}$, where
$g:I\to G$ is a suitable continuous curve.

As a consequence of the uniformity of Theorem 5.2, we obtain a lifting result
for maps in several variables.

######  Theorem 5.5 ([82, Corollary 2]).

Let $(G\circlearrowleft V,d,\sigma)$ be a real finite dimensional orthogonal
representation of a finite group. Let $U\subseteq\mathbb{R}^{m}$ be open and
$f\in C^{d-1,1}(U,\sigma(V))$. If $\overline{f}\in C^{0}(\Omega,V)$ is a
continuous lifting of $f$ on an open subset $\Omega\subseteq U$, then
$\overline{f}$ is locally Lipschitz and, for any pair of relatively compact
open subsets $\Omega_{0}\Subset\Omega_{1}\Subset\Omega$,

(43) $\operatorname{Lip}_{\Omega_{0}}(\overline{f})\leq C\,\max_{1\leq i\leq
n}\|f_{i}\|_{C^{d-1,1}(\overline{\Omega}_{1})}^{1/d_{i}}.$

The constant $C>0$ depends only on $\Omega_{0}$, $\Omega_{1}$, $m$, and on the
isomorphism classes of the representation $G\circlearrowleft V$ and respective
minimal systems of basic invariants.

Not every representation admits unrestricted continuous lifting. For instance,
the orbit space of a finite rotation group acting in the standard way on
$\mathbb{R}^{2}$ is homeomorphic to the set $C$ obtained from the sector
$\\{re^{i\varphi}\in\mathbb{C}:r\in[0,\infty),0\leq\varphi\leq\varphi_{0}\\}$
by identifying the rays that constitute its boundary. A loop on $C$ cannot be
lifted to a loop in $\mathbb{R}^{2}$ unless it is homotopically trivial in
$C\setminus\\{0\\}$. Reflection groups admit continuous lifting defined
everywhere, e.g., the hyperbolic case
$\operatorname{S}_{d}\circlearrowleft\mathbb{R}^{d}$.

In view of the examples for hyperbolic polynomials, these lifting results are
generally optimal.

####  5.2.2. Real analytic lifting

Real analytic curves can always be lifted.

######  Theorem 5.6 ([3, Lemma 3.8 and Theorem 4.2], [82, Theorem 4]).

Let $(G\circlearrowleft V,d,\sigma)$ be a real finite dimensional orthogonal
representation of a compact Lie group. Let $I\subseteq\mathbb{R}$ be an open
interval. Each curve $c\in C^{\omega}(I,\sigma(V))$ has a lifting
$\overline{c}\in C^{\omega}(I,V)$.

That there always exist local real analytic liftings was shown in [3]. As
observed in [82], the local liftings can be glued to a global real analytic
lifting, since the first Čech cohomology group $\check{H}^{1}(I,G^{a})$
vanishes, where $G^{a}$ denotes the sheaf of real analytic maps $I\supseteq
U\to G$, by [111].

Similarly, one sees that _generic_ $C^{\infty}$ curves admit liftings. The
genericity condition is a generalization of normal nonflatness considered for
polynomials in Section 3.3.3: for integer $s\geq 0$ let $A_{s}$ be the union
of all orbit type strata $S$ of $V/G$ (cf. Section 5.3.4) of dimension $\dim
S\leq s$ and let $I_{s}\subseteq\mathbb{R}[V]^{G}$ be the ideal of invariant
polynomials vanishing on $A_{s-1}$. Now $c\in C^{\infty}(I,\sigma(V))$ is
called _normally nonflat_ if, for each $t_{0}\in I$, there exists $f\in I_{s}$
such that the Taylor series of $f\circ c$ at $t_{0}$ is not zero, where $s$ is
minimal with the property that the germ of $c$ at $t_{0}$ is contained in
$A_{s}$. Roughly speaking, it means that lower dimensional orbit type strata
are not met with infinite order of flatness.

######  Theorem 5.7 ([3, Theorem 4.1]).

Let $(G\circlearrowleft V,d,\sigma)$ be a real finite dimensional orthogonal
representation of a compact Lie group. Let $I\subseteq\mathbb{R}$ be an open
interval. Each normally nonflat curve $c\in C^{\infty}(I,\sigma(V))$ has a
lifting $\overline{c}\in C^{\infty}(I,V)$.

In the $C^{\infty}$ case, the gluing of local liftings is much easier; cf. [3,
Lemma 3.8].

Real analytic maps depending on several variables cannot be lifted in general,
but they admit liftings after composition with local blowings-up (i.e.,
blowings-up over an open subset composed with the inclusion of this subset).

######  Theorem 5.8 ([93, Theorem 5.4]).

Let $(G\circlearrowleft V,d,\sigma)$ be a real finite dimensional orthogonal
representation of a compact Lie group. Let $M$ be a real analytic manifold and
$f\in C^{\omega}(M,\sigma(V))$. For any compact subset $K\subseteq M$ there
exist a neighborhood $U$ of $K$ and a finite covering $\\{\pi_{j}:U_{j}\to
U\\}$ of $U$, where each $\pi_{j}$ is a composite of finitely many local
blowings-up with smooth center, such that, for all $j$, the map
$f\circ\pi_{j}$ has a real analytic lifting on $U_{j}$.

Actually, this continues to hold in suitable quasianalytic classes of
$C^{\infty}$ functions (instead of $C^{\omega}$); cf. [93].

###  5.3. The main tools

Let us briefly discuss the main tools used for the lifting problem and
identify their counterpart in the study of hyperbolic polynomials.

####  5.3.1. Removing fixed points

Let $V^{G}:=\\{v\in V:Gv=v\\}$ be the linear subspace of fixed points and let
$V^{\prime}$ be its orthogonal complement in $V$. Then $V=V^{G}\oplus
V^{\prime}$,
$\mathbb{R}[V]^{G}=\mathbb{R}[V^{G}]\otimes\mathbb{R}[V^{\prime}]^{G}$, and
$V/G=V^{G}\times V^{\prime}/G$. Thus it suffices to study the lifting problem
for $G\circlearrowleft V^{\prime}$ or equivalently to assume $V^{G}=\\{0\\}$.
This corresponds to the Tschirnhausen transformation (see Section 2.2.1);
indeed,
$(\mathbb{R}^{d})^{\operatorname{S}_{d}}=\\{x\in\mathbb{R}^{d}:x_{1}=x_{2}=\cdots=x_{d}\\}$
and
$(\mathbb{R}^{d})^{\prime}=\\{x\in\mathbb{R}^{d}:x_{1}+x_{2}+\cdots+x_{d}=0\\}$.

####  5.3.2. Dominant invariant

We may assume that $v\mapsto\langle v\mid v\rangle=\|v\|^{2}$ belongs to the
system of basic invariants $\\{\sigma_{i}\\}_{i=1}^{n}$, say it is
$\sigma_{1}$. This does not change $d$; in fact, $V^{G}=\\{0\\}$ implies
$d\geq 2$. Then $\sigma_{1}$ is dominant in the sense that

$|\sigma_{i}|^{1/d_{i}}\leq C(\sigma)\,|\sigma_{1}|^{1/d_{1}},\quad 1\leq
i\leq n,$

by homogeneity. This corresponds to the dominance of the second coefficient of
monic hyperbolic polynomials in Tschirnhausen form (see Lemma 2.4).

####  5.3.3. The slice theorem

For $v\in V$, let $N_{v}:=T_{v}(Gv)^{\bot}$ be the normal subspace of the
orbit $Gv$ at $v$. It carries a natural action $G_{v}\circlearrowleft N_{v}$
of the _isotropy subgroup_ $G_{v}:=\\{g\in G:gv=v\\}$. The crossed product (or
associated bundle) $G\times_{G_{v}}N_{v}$ carries the structure of an affine
real algebraic variety as the categorical (and geometric) quotient $G\times
N_{v}{/\\!\\!/}G_{v}$ with respect to the action
$G_{v}\circlearrowleft(G\times N_{v})$, $h(g,x):=(gh^{-1},hx)$. The
$G$-equivariant polynomial mapping $\phi:G\times_{G_{v}}N_{v}\to V$,
$[g,x]\mapsto g(v+x)$, induces a polynomial mapping
$\psi:(G\times_{G_{v}}N_{v}){/\\!\\!/}G\to V{/\\!\\!/}G$ sending
$(G\times_{G_{v}}N_{v})/G$ into $V/G$. The $G_{v}$-equivariant embedding
$\alpha:N_{v}\hookrightarrow G\times_{G_{v}}N_{v}$, $x\mapsto[e,x]$, induces
an isomorphism
$\beta:N_{v}{/\\!\\!/}G_{v}\to(G\times_{G_{v}}N_{v}){/\\!\\!/}G$ mapping
$N_{v}/G_{v}$ onto $(G\times_{G_{v}}N_{v})/G$.

$\textstyle{N_{v}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\tau}$$\scriptstyle{\eta}$$\textstyle{G\times_{G_{v}}N_{v}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{N_{v}/G_{v}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(G\times_{G_{v}}N_{v})/G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V/G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N_{v}{/\\!\\!/}G_{v}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\scriptstyle{\theta}$$\textstyle{(G\times_{G_{v}}N_{v}){/\\!\\!/}G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{V{/\\!\\!/}G}$

######  Theorem 5.9 ([72], [102]).

There is an open ball $B_{v}\subseteq N_{v}$ centered at the origin such that
the restriction of $\phi$ to $G\times_{G_{v}}B_{v}$ is an analytic
$G$-isomorphism onto a $G$-invariant neighborhood of $v$ in $V$. The mapping
$\theta$ is a local analytic isomorphism at $0$ which induces a local
homeomorphism of $N_{v}/G_{v}$ and $V/G$.

The slice theorem allows to reduce the lifting problem locally to the slice
representation $G_{v}\circlearrowleft N_{v}$. This reduction replaces the
splitting principle (see Lemma 2.5 and Section 3.2.1) for polynomials. (The
passage to the slice representation at $v=(\lambda_{1},\ldots,\lambda_{d})$
for the standard action $\operatorname{S}_{d}\circlearrowleft\mathbb{R}^{d}$
corresponds to a _full_ splitting $P_{\tilde{a}}=P_{b^{1}}P_{b^{2}}\cdots
P_{b^{k}}$, where each factor $P_{b^{i}}$ corresponds to precisely one of the
distinct elements among the $\lambda_{1},\ldots,\lambda_{d}$ and $\deg
P_{b^{i}}$ is its multiplicity.)

The reduction and the assumption $V^{G}=\\{0\\}$ allow to proceed by induction
on the “size” of $G$: if $H$ and $G$ are compact Lie groups, then $H$ is
_smaller_ than $G$ if either $\dim H<\dim G$ or, given that $\dim H=\dim G$,
$H$ has fewer connected components than $G$. For finite groups this reduces to
induction on the group order.

####  5.3.4. Orbit type stratification

The conjugacy class $(H)$ of $H=G_{v}$ in $G$ is called the _type_ of the
orbit $Gv$. Let $V_{(H)}$ be the union of all orbits of type $(H)$. The
collection of the connected components of the smooth manifolds $V_{(H)}/G$
forms the _orbit type stratification of $V/G$_; cf. [102]. By [13], there is
an identification between the orbit type stratification of $V/G$ and the
natural stratification of $\sigma(V)$ as a semialgebraic set (for the
construction of the latter see [13, p. 246] or [69, Section 25]); it is
analytically locally trivial and thus satisfies Whitney’s conditions $(a)$ and
$(b)$. The inclusion relation on the set of subgroups of $G$ induces a partial
ordering on the family of orbit types. There is a unique minimal orbit type,
the _principal orbit type_ , corresponding to the open and dense submanifold
$V_{\operatorname{reg}}$ consisting of points $v$, where the slice
representation $G_{v}\circlearrowleft N_{v}$ is trivial. The projection
$V_{\operatorname{reg}}\to V_{\operatorname{reg}}/G$ is a locally trivial
fiber bundle. There are only finitely many isomorphism classes of slice
representations.

###  5.4. Examples and applications

####  5.4.1. Differentiable eigenvalues of real symmetric matrices

Let the orthogonal group
$\operatorname{O}(d)=\operatorname{O}(\mathbb{R}^{d})$ act by conjugation on
the real vector space $\operatorname{Sym}(d)$ of real symmetric $d\times d$
matrices,

$\operatorname{O}(d)\times\operatorname{Sym}(d)\ni(S,A)\mapsto
SAS^{-1}=SAS^{t}\in\operatorname{Sym}(d).$

The algebra of invariant polynomials
$\mathbb{R}[\operatorname{Sym}(d)]^{\operatorname{O}(d)}$ is isomorphic to
$\mathbb{R}[\operatorname{Diag}(d)]^{\operatorname{S}_{d}}$ by restriction,
where $\operatorname{Diag}(d)$ is the vector space of real diagonal $d\times
d$ matrices upon which $\operatorname{S}_{d}$ acts by permuting the diagonal
entries. More precisely,
$\mathbb{R}[\operatorname{Sym}(d)]^{\operatorname{O}(d)}=\mathbb{R}[\Sigma_{1},\ldots,\Sigma_{d}]$,
where

$\Sigma_{i}(A)=\textstyle{\operatorname{Trace}(\bigwedge^{i}A:\bigwedge^{i}\mathbb{R}^{d}\to\bigwedge^{i}\mathbb{R}^{d})}$

is the $i$-th characteristic coefficient of $A$ and
$\Sigma_{i}|_{\operatorname{Diag}(d)}=\sigma_{i}$, where $\sigma_{i}$ is the
$i$-th elementary symmetric polynomial and we identify
$\operatorname{Diag}(d)\cong\mathbb{R}^{d}$ (cf. [76, 7.1]). This means that
the representation $\operatorname{O}(d)\circlearrowleft\operatorname{Sym}(d)$
is polar and $\operatorname{Diag}(d)$ forms a section.

Let $A:\mathbb{R}\to\operatorname{Sym}(d)$ be a curve of symmetric matrices of
some regularity. The characteristic polynomial $\chi_{A}$ is a curve of monic
hyperbolic polynomials of the same regularity. In this case, as we already
know from Section 3.5, the regularity results provided by the general theory
for hyperbolic polynomials are not optimal.

A heuristic explanation is that in this problem only a “partial” lifting is
necessary: the curve $\chi_{A}$ in the orbit space is the projection of the
curve $A$ under
$\operatorname{Sym}(d)\to\operatorname{Sym}(d)/\operatorname{O}(d)$ and is
then lifted over
$\operatorname{Diag}(d)\to\operatorname{Diag}(d)/\operatorname{S}_{d}$.

$\textstyle{\mathbb{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{A}$$\scriptstyle{\chi_{A}}$$\textstyle{\operatorname{Diag}(d)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Sym}(d)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Diag}(d)/\operatorname{S}_{d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Sym}(d)/\operatorname{O}(d)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\sigma(\mathbb{R}^{d})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{R}^{d}}$

####  5.4.2. Differentiable decomposition of nonnegative functions into sums
of squares

Let the orthogonal group $\operatorname{O}(n)$ act in the standard way on
$\mathbb{R}^{n}$. The algebra of invariant polynomials
$\mathbb{R}[\mathbb{R}^{n}]^{\operatorname{O}(n)}$ is generated by
$\sigma=\sum_{i=1}^{n}x_{i}^{2}$; note that $d=2$. The orbit space
$\mathbb{R}^{n}/\operatorname{O}(n)$ can be identified with the half-line
$[0,\infty)=\sigma(\mathbb{R}^{n})$. Each line through the origin of
$\mathbb{R}^{n}$ forms a section of
$\operatorname{O}(n)\circlearrowleft\mathbb{R}^{n}$.

Lifting a nonnegative function $f$ over $\sigma$ means decomposing $f$ into a
sum of $n$ squares. We conclude (from Theorem 5.5) that any nonnegative
$C^{1,1}$ function $f:\mathbb{R}^{m}\to[0,\infty)$ is the square of a
$C^{0,1}$ function $g$. The image of the lifting $(g,0,\ldots,0)$ lies in the
section $\mathbb{R}(1,0,\ldots,0)$ of
$\operatorname{O}(n)\circlearrowleft\mathbb{R}^{n}$.

There are stronger results that benefit from the additionally available space:

######  Theorem 5.10 ([34]).

Any nonnegative $C^{3,1}$ function $f:\mathbb{R}^{m}\to[0,\infty)$ is a sum of
$n=n(m)$ squares of $C^{1,1}$ functions.

This is sharp in the sense that there exist $C^{\infty}$ functions
$f:\mathbb{R}^{m}\to[0,\infty)$, for $m\geq 4$, that are not sums of squares
of $C^{2}$ functions; see [18]. In fact, a real nonnegative homogeneous
polynomial $p$ of degree $2d$ on $\mathbb{R}^{m}$ that is a sum of squares of
$C^{d}$ functions $f_{i}$ is necessarily a sum of squares of polynomials,
since Taylor expansion of $f_{i}$ gives $f_{i}(x)=q_{i}(x)+o(|x|^{d})$ at
$x=0$, where $q_{i}$ is a homogeneous polynomial of degree $d$, so that
$p=\sum_{i}q_{i}^{2}$. But there exist nonnegative homogeneous polynomials of
degree $4$ on $\mathbb{R}^{4}$ that are not sums of squares of polynomials;
cf. [16, Section 6.3]. For the same reason, there are nonnegative $C^{\infty}$
functions on $\mathbb{R}^{3}$ that are not sums of squares of $C^{3}$
functions.

In dimension $m=1$ (where there are no algebraic obstructions) there is the
following result.

######  Theorem 5.11 ([17]).

Let $p\in\mathbb{N}$. Any nonnegative $C^{2p}$ function
$f:\mathbb{R}\to[0,\infty)$ is the sum of two squares of $C^{p}$ functions.

The decomposition depends on $p$.

####  5.4.3. Polynomials with symmetries

In [71] the lifting problem was used to improve upon the regularity problem
for hyperbolic polynomials with symmetries. The idea is the following. A curve
of monic hyperbolic polynomials of degree $d$ is a curve $p$ in the
semialgebraic set
$\operatorname{Hyp}(d)=\sigma(\mathbb{R}^{d})\subseteq\mathbb{R}^{d}$, where
$\sigma$ consists of the elementary symmetric functions $\sigma_{i}$. If $p$
lies in some proper semialgebraic subset of $\operatorname{Hyp}(d)$, then $p$
has to fulfill more constraints and thus, in general, the conditions that
guarantee the existence of differentiable systems of the roots are weaker. For
instance, if we know that the roots of $p$ fulfill some linear relations,
i.e., they lie in some linear subspace $V$ of $\mathbb{R}^{d}$, then $p$ lies
in $\sigma(V)\subseteq\operatorname{Hyp}(d)$. The symmetries of the roots are
represented by the induced action $G\circlearrowleft V$ of

$G:=\frac{\\{\tau\in\operatorname{S}_{d}:\tau
V=V\\}}{\\{\tau\in\operatorname{S}_{d}:\tau v=v\text{ for all }v\in V\\}}.$

The solution of the lifting problem can be brought to bear, if the
restrictions $\sigma_{i}|_{V}$, $1\leq i\leq d$, generate $\mathbb{R}[V]^{G}$.
Indeed, the orbit type stratification of $G\circlearrowleft V$ is generally
coarser than the restriction to $V$ of the orbit type stratification of
$\operatorname{S}_{d}\circlearrowleft\mathbb{R}^{d}$.

###  5.5. A reformulation of the regularity problem for general polynomials

If we let the symmetric group $\operatorname{S}_{d}$ act on $\mathbb{C}^{d}$,
by permuting the coordinates, then again
$\mathbb{C}[\mathbb{R}^{d}]^{\operatorname{S}_{d}}$ is generated by the
elementary symmetric functions $\sigma_{i}$, $1\leq i\leq d$. The map
$\sigma=(\sigma_{1},\ldots,\sigma_{d}):\mathbb{C}^{d}\to\mathbb{C}^{d}$ is
onto. Each point in $\mathbb{C}^{d}$ represents a monic polynomial of degree
$d$ with complex coefficients and the $\sigma$-fiber over this point is the
$\operatorname{S}_{d}$-orbit through the $d$-tuple of its roots (with
multiplicities).

A family of monic polynomials $P_{a}(x)$, $x\in U\subseteq\mathbb{R}^{m}$, is
a map $p:U\to\mathbb{C}^{d}$ and a system of roots for the family $P_{a}$ is a
lifting $\overline{p}:U\to\mathbb{C}^{d}$ of $p$ over $\sigma$, i.e.,
$\sigma\circ\overline{p}=p$. Given that $p$ is regular of some kind, how
regular can a lifting of $p$ be?

$\textstyle{\mathbb{C}^{d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{p}}$$\scriptstyle{p}$$\textstyle{\mathbb{C}^{d}}$

Note that, in contrast to the hyperbolic case (cf. Section 5.1), there are no
constraints for the map $p$, since $\sigma(\mathbb{C}^{d})=\mathbb{C}^{d}$.

Let us see now how this problem can be studied in greater generality. From now
on, representation spaces $V$ will be complex and finite dimensional.

###  5.6. Representations of linearly reductive groups

An algebraic group $G$ is called _linearly reductive_ if for each rational
representation $V$ and each subrepresentation $W\subseteq V$ there is a
subrepresentation $W^{\prime}\subseteq V$ such that $V=W\oplus W^{\prime}$.

By Hilbert’s finiteness theorem, for rational representations $V$ of linearly
reductive groups $G$, the algebra of $G$-invariant polynomials
$\mathbb{C}[V]^{G}$ is finitely generated. Let $\\{\sigma_{i}\\}_{i=1}^{n}$ be
a system of homogeneous generators which we call _basic invariants_. In a
_minimal_ system of basic invariants the number $n$ of elements $\sigma_{i}$
and their degrees $d_{i}:=\deg\sigma_{i}$ are uniquely determined; set

$d:=\max_{1\leq i\leq n}d_{i}.$

The map
$\sigma=(\sigma_{1},\ldots,\sigma_{n}):V\to\sigma(V)\subseteq\mathbb{C}^{n}$
can be identified with the morphism $V\to V{/\\!\\!/}G$ induced by the
inclusion $\mathbb{C}[V]^{G}\to\mathbb{C}[V]$; the _categorical quotient_
$V{/\\!\\!/}G$ is the affine variety with coordinate ring $\mathbb{C}[V]^{G}$.
In this setting, $V{/\\!\\!/}G$ is generally not a _geometric quotient_ : the
$G$-orbits in $V$ are not in a one-to-one correspondence with the points in
$V{/\\!\\!/}G$. In fact, for every point $z\in V{/\\!\\!/}G$ there is a unique
closed orbit in the fiber $\sigma^{-1}(z)$ which lies in the closure of every
other orbit in this fiber. On the other hand, if $G$ is a finite group, then
all $G$-orbits are closed and thus $V{/\\!\\!/}G$ is a geometric quotient
isomorphic to the orbit space $V/G$.

We write again $(G\circlearrowleft V,d,\sigma)$ for this setup. If
$f:U\to\mathbb{C}^{n}$, $U\subseteq\mathbb{R}^{m}$, is a map with some
regularity and image contained in $\sigma(V)$ (which in general is a proper
subset of $\mathbb{C}^{n}$), we ask how regular a lifting $\overline{f}$ of
$f$ can be. By a _lifting_ of $f$ we mean a map $\overline{f}:U\to V$ such
that $f=\sigma\circ\overline{f}$ and, for all $x\in U$, $\overline{f}(x)$ lies
in the unique closed orbit in the fiber $\sigma^{-1}(f(x))$.

$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{f}}$$\scriptstyle{f}$$\textstyle{\sigma(V)}$

The problem is independent of the choice of the basic invariants, since any
two choices differ by a polynomial diffeomorphism.

The main difference to the real problem introduced in Section 5.2 is that
there is no invariant polynomial that dominates all the others. Even for
finite $G$, where we can always choose an invariant Hermitian inner product on
$V$ (by averaging over $G$) and hence assume that the representation is
unitary, the invariant form $v\mapsto\|v\|^{2}$ is not a member of
$\mathbb{C}[V]^{G}$. This makes the complex case much more difficult and
causes a larger loss of regularity.

####  5.6.1. Sobolev lifting

Let us first assume that $G$ is a finite group. Then each continuous curve
$c:I\to\sigma(V)$, where $I\subseteq\mathbb{R}$ is an interval, has a
continuous lifting $\overline{c}:I\to V$, by [70, Theorem 5.1]. In Parusiński
and Rainer [86], the optimal Sobolev regularity for the roots of polynomials
was extended to group representations. The following result generalizes
Theorem 4.2.

######  Theorem 5.12 ([86, Theorem 1.1]).

Let $(G\circlearrowleft V,d,\sigma)$ be a complex finite dimensional
representation of a finite group $G$. Let $c\in
C^{d-1,1}([\alpha,\beta],\sigma(V))$ be a curve defined on an open bounded
interval $(\alpha,\beta)$. Then each continuous lifting
$\overline{c}:(\alpha,\beta)\to V$ of $c$ over $\sigma$ is absolutely
continuous and belongs to $W^{1,p}((\alpha,\beta),V)$ with

(44) $\|\overline{c}^{\prime}\|_{L^{p}((\alpha,\beta))}\leq C\,\max_{1\leq
j\leq n}\|c_{j}\|^{1/d_{j}}_{C^{d-1,1}([\alpha,\beta])}$

for all $1\leq p<d/(d-1)$, where $C$ is a constant which depends only on the
representation $G\circlearrowleft V$, the length of the interval
$(\alpha,\beta)$, and $p$.

The bound (44), similarly as (34), is not scale invariant.

######  Remark 5.13.

Let us comment on how the constant $C$ in (44) depends on the length of the
interval $(\alpha,\beta)$.

(a) In general, the constant $C$ is of the form

$C(G\circlearrowleft
V,p)\,\max\\{1,(\beta-\alpha)^{1/p},(\beta-\alpha)^{-1+1/p}\\}.$

(b) If the curve $c$ starts, ends, or passes through $0$ (the most singular
point in $\sigma(V)$), then $C$ is of the form

(45) $C(G\circlearrowleft V,p)\,\max\\{1,(\beta-\alpha)^{1/p}\\}.$

(c) If the representation is coregular, then again the constant is of the form
(45). A representation $G\circlearrowleft V$ is called _coregular_ if
$\mathbb{C}[V]^{G}$ is isomorphic to a polynomial algebra, i.e., there is a
system of basic invariants without polynomial relations among them. By the
Shephard–Todd–Chevalley theorem ([104], [23], [103]), this is the case if and
only if $G$ is generated by pseudoreflections (i.e., invertible linear
transformations of finite order and fixed point space a hyperplane).

(d) The constant is also of the form (45) if the curve $c$ satisfies
$c^{(j)}(\alpha)=c^{(j)}(\beta)=0$ for all $j=1,\ldots,d-1$.

Note that, in view of the counterexamples for the regularity of the roots of
polynomials, the results of Theorem 5.12 (as most of the subsequent results)
are optimal.

The lifting of mappings defined in open domains of dimension $m>1$ essentially
admits the same regularity as for curves, provided that continuous lifting is
possible. But, in general, there are topological obstructions for continuous
lifting.

######  Theorem 5.14 ([86, Theorem 1.4]).

Let $(G\circlearrowleft V,d,\sigma)$ be a complex finite dimensional
representation of a finite group $G$. Let $f\in
C^{d-1,1}(\overline{\Omega},\sigma(V))$, where $\Omega\subseteq\mathbb{R}^{m}$
is an open bounded box $\Omega=I_{1}\times\cdots\times I_{m}$. Then each
continuous lifting $\overline{f}:U\to V$ of $f$ over $\sigma$ defined on an
open subset $U\subseteq\Omega$ belongs to $W^{1,p}(U,V)$ and satisfies

(46) $\|\nabla\overline{f}\|_{L^{p}(U)}\leq C(G\circlearrowleft
V,\Omega,m,p)\,\max_{1\leq j\leq
n}\|f_{j}\|^{1/d_{j}}_{C^{d-1,1}(\overline{\Omega})}$

for all $1\leq p<d/(d-1)$.

The case $U=\Omega$ is not excluded. It is clear that Theorem 5.14 implies a
version of the statement, where $\Omega\subseteq\mathbb{R}^{m}$ is any bounded
open set, $U\Subset\Omega$ is relatively compact open in $\Omega$, and the
constant also depends on $U$ (or more precisely on a cover of $U$ by boxes
contained in $\Omega$). Concerning a global result we have the following.

######  Remark 5.15.

If $G\circlearrowleft V$ is coregular, then Theorem 5.14 holds as stated for
any bounded Lipschitz domain $\Omega$; cf. [86, Remark 1.5].

If there is no continuous lifting, it is natural to ask how bad the
discontinuities can be:

######  Open Problem 5.16.

When continuous lifting is impossible, do there exist liftings of bounded
variation?

See Theorem 5.23 for a partial answer.

In the general case, for an infinite linearly reductive group $G$, it is not
clear if a continuous curve in $\sigma(V)$ admits a continuous lifting to $V$.
The notion of _stability_ in geometric invariant theory provides a remedy. A
point $v\in V$ is called _stable_ if the orbit $Gv$ is closed and the isotropy
group $G_{v}$ is finite. The subset $V^{s}\subseteq V$ of stable points is
$G$-invariant and open in $V$, and its image $\sigma(V^{s})$ is open in
$V{/\\!\\!/}G\cong\sigma(V)$ (cf. [78, Proposition 5.15]). The restriction
$\sigma:V^{s}\to\sigma(V^{s})$ of the map $\sigma$ provides a one-to-one
correspondence between points in $\sigma(V^{s})\cong V^{s}/G$ and $G$-orbits
in $V^{s}$, that is $V^{s}/G$ is a geometric quotient. A continuous curve
$c:I\to\sigma(V^{s})$ has a continuous lifting $\overline{c}:I\to V^{s}$, cf.
[86, Lemma 1.6]. Theorem 5.12 has the following corollary.

######  Corollary 5.17 ([86, Theorem 1.7]).

Let $(G\circlearrowleft V,d,\sigma)$ be a rational complex finite dimensional
representation of a linearly reductive group $G$. Let $c\in
C^{d-1,1}([\alpha,\beta],\sigma(V^{s}))$ be a curve defined on a compact
interval $[\alpha,\beta]$ with $c([\alpha,\beta])\subseteq\sigma(V^{s})$. Then
there exists an absolutely continuous lifting $\overline{c}:[\alpha,\beta]\to
V^{s}$ of $c$ over $\sigma$ which belongs to $W^{1,p}([\alpha,\beta],V^{s})$
with

(47) $\|\overline{c}^{\prime}\|_{L^{p}([\alpha,\beta])}\leq
C(G\circlearrowleft V,[\alpha,\beta],p)\,\max_{1\leq j\leq
n}\|c_{j}\|^{1/d_{j}}_{C^{d-1,1}([\alpha,\beta])}$

for all $1\leq p<d/(d-1)$.

For a mapping $f$ defined on a compact subset $K$ of $\mathbb{R}^{m}$ with
$f(K)\subseteq\sigma(V^{s})$ the situation is more complicated: we can apply
Theorem 5.14 to the slice representations at any point $v\in V^{s}$, but it is
not clear if these local (and partial) lifts can be glued in a continuous
fashion.

More can be said for polar representations (which include e.g. the adjoint
actions). We encountered polar representations already in Section 5.2 in the
real setup. Let us briefly discuss them in the framework of rational
representations $V$ of linearly reductive groups $G$; we follow [29]. Let
$v\in V$ be such that the orbit $Gv$ is closed and consider the linear
subspace $\Sigma_{v}=\\{x\in V:\mathfrak{g}x\subseteq\mathfrak{g}v\\}$, where
$\mathfrak{g}$ is the Lie algebra of $G$. All orbits that intersect
$\Sigma_{v}$ are closed, whence $\dim\Sigma_{v}\leq\dim V{/\\!\\!/}G$. The
representation $G\circlearrowleft V$ is said to be _polar_ if there exists
$v\in V$ with closed orbit $Gv$ and $\dim\Sigma_{v}=\dim V{/\\!\\!/}G$. Then
$\Sigma_{v}$ is called a _section_ or _Cartan subspace_ of $V$. Any two
sections are $G$-conjugate. Let us fix a section $\Sigma$. All closed orbits
in $V$ intersect $\Sigma$. The _Weyl group_ $W:=N_{G}(\Sigma)/Z_{G}(\Sigma)$,
where $N_{G}(\Sigma)=\\{g\in G:g\Sigma=\Sigma\\}$ is the normalizer and
$Z_{G}(\Sigma)=\\{g\in G:gx=x\text{ for all }x\in\Sigma\\}$ is the centralizer
of $\Sigma$ in $G$, is finite and the intersection of any closed $G$-orbit in
$V$ with $\Sigma$ is precisely one $W$-orbit. The ring $\mathbb{C}[V]^{G}$ is
isomorphic via restriction to the ring $\mathbb{C}[\Sigma]^{W}$. If $G$ is
connected, then $W$ is a pseudoreflection group and hence
$\mathbb{C}[V]^{G}\cong\mathbb{C}[\Sigma]^{W}$ is a polynomial ring, by the
Shephard–Todd–Chevalley theorem.

######  Corollary 5.18 ([86, Theorem 1.8]).

Let $(G\circlearrowleft V,d,\sigma)$ be a polar representation of a linearly
reductive group $G$. Then:

  1. (1)

Each curve $c\in C^{d-1,1}([\alpha,\beta],\sigma(V))$ has an absolutely
continuous lifting $\overline{c}:(\alpha,\beta)\to V$ of $c$ over $\sigma$
which belongs to $W^{1,p}((\alpha,\beta),V)$ for all $1\leq p<d/(d-1)$ and
satisfies (44).

  2. (2)

Let $f\in C^{d-1,1}(\overline{\Omega},\sigma(V))$, where
$\Omega\subseteq\mathbb{R}^{m}$ is an open bounded box
$\Omega=I_{1}\times\cdots\times I_{m}$. Each continuous lifting $\overline{f}$
defined on an open subset $U\subseteq\Omega$ with values in a section $\Sigma$
is of class $W^{1,p}$ on $U$ for all $1\leq p<d/(d-1)$ and satisfies (46).

  3. (3)

If $G$ is connected, then the constant in (44) is of the form (45) and
$\Omega$ can be any bounded Lipschitz domain.

This follows from applying Theorem 5.12 and Theorem 5.14 to
$W\circlearrowleft\Sigma$.

######  Open Problem 5.19.

To what extent are the above results true for general (nonpolar)
representation?

####  5.6.2. Analytic lifting

Let $(G\circlearrowleft V,d,\sigma)$ be a complex finite dimensional
representation of a finite group $G$. Then the orbit space coincides with the
categorical quotient $V{/\\!\\!/}G$ which is a normal affine variety. Thus the
orbit space has the natural structure of a complex analytic set and there are
several types of morphisms into $V{/\\!\\!/}G$, like regular, rational, or
holomorphic. In [58] those regular, holomorphic, or formal maps into
$V{/\\!\\!/}G$ that admit a regular, holomorphic, or formal lifting to $V$ are
characterized. The conditions are formulated in terms of the orbit type
stratification of the orbit space which, in this case (compare with Section
5.3.4), is finer than its stratification as affine variety and with the use of
spaces of jets at $0\in\mathbb{C}^{m}$ of morphisms $\mathbb{C}^{m}\to
V{/\\!\\!/}G$ of finite and infinite order.

Now let $(G\circlearrowleft V,d,\sigma)$ be a rational representation of a
linearly reductive group $G$. Suppose that $c:\mathbb{R}\to
V{/\\!\\!/}G\cong\sigma(V)\subseteq\mathbb{C}^{n}$ is real analytic. In
contrast to the real case, there does not always exist a real analytic lifting
of $c$ over $\sigma$. But there is the following generalization of Puiseux’s
theorem.

######  Theorem 5.20 ([70, Theorem 3.3 and 3.4]).

Let $(G\circlearrowleft V,d,\sigma)$ be a rational representation of a
linearly reductive group $G$. Then:

  1. (1)

Let $I\subseteq\mathbb{R}$ be an open interval, $t_{0}\in I$, and $c\in
C^{\omega}(I,\sigma(V))$. Then there exists a positive integer $N$ such that,
locally near $t_{0}$, $t\mapsto c(t_{0}\pm(t-t_{0})^{N})$ admits a real
analytic lifting $\overline{c}_{\pm}$ to $V$.

  2. (2)

Let $U\subseteq\mathbb{C}$ be open and connected, $z_{0}\in U$, and
$c\in\mathcal{H}(U,\sigma(V))$, i.e., $c$ is holomorphic. Then there exists a
positive integer $N$ such that, locally near $z_{0}$, $z\mapsto
c(z_{0}+(z-z_{0})^{N})$ admits a holomorphic lifting $\overline{c}$ to $V$.

In analogy to Section 5.2.2, one defines that $c$ is normally nonflat at
$t_{0}$, see [70, Section 2.5]; roughly speaking, it means that $c$ does not
meet lower dimensional strata of the orbit type stratification of
$V{/\\!\\!/}G$ with infinite order of flatness at $t_{0}$. Then in (1) we may
take $c\in C^{\infty}(I,\sigma(V))$ if $c$ is normally nonflat at $t_{0}$ and
conclude that $\overline{c}_{\pm}$ is of class $C^{\infty}$. Or we may work
with suitable quasianalytic subclasses of $C^{\infty}$.

As a consequence, we may conclude that real analytic curves in $\sigma(V)$
have liftings that are locally absolutely continuous.

######  Theorem 5.21 ([70, Theorem 5.4]).

Let $(G\circlearrowleft V,d,\sigma)$ be a rational representation of a
linearly reductive group $G$. Let $I\subseteq\mathbb{R}$ be an open interval.
Each $c\in C^{\omega}(I,\sigma(V))$ admits a lifting $\overline{c}:I\to V$
that is locally absolutely continuous.

Again this remains true for smooth curves in $\sigma(V)$ that are normally
nonflat.

For maps in several variables into $\sigma(V)$ there is an analogue of Theorem
5.8, where, however, local blowings-up _and_ local power substitutions must be
used. A _local power substitution_ is the composite of the inclusion of a
coordinate chart $U$ and a map $W\to U$ given in local coordinates by

$(x_{1},\ldots,x_{m})\mapsto((-1)^{\epsilon_{1}}x_{1}^{\gamma_{1}},\ldots,(-1)^{\epsilon_{m}}x_{m}^{\gamma_{m}}),$

for some integers $\gamma_{i}\geq 1$ and $\epsilon_{i}\in\\{0,1\\}$.

######  Theorem 5.22 ([93, Theorem 4.6]).

Let $(G\circlearrowleft V,d,\sigma)$ be a rational representation of a
linearly reductive group $G$. Let $M$ be a real analytic manifold and $f\in
C^{\omega}(M,\sigma(V))$. For each compact $K\subseteq M$, there exist a
neighborhood $U$ of $K$ in $M$ and a finite covering $\\{\pi_{j}:U_{j}\to
U\\}$ of $U$, where each $\pi_{j}$ is a composite of finitely many maps each
of which is either a local blowing-up with smooth center or a local power
substitution, such that, for all $j$, the map $f\circ\pi_{j}$ has a real
analytic lifting on $U_{j}$.

The theorem remains true for suitable quasianalytic subclasses of $C^{\infty}$
as well as for holomorphic maps. In the latter case, a local power
substitution is simply a map of the form
$(z_{1},\ldots,z_{m})\mapsto(z_{1}^{\gamma_{1}},\ldots,z_{m}^{\gamma_{m}})$.

It is possible to extract information about the existence of weak liftings of
$f$ (without modification of its domain). For instance, there is the following
result which gives a partial answer to Open Problem 5.16.

######  Theorem 5.23 ([93, Theorem 6.11]).

Let $(G\circlearrowleft V,d,\sigma)$ be a rational representation of a
linearly reductive group $G$. Let $U\subseteq\mathbb{R}^{m}$ be open and $f\in
C^{\omega}(U,\sigma(V))$. For each compact $K\subseteq U$, there exist a
neighborhood $W$ of $K$ in $U$ and a lifting $\overline{f}:W\to V$ of $f$ of
class SBV (i.e., special functions of bounded variation).

Actually, $f$ can be taken in a suitable quasianalytic class.

###  5.7. Some remarks on the proofs

We focus on the proof of Theorem 5.12. The foundation of the proof is that the
result holds for finite rotation groups $C_{d}\cong\mathbb{Z}/d\mathbb{Z}$
acting in the standard way on $\mathbb{C}$, where
$\mathbb{C}[\mathbb{C}]^{C_{d}}$ is generated by $z\mapsto z^{d}$ and a
lifting of a map $f$ is a solution of the radical equation $z^{d}=f$. This
follows from Theorem 4.1. Among all representations of finite groups $G$ of a
fixed order $|G|$, it is the one with the worst loss of regularity, since in
general $d\leq|G|$, by Noether’s degree bound, and equality can only happen
for cyclic groups.

The strategy in the general case is similar to the one described in Section
5.3. Evidently, one may reduce to the case that the linear subspace $V^{G}$ of
invariant vectors is trivial. Then Luna’s slice theorem (see [72] or [102,
Theorem 5.3]) allows us to reduce the problem locally to the slice
representation $G_{v}\circlearrowleft N_{v}$ of the isotropy group $G_{v}$ on
$N_{v}$, where $T_{v}V\cong T_{v}(Gv)\oplus N_{v}$ is a $G_{v}$-splitting.
Since in our case $G$ is finite, we have $N_{v}\cong V$. The assumption
$V^{G}=\\{0\\}$ entails that for all $v\in V\setminus\\{0\\}$ the isotropy
group $G_{v}$ is a proper subgroup of $G$ which suggests to use induction on
the group order. For this induction scheme to work it is important that the
slice reduction is uniform in the sense that it does not depend on the
parameter $t$ of the curve $c$ in $\sigma(V)\subseteq\mathbb{C}^{n}$.

The lack of a dominant invariant polynomial (in contrast to the real case)
forces one to choose at points $t_{0}$ with $c(t_{0})\neq 0$ a _dominant_
component $c_{k}$ of $c$ such that

$|c_{k}^{1/d_{k}}(t_{0})|=\max_{1\leq j\leq n}|c_{j}^{1/d_{j}}(t_{0})|$

and to work on small neighborhoods of $t_{0}$ with this distinguished
component $c_{k}$.

The better bound in the case that $G\circlearrowleft V$ is coregular (cf.
Remark 5.13; see also Remark 5.15 and Corollary 5.18(3)) is achieved by an
application of Whitney’s extension theorem to extend the curve $c$ to a larger
interval on which it can be suitably modified without changing it on the
original interval $(\alpha,\beta)$. Coregularity guarantees that the extended
curve still lies in $\sigma(V)$ (because $\sigma(V)=\mathbb{C}^{n}$).

##  6\. Applications

###  6.1. Zero sets of smooth functions

It is well-known that every closed subset of $\mathbb{R}^{n}$ is the zero set
$Z_{f}$ of some real valued $C^{\infty}$ function
$f:\mathbb{R}^{n}\to\mathbb{R}$. So in general $Z_{f}$ can be arbitrarily
irregular. On the other hand, constraints for the derivatives of $f$ entail
regularity properties of $Z_{f}$; think of the implicit function theorem or of
the solutions of elliptic and parabolic PDEs.

Let us assume here that our function $f$ vanishes to some finite order
$\gamma$ at a point $x_{0}$. Then Malgrange’s preparation theorem puts one in
the position to study $Z_{f}$ in a neighborhood of $x_{0}$, by using the
regularity theorems for the roots of polynomials, that is Theorem 4.2 or
Theorem 4.9. Indeed, in suitable local coordinates at $x_{0}$, we have
$f=p\cdot u$, where $p$ is a monic polynomial in one of the coordinates with
$C^{\infty}$ coefficients in the other coordinates and $u$ is a $C^{\infty}$
unit.

This setting is widely applicable, for instance, to solutions of second order
elliptic equations, Laplace eigenfunctions and finite linear combinations of
such ([32], [68], [50]), etc. Often $Z_{f}$ is called the _nodal set_ of $f$,
but we will stick to the term _zero set_.

The following result is taken from Bär [8].

######  Theorem 6.1 ([8, Lemma 3]).

Let $f:\mathbb{R}^{n}\to\mathbb{R}$ be a $C^{\infty}$ function that vanishes
to order $\gamma$ at $x_{0}$. Then there exist $r_{0}>0$ and a affine
hyperplane $H$ in $\mathbb{R}^{n}$ such that $Z_{f}\cap B(x_{0},r_{0})$ is
contained in the union of countably many graphs of $C^{\infty}$ functions from
$H\cap B(x_{0},r_{0})$ to $H^{\bot}$. Moreover, for all $r<r_{0}$,

$\mathcal{H}^{n-1}(Z_{f}\cap B(x_{0},r))\leq n2^{n-1}\gamma\,r^{n-1}.$

Note that Yomdin [116] obtained a similar result; he assumed that the partial
derivatives of some order $j$ of $f$ are “small” in a certain precise sense
and concluded that $f$ behaves in many regards like a polynomial of degree
$j-1$, in particular, it has vanishing order at most $j-1$.

The following theorem is a consequence of Theorem 4.2 or Theorem 4.9 (together
with Malgrange’s preparation theorem), due to Beck, Becker-Kahn, and Hanin
[10].

######  Theorem 6.2 ([10, Theorem 1]).

Let $U\subseteq\mathbb{R}^{n}$ be open and let $\Theta$ be a compact smooth
manifold (possibly with boundary). Let $f\in C^{\infty}(U\times\Theta)$ and
write $f^{\theta}:=f(\cdot,\theta)$ for $\theta\in\Theta$. Suppose that
$f^{\theta_{0}}$ has finite vanishing order $\gamma$ at $x_{0}\in U$. Then
there exist $p>1$, $r_{0}>0$, a neighborhood $V$ of $\theta_{0}$ in $\Theta$,
an affine hyperplane $H\subseteq\mathbb{R}^{n}$, and functions
$f_{i}^{\theta}\in W^{1,p}(H,H^{\bot})$, $1\leq i\leq\gamma$, $\theta\in V$,
with

$\sup_{1\leq i\leq\gamma,\,\theta\in V}\|f_{i}^{\theta}\|_{W^{1,p}}<\infty$

such that

$Z_{f^{\theta}}\cap
B(x_{0},r_{0})\subseteq\bigcup_{i=1}^{\gamma}\Gamma_{f_{i}^{\theta}},\quad\theta\in
V,$

where $\Gamma_{f^{\theta}_{i}}$ is the graph of $f^{\theta}_{i}$.

######  Remark 6.3.

Theorem 6.2 was proved in [10] based on Theorem 4.9. Using the optimal result,
Theorem 4.2, in the proof, one can get more precise information on $p$: the
statement is true for all $1\leq p<\gamma/(\gamma-1)$.

By the same strategy, one gets similar parameterization results in the setting
studied by Yomdin [116] (where the partial derivatives of some order are
small) as well as for $C^{\infty}$ functions all of whose derivatives are
suitably controlled by a weight sequence (e.g. functions in a quasianalytic
Denjoy–Carleman class), see Rainer [97]. In the latter case, the vanishing
order is bounded (globally on a convex body) by a quantity computed from the
weight sequence which has similarities with the polynomial degree.

Note that, if one is interested only in the properties of the zero set of a
function near a point of finite vanishing order, then there is some freedom in
the choice of the local coordinates (i.e., the hyperplane $H$ in Theorem 6.2).

######  Open Problem 6.4.

Is is possible to improve the above results by choosing the local system of
coordinates wisely?

As a consequence of Theorem 6.1 or Theorem 6.2, one gets a nonconcentration
estimate [10, Proposition 4]: suppose that the vanishing order of $f^{\theta}$
is at most $\gamma$ for all $\theta\in\Theta$. For compact $K\subseteq U$ and
$m$-rectifiable $E\subseteq\mathbb{R}^{n}$, for some $m\leq n-1$, there exists
$r_{0}>0$ such that

$\mathcal{H}^{n-1}(Z_{f^{\theta}}\cap E_{r}^{\prime}\cap K)\leq
C(n)\,r^{-1}\mathcal{H}^{n}(E_{r}),\quad r\leq r_{0},$

for all $\theta\in\Theta$ and all images $E^{\prime}$ of Lipschitz maps $E\to
K$ with Lipschitz constant $\leq 1$, where $E_{r},E_{r}^{\prime}$ denote the
$r$-neighborhoods of $E,E^{\prime}$ in $\mathbb{R}^{n}$. This entails the
following continuity result. Let $\operatorname{Sing}_{f}:=\\{x\in
Z_{f}:\nabla f(x)=0\\}$.

######  Corollary 6.5 ([10, Corollary 2]).

Let $U\subseteq\mathbb{R}^{n}$ be open and $f\in C^{\infty}(U\times[0,1])$
such that $f^{1}:=f(\cdot,1)$ has finite vanishing order on $U$. Then for any
compact $K\subseteq U$ with
$\mathcal{H}^{n-1}(K\cap\operatorname{Sing}_{f})=0$ we have

$\lim_{\theta\to 1}\mathcal{H}^{n-1}(Z_{f^{\theta}}\cap
K)=\mathcal{H}^{n-1}(Z_{f^{1}}\cap K).$

In particular, the authors of [10] conclude a continuity result for the zero
sets of the heat flow on a compact smooth Riemannian manifold $(M,g)$: if
$u:M\times(0,\infty)\to\mathbb{R}$ solves

$(\partial_{t}+\Delta_{g})u(x,t)=0$

with initial data $u^{0}\in L^{2}(M)$, then

$\lim_{t\to 1}\mathcal{H}^{n-1}(Z_{u^{t}})=\mathcal{H}^{n-1}(Z_{\psi}),$

where $\psi$ is the first nonzero eigenspace projection of $u^{0}$ and the
time variable underwent a change from $t$ to $\frac{2}{\pi}\arctan(t)$.

Another application of Theorem 6.2 presented in [10] is that, given
$C^{\infty}$ functions $f_{0},f_{1},\ldots,f_{p}$ on a compact Riemannian
manifold $(M,g)$ of dimension $n$ such that for all
$(\theta_{0},\theta_{1},\ldots,\theta_{p})\in\mathbb{R}^{p+1}\setminus\\{0\\}$
the vanishing order of $\theta_{0}f_{0}+\cdots+\theta_{p}f_{p}$ is at most
$\gamma$ on $M$, the map

$\mathbb{R}\mathbb{P}^{p}\ni[\theta_{0}:\cdots:\theta_{p}]\mapsto\partial\\{\theta_{0}f_{0}+\cdots+\theta_{p}f_{p}<0\\}$

into the space of modulo 2 flat $(n-1)$-cycles in $M$ is an admissible
$p$-sweepout (see [10] for the definition). This was proposed by Marques and
Neves in [74], where they settled a conjecture of Yau on the existence of
infinitely many minimal surfaces in the setting of positive Ricci curvature.
This proof uses an argument involving the growth in $p$ of the min-max width
$\omega_{p}(M)$ which is defined in terms of admissible $p$-sweepouts and is a
nonlinear version of the spectrum of the Laplacian (cf. Gromov [41]). In fact,
by Guth [43], one has

$\omega_{p}(M)\leq C\,p^{1/n},$

which is reproved in [10, Theorem 3] as a corollary of Theorem 6.2.

Let us finish this section with a few consequences of the results for
radicals, i.e. Theorem 4.1, due to Ghisi and Gobbino [37] and the co-area
formula. These facts play an essential role in the proof of the existence of
$BV$ roots, Theorem 4.11, in [85]. They concern the behavior of differentiable
functions near their zero set and the level sets of their sign; they are
interesting in their own right. Note that there is no assumption on the
vanishing order.

######  Theorem 6.6 ([85, Theorem 3.5]).

Let $k\in\mathbb{N}_{\geq 1}$, $\alpha\in(0,1]$, and set $s=k+\alpha$. Let
$\Omega\subseteq\mathbb{R}^{n}$ be a bounded Lipschitz domain and $f\in
C^{k,\alpha}(\overline{\Omega},\mathbb{R}^{\ell+1})$, $f\not\equiv 0$. Then
there is a constant $C=C(n,\ell,k,\alpha,\Omega)$ such that for all
$0<\epsilon\leq 1$ and all small $\delta>0$ we have

$\big{|}\big{\\{}y\in(0,\delta):y^{1/s}\,\mathcal{H}^{n-1}(|f|^{-1}(y))\geq\epsilon^{-1}C\,\|f\|^{1/s}_{C^{k,\alpha}(\overline{\Omega})}\big{\\}}\big{|}\leq\epsilon\delta.$

In particular, if $A\subseteq[0,\infty)$ has full measure near $0$, i.e.,
$|A\cap[0,\epsilon)|=\epsilon$ for some $\epsilon>0$, then there is a sequence
$A\ni y_{j}\to 0$ with

$\sup_{j}\Big{(}y_{j}^{1/s}\,\mathcal{H}^{n-1}(|f|^{-1}(y_{j}))\Big{)}\leq
C\,\|f\|^{1/s}_{C^{k,\alpha}(\overline{\Omega})}.$

That means that the $\mathcal{H}^{n-1}$-measures of the level sets
$\\{|f|=y_{j}\\}$ do not grow faster than $y_{j}^{-1/s}$ as $y_{j}\to 0$.

The _sign_ of $f:\Omega\to\mathbb{R}^{\ell+1}$ is the map
$\operatorname{sgn}(f):\Omega\setminus f^{-1}(0)\to\mathbb{S}^{\ell}$ given by

$\operatorname{sgn}(f):=\frac{f}{|f|}.$

######  Theorem 6.7 ([85, Theorem 3.3]).

Let $k\in\mathbb{N}_{\geq 1}$, $\alpha\in(0,1]$, and set $s=k+\alpha$. Let
$\Omega\subseteq\mathbb{R}^{n}$ be a bounded Lipschitz domain and $f\in
C^{k,\alpha}(\overline{\Omega},\mathbb{R}^{\ell+1})$, where $n\geq\ell\geq 1$.
Then there is a constant $C=C(n,\ell,k,\alpha,\Omega)$ such that for each
small $\epsilon>0$

$\mathcal{H}^{\ell}\Big{(}\Big{\\{}y\in\mathbb{S}^{\ell}:\int_{\operatorname{sgn}(f)^{-1}(y)}|f|^{\ell/s}\,d\mathcal{H}^{n-\ell}\geq\epsilon^{-1}\,C\,\|f\|^{\ell/s}_{C^{k,\alpha}(\overline{\Omega})}\Big{\\}}\Big{)}\leq\epsilon.$

In particular, we have $\ell/s$-integrability with respect to
$\mathcal{H}^{n-\ell}$ of $f$ along almost every level set of the sign of $f$:

$\int_{\operatorname{sgn}(f)^{-1}(y)}|f|^{\ell/s}\,d\mathcal{H}^{n-\ell}<\infty,\quad\text{
for $\mathcal{H}^{\ell}$-a.e. }y\in\mathbb{S}^{\ell}.$

Moreover, for every relatively compact open $K\Subset\Omega\setminus
f^{-1}(0)$,

$\mathcal{H}^{n-\ell}\big{(}K\cap\operatorname{sgn}(f)^{-1}(y)\big{)}<\infty,\quad\text{
for $\mathcal{H}^{\ell}$-a.e. }y\in\mathbb{S}^{\ell}.$

###  6.2. Extension to the optimal transport between algebraic hypersurfaces

A recent paper [5], by Antonini, Cavalletti, and Lerario, contains an
interesting reinterpretation and extension of the regularity problem of the
roots, Section 4, to the study of the Wasserstein distance on the space of
$d$-degree hypersurfaces in $\mathbb{C}\mathbb{P}^{n}$.

Let $(M,g)$ be a smooth compact Riemannian manifold and let $\mathscr{P}(M)$
denote the space of Borel probability measures on $M$. Fix $q\geq 1$. For any
$\mu_{0},\mu_{1}\in\mathscr{P}(M)$, the _$q$ -Wasserstein distance_ is defined
by

$W_{q}(\mu_{0},\mu_{1}):=\left(\inf_{\xi\in\Pi(\mu_{0},\mu_{1})}\int_{M\times
M}\mathsf{d}_{g}(x,y)^{q}\,\xi(dxdy)\right)^{\frac{1}{q}},$

where $\mathsf{d}_{g}$ is the geodesic distance on $M$ and
$\Pi(\mu_{0},\mu_{1})$ is the set of transport plans between $\mu_{0}$ and
$\mu_{1}$, i.e., the set of probability measures $\xi$ on $M\times M$ with
marginals $\mu_{0}$ and $\mu_{1}$ (the push-forward measures of $\xi$ with
respect to the natural projections). Intuitively, this distance minimizes the
transport cost from $\mu_{0}$ to $\mu_{1}$ over all such plans.

Denote by $H_{n,d}$ the space of complex homogeneous polynomials of degree $d$
in $n+1$ variables. Let $P_{n,d}:=\mathbb{P}(H_{n,d})$ be the
projectivization. Note that $P_{n,d}$ can be identified with
$\mathbb{C}\mathbb{P}^{N},N=\binom{n+d}{d}-1$. Let $Z_{P}$ denote the zero set
of $P\in H_{n,d}$. This set depends only on the class of $P$ in $P_{n,d}$.
Therefore, to simplify the notation, we will consider the zero sets $Z_{P}$ of
$P\in P_{n,d}$, abusing the terminology slightly.

Let us associate to each $P\in P_{n,d}$ a measure
$\mu(P)\in\mathscr{P}(\mathbb{C}\mathbb{P}^{n})$ as follows. If $Z_{P}$ is
nonsingular, then we take the restriction to ${Z_{P}}$ of the Hausdorff
measure $\mathcal{H}^{2n-2}$ (with respect to the Fubini–Study metric of
$\mathbb{C}\mathbb{P}^{n}$), normalized:

$\mu(P):=\frac{1}{\mathrm{vol}(Z_{P})}\,\mathcal{H}^{2n-2}\llcorner{Z_{P}},$

where $\mathrm{vol}(Z_{P}):=(\mathcal{H}^{2n-2}\llcorner{Z_{P}})({Z_{P}})$.
(Note that all nonsingular hypersurfaces of degree $d$ have the same volume
$\mathrm{vol}(Z_{P})=d\,\mathrm{vol}(\mathbb{C}\mathbb{P}^{n-1})$ so that the
normalization factor does not depend on $P$.)

Denote by $\Delta_{n,d}\subseteq P_{n,d}$ the discriminant locus, i.e., the
set of polynomials $P\in P_{n,d}$ such that $Z_{P}$ is singular. It is a
hypersurface in $P_{n,d}=\mathbb{C}\mathbb{P}^{N}$ given by

$\Delta_{n,d}:=\left\\{P\in P_{n,d}:\exists
z\in\mathbb{C}^{n+1}\setminus\\{0\\},\,P(z)=\frac{\partial P}{\partial
z_{0}}(z)=\cdots=\frac{\partial P}{\partial z_{n}}(z)=0\right\\},$

Using integral geometry (a Cauchy–Crofton kind of argument) one can extend the
definition of $\mu(P)$ to singular hypersurfaces, that is for
$P\in\Delta_{n,d}$, as follows, see [5, Theorem 1.1],

$\int_{\mathbb{C}\mathbb{P}^{n}}f\,d\mu(P):=\frac{1}{d}\int_{\mathbb{G}(1,n)}\left(\sum_{z\in
Z_{P}\cap\ell}m_{z}(P|_{\ell})f(z)\right)\mathrm{vol}_{\mathbb{G}(1,n)}(d\ell),\quad
f\in C^{0}(\mathbb{C}\mathbb{P}^{n}),$

where ${\mathbb{G}(1,n)}$ stands for the set of projective lines in projective
$n$-space, which is equivalent to the Grassmannian $\operatorname{Gr}(2,n+1)$,
and $m_{z}(P|_{\ell})$ is the multiplicity of $z$ as a zero of $P|_{\ell}$.
Then for every $q\geq 1$, the map
$\mu:P_{n,d}\to\mathscr{P}_{q}(\mathbb{C}\mathbb{P}^{n})$ is continuous and
injective. (Here by $\mathscr{P}_{q}(\mathbb{C}\mathbb{P}^{n})$ we mean
$\mathscr{P}(\mathbb{C}\mathbb{P}^{n})$ with the Wasserstein distance
$W_{q}$.)

######  Example 6.8 ($n=1$).

For simplicity, we prefer to work with a polynomial of one complex variable
$P\in\mathbb{C}[z]$. The passage to homogeneous polynomials of two complex
variables $z_{0},z_{1}$ is classical be means of homogenization, see e.g. [5,
Remark 3.2].

Consider the space of monic polynomials of degree $d$ in one complex variable,
$P_{a}(z)=z^{d}+a_{1}z^{d-1}+\cdots+a_{d-1}z+a_{d}$,
$a=(a_{1},\ldots,a_{d})\in\mathbb{C}^{d}$. To each polynomial
$P=P_{a}\in\mathbb{C}^{d}$ one associates a probability measure
$\mu(P)\in\mathscr{P}(\mathbb{C})$ defined by

$\mu(P):=\frac{1}{d}\sum_{P(z)=0}m_{z}(P)\cdot\delta_{z},$

where $m_{z}(P)\in\mathbb{N}$ denotes the multiplicity of $z$ as a zero of $P$
and $\delta_{z}$ is the Dirac measure. This defines a map

$\mu:\mathbb{C}^{d}\to\mathscr{P}(\mathbb{C}).$

Geometrically, the image of $\mu$ can be identified with the quotient of
$\mathbb{C}^{d}$ under the action of the symmetric group
$\operatorname{S}_{d}$. The $q$-th Wasserstein metric on
$\mathscr{P}(\mathbb{C})$ is given by

$W_{q}([x],[y])=\Big{(}{\frac{1}{d}}\min_{\sigma\in\operatorname{S}_{d}}\sum_{j=1}^{d}|x_{j}-y_{\sigma(j)}|^{q}\Big{)}^{1/q},$

where $[x]=[x_{1},\ldots,x_{d}]$ and $[y]=[y_{1},\ldots,y_{d}]$ are unordered
$d$-tuples. It is geodesically convex and coincides with the restriction of
the $q$-Wasserstein metric of $\mathscr{P}_{q}(\mathbb{C})$.

Contrary to the case $n=1$, in general, the image of $\mu$ is not geodesically
convex for the $W_{q}$-metric, that is the geodesic joining $\mu(P_{0})$ to
$\mu(P_{1})$ in $\mathscr{P}_{q}(\mathbb{C}\mathbb{P}^{n})$ will in general
not stay on $\mu(P_{n,d})$. To overcome this issue it is proposed in [5] to
consider the inner $W_{q}$-distance, adopting the approach of J.-D. Benamou
and Y. Brenier. For this one needs the notion of a $q$-absolutely continuous
curve in this setup. A curve
$\mu_{t}:I\to\mathscr{P}_{q}(\mathbb{C}\mathbb{P}^{n})$ belongs to
$AC^{q}(I,\mathscr{P}_{q}(\mathbb{C}\mathbb{P}^{n}))$ and is called _$q$
-absolutely continuous_ if there is $f\in L^{q}(I)$ such that

$W_{q}(\mu_{s},\mu_{t})\leq\int_{s}^{t}f(\tau)\,d\tau,\quad\text{for all
}s\leq t.$

In that case, for $\mathcal{L}^{1}$-a.e. $t\in I$ the limit $\lim_{h\to
0}W_{q}(\mu_{t+h},\mu_{t})/|h|$ exists, is denoted by $|\dot{\mu}_{t}|_{q}$,
and called _metric speed_. Then the _inner $W_{q}$-distance on $P_{n,d}$_ is
defined by

$W_{q}^{\textrm{in}}(P_{0},P_{1}):=\inf_{\mu_{t}:=\mu(\gamma(t))}\Big{(}\int_{I}|\dot{\mu}_{t}|_{q}^{q}\,dt\Big{)}^{\frac{1}{q}},$

where the infimum is taken over all $q$-absolutely continuous curves
$\gamma:I=[0,1]\to P_{n,d}$ joining $P_{0}$ and $P_{1}$.

It is shown in Section 5.2 of [5], see Theorem 5.13 and Remark 6.10 therein,
that the metric space $(P_{n,d},W_{q}^{\textrm{in}})$ is complete and
geodesic. Moreover, away of the discriminant locus $\Delta_{n,d}$ the
$W_{q}^{\textrm{in}}$ distance and the Fubini–Study distance
$\mathrm{d}_{\mathrm{FS}}$ on $P_{n,d}$ defined via the identification
$P_{n,d}=\mathbb{C}\mathbb{P}^{N}$ are related as follows.

######  Theorem 6.9.

Let $\epsilon>0$ and denote $P_{n,d}(\epsilon):=\\{p\in
P_{n,d}:\mathrm{d}_{\mathrm{FS}}(p,\Delta_{n,d})\geq\epsilon\\}$. Then the
identity map

$\mathrm{id}:(P_{n,d}(\epsilon),\mathrm{d}_{\mathrm{FS}})\to(P_{n,d}(\epsilon),W_{q}^{\mathrm{in}})$

is Lipschitz. In particular, $P_{n,d}(\epsilon)$ is compact in the
$W_{q}^{\mathrm{in}}$–topology.

In the proof, continuous semialgebraic curves are used. By Puiseux’s theorem,
they can be reparametrized, near their singular points, to get $C^{m}$
regularity, for arbitrary finite $m$.

Another important question is the compactness of
$(P_{n,d},W_{q}^{\mathrm{in}})$. This problem was only partially solved in
[5].

######  Theorem 6.10 ([5, Theorem 6.11]).

There is $e(n)\in\mathbb{N}$ such that, for every $1\leq q<e(n)d/(e(n)d-1)$,
the complete and separable geodesic space
$(P_{n,d},W_{q}^{\operatorname{in}})$ is compact.

Theorem 4.2 is used in the proof of the above theorem to provide
$q$-absolutely continuous curves as follows. This is fairly straightforward
for $n=1$ because, any curve of polynomials

$P_{a}(t)(z)=z^{d}+\sum_{j=1}^{d}a_{j}(t)z^{d-j},\quad t\in[0,1],$

with $a_{j}\in C^{d-1,1}([0,1])$, defines, after homogenization, an element of
$AC^{q}([0,1];\mathscr{P}_{q}(\mathbb{C}\mathbb{P}^{1}))$ for every $1\leq
q<d/(d-1)$. Thus one may take $e(1)=1$.

Thanks to the uniformity of the bound of Theorem 4.2, the general case can be
reduced to the case $n=1$ by restriction to all affine lines parameterized by
${\mathbb{G}(1,n)}$ and then integration over this Grassmannian, provided we
can avoid the lines entirely included in $Z_{P}$. This is always possible for
generic $P$ if $d>2n-3$, [5, Theorem 6.3], but not in general. Therefore it is
proposed in [5] to use the integration over the group of unitary
transformations of a rational curve of sufficiently high degree (instead of a
line). It follows from a recent result of B. Lehmann, E. Riedl and S. Tanimoto
that for a given curve of polynomials $P_{t}\in P_{n,d}$ there is a rational
curve of degree $e(n)$, given by an explicit formula, such that none of its
unitary transformations can be entirely included in $Z_{P_{t}}$. For all
details of the proof we refer the reader to [5].

######  Open Problem 6.11.

For which $q\geq 1$ is the space $(P_{n,d},W_{q}^{\operatorname{in}})$
compact? In particular, is it always compact for $q=2$?

##  7\. Appendix A. Function spaces

###  7.1. Hölder spaces

Let $\Omega\subseteq\mathbb{R}^{n}$ be open. We denote by $C^{0}(\Omega)$ the
space of continuous complex valued functions on $\Omega$. For
$k\in\mathbb{N}_{\geq 1}\cup\\{\infty\\}$, $C^{k}(\Omega)$ is the space of
$k$-times continuously differentiable functions,

$C^{k}(\Omega)=\\{f\in\mathbb{C}^{\Omega}:\partial^{\alpha}f\in
C^{0}(\Omega)\text{ for }0\leq|\alpha|\leq k\\},\quad k\in\mathbb{N},$

and

$C^{\infty}(\Omega)=\bigcap_{k\in\mathbb{N}}C^{k}(\Omega).$

The space of real analytic functions on $\Omega$ is denoted by
$C^{\omega}(\Omega)$. If $\Omega\subseteq\mathbb{C}^{n}$, then
$\mathcal{H}(\Omega)$ is the space of holomorphic functions on $\Omega$.

If the open set $\Omega\subseteq\mathbb{R}^{n}$ is bounded, we write
$C^{k}(\overline{\Omega})$ for the space of all $f\in C^{k}(\Omega)$ such that
$\partial^{\alpha}f$, $0\leq|\alpha|\leq k$, has a continuous extension to the
closure $\overline{\Omega}$. Then $C^{k}(\overline{\Omega})$ endowed with the
norm

$\|f\|_{C^{k}(\overline{\Omega})}:=\sup_{|\alpha|\leq
k}\sup_{x\in\Omega}|\partial^{\alpha}f(x)|$

is a Banach space. The spaces $C^{k}(\Omega)$, $C^{\infty}(\Omega)$,
$C^{\omega}(\Omega)$, and $\mathcal{H}(\Omega)$ carry their natural locally
convex topologies, where $\Omega$ is not necessarily bounded; note that
$C^{k}(\Omega)$, $C^{\infty}(\Omega)$, and $\mathcal{H}(\Omega)$ are Fréchet
spaces.

Let $\alpha\in(0,1]$ and let $\Omega\subseteq\mathbb{R}^{n}$ be open and
bounded. A function is _$\alpha$ -Hölder continuous in $\Omega$_ if

$\operatorname{H\ddot{o}ld}_{\alpha,\Omega}(f):=\sup_{x,y\in\Omega,\,x\neq
y}\frac{|f(x)-f(y)|}{|x-y|^{\alpha}}<\infty.$

It is _Lipschitz continuous in $\Omega$_ if it is $1$-Hölder continuous in
$\Omega$; in that case, we write

$\operatorname{Lip}_{\Omega}(f):=\operatorname{H\ddot{o}ld}_{1,\Omega}(f)<\infty.$

For $k\in\mathbb{N}$ and $\alpha\in(0,1]$, we have the scale of
_Hölder(–Lipschitz) spaces_

$C^{k,\alpha}(\overline{\Omega}):=\\{f\in
C^{k}(\overline{\Omega}):\partial^{\beta}f\text{ is $\alpha$-H\"{o}lder
continuous for }|\beta|=k\\}$

and we equip them with the norm

$\|f\|_{C^{k,\alpha}(\overline{\Omega})}:=\|f\|_{C^{k}(\overline{\Omega})}+\sup_{|\beta|=k}\operatorname{H\ddot{o}ld}_{\alpha,\Omega}(\partial^{\beta}f).$

Then $C^{k,\alpha}(\overline{\Omega})$ is a Banach space. We write
$C^{k,\alpha}(\Omega)$ (where $\Omega$ is not necessarily bounded) for the
space of all functions $f\in C^{k}(\Omega)$ such that
$\|f|_{\Omega^{\prime}}\|_{C^{k,\alpha}(\overline{\Omega}^{\prime})}<\infty$
for all relatively compact $\Omega^{\prime}\Subset\Omega$ with its natural
Fréchet topology.

###  7.2. Lebesgue spaces and weak Lebesgue spaces

Let $\Omega\subseteq\mathbb{R}^{n}$ be open and $1\leq p\leq\infty$. The
_Lebesgue space_ $L^{p}(\Omega)$ is the space of measurable complex valued
functions $f:\Omega\to\mathbb{C}$ that are $p$-integrable with respect to the
$n$-dimensional Lebesgue measure $\mathcal{L}^{n}$; actually, the elements of
$L^{p}(\Omega)$ are equivalence classes of functions that coincide
$\mathcal{L}^{n}$-almost everywhere. If not stated otherwise, “measurable”
always means “Lebesgue measurable”. The $L^{p}$-norms make the Lebesgue spaces
to Banach spaces:

$\displaystyle\|f\|_{L^{p}(\Omega)}$
$\displaystyle:=\Big{(}\int_{\Omega}|f(x)|^{p}dx\Big{)}^{1/p},\quad 1\leq
p<\infty,$ $\displaystyle\|f\|_{L^{\infty}(\Omega)}$
$\displaystyle:=\operatorname{ess\,sup}_{x\in\Omega}|f(x)|,\quad p=\infty.$

We denote by $L^{p}_{\operatorname{loc}}(\Omega)$ the space of measurable
functions $f:\Omega\to\mathbb{C}$ such that $\|f\|_{L^{p}(K)}<\infty$ for each
compact subset $K\subseteq\Omega$.

Let $\Omega\subseteq\mathbb{R}^{n}$ be open and bounded. For $1\leq p<\infty$,
the _weak $L^{p}$-space_ is the space of all measurable functions
$f:\Omega\to\mathbb{C}$ such that

$\|f\|_{p,w,\Omega}:=\sup_{r>0}\big{(}r\cdot\mathcal{L}^{n}(\\{x\in\Omega:|f(x)|>r\\})^{1/p}\big{)}<\infty;$

again, we identify functions that coincide $\mathcal{L}^{n}$-almost
everywhere. Note that $\|\cdot\|_{p,w,\Omega}$ is only a quasinorm: the
triangle inequality fails, but for $f_{j}\in L^{p}_{w}(\Omega)$ one still has

$\Big{\|}\sum_{j=1}^{m}f_{j}\Big{\|}_{p,w,\Omega}\leq
m\sum_{j=1}^{m}\|f_{j}\|_{p,w,\Omega}.$

The spaces $L^{p}_{w}(\Omega)$ are complete with respect to the quasinorms
$\|\cdot\|_{p,w,\Omega}$, i.e., they are quasi-Banach spaces. If $p>1$, then
there exists a norm equivalent to $\|\cdot\|_{p,w,\Omega}$ which makes
$L^{p}_{w}(\Omega)$ into a Banach space. For $1\leq q<p<\infty$, we have

$\|f\|_{q,w,\Omega}\leq\|f\|_{L^{q}(\Omega)}\leq\Big{(}\frac{p}{p-q}\Big{)}^{1/q}\mathcal{L}^{n}(\Omega)^{1/q-1/p}\,\|f\|_{p,w,\Omega}$

and thus there are continuous strict inclusions $L^{p}_{w}(\Omega)\subseteq
L^{q}(\Omega)\subseteq L^{q}_{w}(\Omega)$. The $L^{p}_{w}$-quasinorm is
$\sigma$-subadditive (i.e.,
$\|f\|^{p}_{p,w,\Omega}\leq\sum_{j}\|f\|^{p}_{p,w,\Omega_{j}}$ for countable
open covers $\Omega=\bigcup_{j}\Omega_{j}$) but not $\sigma$-additive.

###  7.3. Sobolev spaces

Let $\Omega\subseteq\mathbb{R}^{n}$ be open. For $k\in\mathbb{N}_{\geq 1}$ and
$1\leq p\leq\infty$, we consider the _Sobolev space_

$W^{k,p}(\Omega):=\\{f\in L^{p}(\Omega):\partial^{\alpha}f\in
L^{p}(\Omega)\text{ for }|\alpha|\leq k\\},$

where $\partial^{\alpha}f$ are distributional derivatives, with the norm

$\|f\|_{W^{k,p}(\Omega)}:=\sum_{|\alpha|\leq
k}\|\partial^{\alpha}f\|_{L^{p}(\Omega)}.$

Then $W^{k,p}(\Omega)$ are Banach spaces. We denote by
$W^{k,p}_{\operatorname{loc}}(\Omega)$ the space of functions $f\in
L^{p}_{\operatorname{loc}}(\Omega)$ such that $\partial^{\alpha}f\in
L^{p}_{\operatorname{loc}}(\Omega)$ for $|\alpha|\leq k$.

Let $k=1$ and suppose $f\in L^{p}(\Omega)$. Then $f\in W^{1,p}(\Omega)$ if and
only if $f$ has a representative $\overline{f}$ that is absolutely continuous
on almost all line segments in $\Omega$ parallel to the coordinate axes and
whose classical derivatives $\partial_{i}\overline{f}$ belong to
$L^{p}(\Omega)$. In particular, for a bounded interval $I\subseteq\mathbb{R}$,
$W^{1,1}(I)$ can be identified with the space $AC(I)$ of absolutely continuous
functions on $I$. We recall that, by definition, $f\in AC(I)$ if for every
$\epsilon>0$ there exists $\delta>0$ such that
$\sum_{j=1}^{k}|f(b_{j})-f(a_{j})|\leq\epsilon$ for every finite number of
disjoint intervals $(a_{j},b_{j})$, $j=1,\ldots,k$, satisfying
$[a_{j},b_{j}]\subseteq I$ and $\sum_{j=1}^{k}(b_{j}-a_{j})\leq\delta$.

In the case that $p=\infty$ and that $\Omega$ is a bounded Lipschitz domain,
$f\in W^{1,\infty}(\Omega)$ if and only if $f$ has a representative that is
Lipschitz continuous on $\Omega$.

###  7.4. Functions of bounded variation

Let $\Omega\subseteq\mathbb{R}^{n}$ be open. A real valued $f\in
L^{1}(\Omega)$ is a _function of bounded variation in $\Omega$_ if the
distributional derivative of $f$ is representable by a finite Radon measure in
$\Omega$, i.e., for all $C^{\infty}$ functions $\varphi:\Omega\to\mathbb{R}$
with compact support in $\Omega$ and all $1\leq i\leq n$,

$\int_{\Omega}f(x)\partial_{i}\varphi(x)\,dx=-\int_{\Omega}\varphi\,dD_{i}f,$

for some $\mathbb{R}^{n}$-valued Radon measure $Df=(D_{1}f,\ldots,D_{n}f)$ on
$\Omega$. By definition, a complex valued function is of bounded variation in
$\Omega$ if its real and imaginary parts are. The space $BV(\Omega)$ of all
functions of bounded variation in $\Omega$ equipped with the norm

$\|f\|_{BV(\Omega)}:=\|f\|_{L^{1}(\Omega)}+|Df|(\Omega),$

where $|Df|(\Omega)$ is the total variation measure, is a Banach space.

The Sobolev space $W^{1,1}(\Omega)$ is strictly contained in $BV(\Omega)$ (for
$f\in W^{1,1}(\Omega)$ we have $Df=\nabla f\,\mathcal{L}^{n}$).

Functions of bounded variation may have discontinuities. Let $f\in
BV(\Omega)$. Then the Lebesgue decomposition with respect to $\mathcal{L}^{n}$
induces a decomposition

$Df=D^{a}f+D^{j}f+D^{c}f,$

where $D^{a}f=\nabla f\,\mathcal{L}^{n}$ is the _absolutely continuous part_ ,
$D^{j}f$ is the _jump part_ (given by integration of the jump height against
the restriction of $\mathcal{H}^{n-1}$ to the set of approximate jump points
$J_{f}$, i.e.,
$D^{j}f=((f^{+}-f^{-})\otimes\nu_{f})\,(\mathcal{H}^{n-1}\llcorner J_{f})$,
where $\nu_{f}$ is the direction of the jump), and $D^{c}f$ is the _Cantor
part_.

The closed subspace $SBV(\Omega):=\\{f\in BV(\Omega):D^{c}f=0\\}$ of
$BV(\Omega)$ is called the space of _special functions of bounded variation_.
We have continuous strict inclusions $W^{1,1}(\Omega)\subseteq
SBV(\Omega)\subseteq BV(\Omega)$.

##  8\. Appendix B. The space of hyperbolic polynomials

The fact that the regularity of the roots of hyperbolic polynomials is
essentially better than that of the roots of general polynomials is related to
the geometry of the space $\operatorname{Hyp}(d)$ of monic hyperbolic
polynomials of degree $d$ and the constraints a sufficiently differentiable
curve of hyperbolic polynomials is subdued to at the boundary of
$\operatorname{Hyp}(d)$. In this section, we collect some interesting
topological and geometric properties of $\operatorname{Hyp}(d)$.

The following results are due to Kostov [54]; see also Arnol’d [6] and
Givental’ [38]. Let $p=(p_{1},\ldots,p_{d})$ be a probability vector, i.e.,
$p_{i}>0$ and $p_{1}+\cdots+p_{d}=1$. The _Vandermonde mapping_
$W=W(p):\mathbb{R}^{d}\to\mathbb{R}^{d}$ is defined by

$W_{j}(x):=p_{1}x_{1}^{j}+\cdots+p_{d}x_{d}^{j},\quad 1\leq j\leq d.$

If $p_{i}=1/d$ for $1\leq j\leq d$, then $W_{j}$, $1\leq j\leq d$, are the
Newton polynomials (up to a constant factor) which generate the algebra of
symmetric polynomials on $\mathbb{R}^{d}$ and consequently the image
$W(\mathbb{R}^{d})$ is isomorphic to $\operatorname{Hyp}(d)$ in this case.

The set $K:=\\{x\in\mathbb{R}^{d}:x_{1}\leq x_{2}\leq\cdots\leq x_{d}\\}$ is a
fundamental domain for the action of the symmetric group
$\operatorname{S}_{d}$ on $\mathbb{R}^{d}$ by permuting the coordinates. Every
set of the form $\\{x\in K:W_{j}(x)=c_{j},\,1\leq j\leq k\\}$ is called a
_$(d-k)$ -dimensional Vandermonde manifold_.

######  Theorem 8.1 ([54]).

Every Vandermonde manifold is either contractible or empty. The map $W:K\to
W(K)$ is a homeomorphism. Every set of the form $\\{y\in
W(K):y_{j}=c_{j},\,1\leq j\leq k\\}$ is either contractible or empty.

As a consequence, $W^{k}:=(W_{1},\ldots,W_{k})$, $k\leq d$, is a homeomorphism
of the closure of every $k$-dimensional stratum of the polyhedron $K$ onto its
image; indeed, the restriction of $W^{k}$ to such a stratum is the
$k$-dimensional Vandermonde mapping for some probability vector.

Let $K_{0}:=\\{x\in K:W_{1}(x)=0,\,W_{2}(x)\leq 1\\}$.

######  Theorem 8.2 ([54]).

The sets $W(K_{0})$ and $W^{k}(K_{0})$, $1\leq k\leq d-1$, are quasiconvex,
i.e., there is a constant $C>0$ such that any two points can be joint by a
piecewise smooth curve in the set of length not exceeding $C$ times the
Euclidean distance of the two points.

In particular, the space of monic hyperbolic polynomials of degree $d$ in
Tschirnhausen form with $|\tilde{a}_{2}|\leq 1$ is quasiconvex. We refer to
[54] for additional information on the stratification of $W(K)$.

The semialgebraic set $\operatorname{Hyp}(d)\subseteq\mathbb{R}^{d}$ can be
described by explicit inequalities. Let $s_{j}=x_{1}^{j}+\cdots+x_{d}^{j}$,
$j\geq 0$, be the Newton polynomials and consider the Bezoutiant matrix

$B:=\begin{pmatrix}s_{0}&s_{1}&\cdots&s_{d-1}\\\ s_{1}&s_{2}&\cdots&s_{d}\\\
\vdots&\vdots&\ddots&\vdots\\\ s_{d-1}&s_{d}&\cdots&s_{2d-2}\end{pmatrix}.$

We have $B=\tilde{B}(\sigma_{1},\ldots,\sigma_{d})$ for a unique matrix valued
polynomial $\tilde{B}$, where the $\sigma_{i}$ are the elementary symmetric
functions. The following result is Hermite’s [47] and Sylvester’s [107]
version of Sturm’s theorem; the indicated references give a modern account and
a far reaching generalization.

######  Theorem 8.3 ([89, 90]).

The space $\operatorname{Hyp}(d)$ of monic hyperbolic polynomials of degree
$d$ can be identified with $\\{y\in\mathbb{R}^{d}:\tilde{B}(y)\geq 0\\}$. The
rank of $\tilde{B}$ equals the number of distinct roots and its signature
equals the number of distinct real roots.

Since a real symmetric matrix is positive semidefinite if and only if the
determinants of the principal minors are nonnegative, this description yields
explicit inequalities that determine $\operatorname{Hyp}(d)$.

Recall that $\operatorname{Hyp}_{T}(d)=\operatorname{Hyp}(d)\cap\\{y_{1}=0\\}$
is the space of monic hyperbolic polynomials of degree $d$ in Tschirnhausen
form. Then $\operatorname{Hyp}_{T}(1)$ is the origin,
$\operatorname{Hyp}_{T}(2)$ is $(-\infty,0]$, $\operatorname{Hyp}_{T}(3)$ is
the closure of the inside of a cusp, and $\operatorname{Hyp}_{T}(4)$ is the
closure of the inside of the swallowtail.

In [75] Meguerditchian studies the “escape” from the set
$\operatorname{Hyp}(d)$: roughly speaking, for a polynomial $P$ in
$\operatorname{Hyp}(d)$ an integer $s_{P}$, which depends only on the
multiplicity vector of $P$, is introduced such that one can decide whether
$P+Q$ belongs to $\operatorname{Hyp}(d)$ or not in terms of how the degree of
$Q$ relates to $s_{P}$.

By Nuij [79], for $P\in\operatorname{Hyp}(d)$ also
$P+sP^{\prime}\in\operatorname{Hyp}(d)$, for all $s\in\mathbb{R}$. A
generalization of this result is due to Kurdyka and Păunescu [65].

### Acknowledgements

Armin Rainer was supported by the Austrian Science Fund (FWF) Project P
32905-N.

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## Index

  * Abhyankar–Jung theorem Theorem 2.11
  * absolutely continuous §6.2, §7.3
  * $AC$ §7.3
  * $\mathcal{A}_{d}(\mathbb{R}^{m})$ §4.2.1
  * basic invariants §5.2, §5.6
    * minimal system of §5.2, §5.6
  * bounded variation §7.4
  * Bronshtein’s theorem Theorem 3.1, Theorem 3.11, Theorem 3.2
  * $BV$ §7.4
  * Cartan subspace §5.2.1, §5.6.1
  * categorical quotient §5.2, §5.6
  * characteristic map §3.4.2
  * characteristic roots §3.4.2
  * $\operatorname{\chi}_{A}$ Notation
  * $C^{\infty}$ §7.1
  * $C^{k}$ §7.1
  * $C^{k,\alpha}$ §7.1
  * $C^{\omega}$ §7.1
  * coregular representation Remark 5.13
  * $\mathbb{C}[V]^{G}$ §5.6
  * $DC$ §3.4.2
  * definable §3.3.4
  * $\operatorname{\Delta}_{A}$ Notation
  * $\operatorname{\Delta}_{P}$ Notation
  * $\operatorname{Diag}(d)$ §5.4.1, Notation
  * difference-convex §3.4.2
  * elementary symmetric functions §5.1
  * $\mathcal{F}^{\beta}$ §3.3.2
  * fixed points §5.3.1
  * $f_{\operatorname{red}}$ Notation
  * Gårding hyperbolic polynomial §3.4.1
  * geometric quotient §5.6
  * Glaeser’s inequality §3.2.2
  * $\operatorname{GL}_{n}(\mathbb{C})$ Notation
  * $G\circlearrowleft V$ §5.2, §5.6
  * $(G\circlearrowleft V,d,\sigma)$ §5.2, §5.6
  * $\mathcal{H}$ §7.1
  * $\operatorname{Herm}(d)$ §3.5, Notation
  * higher order Glaeser inequalities §4.2
  * $\mathcal{H}^{k}$ Notation
  * $\operatorname{H\ddot{o}ld}_{\alpha,\Omega}(f)$ §7.1
  * Hölder–Lipschitz space §7.1
  * $\operatorname{Hyp}(d)$ §2.2.2, §8, Notation
  * hyperbolic polynomial §1
  * $\operatorname{Hyp}_{T}^{0}(d)$ §2.2.2, Notation
  * $\operatorname{Hyp}_{T}(d)$ §2.2.2, §8, Notation
  * $\mathbb{I}$ Notation
  * inner Wasserstein distance §6.2
  * isotropy subgroup §5.3.3
  * Lax conjecture Remark 3.30
  * Lebesgue space §7.2
  * lifting §5.1, §5.2, §5.5, §5.6
  * linearly reductive group §5.6
  * $\operatorname{Lip}_{\Omega}(f)$ §7.1
  * $\mathcal{L}^{n}$ Notation
  * local blowings-up §5.2.2
  * local power substitution §5.6.2
  * $L^{p}$ §7.2
  * $L^{p}_{w}$ §4.1, §7.2
  * $\operatorname{Mat}_{m,n}(R)$ Notation
  * $\operatorname{Mat}_{n}(R)$ Notation
  * metric speed §6.2
  * multiplicity §3.3.3
  * multivalued Sobolev functions §4.2.1
  * Newton identities §2.2.1
  * Newton polynomials §2.2.1
  * normally nonflat §3.3.3, §5.2.2
  * o-minimal §3.3.4
  * $\operatorname{O}_{n}(\mathbb{R})$ Notation
  * orbit map §5.2
  * orbit space §5.2, §5.6
  * orbit type §5.3.4
    * principal §5.3.4
  * orbit type stratification §5.3.4
  * parameterization of multivalued function §4.2.1
  * parameterization of the roots Notation
  * polar representation §5.2.1, §5.6.1
  * polynomial
    * elementary symmetric §2.2.1, §5.1
    * Gårding hyperbolic §3.4.1
    * hyperbolic §1
    * Newton §2.2.1
    * quasi-ordinary §2.3.1
    * real stable §3.4.1
    * stable §3.4.1
  * pseudoreflection Remark 5.13
  * quasi-ordinary polynomial §2.3.1
  * real stable polynomial §3.4.1
  * Rellich’s theorem Theorem 2.1, Theorem 2.2
  * $\mathbb{R}[V]^{G}$ §5.2
  * $SBV$ §7.4
  * section §5.2.1, §5.6.1
  * sign of a function §6.1
  * singular value decomposition Theorem 2.17
  * singular values §2.4, §3.6
  * slice representation §5.3.3
  * slice theorem §5.3.3
  * $\operatorname{S}_{n}$ Notation
  * Sobolev space §7.3
  * special functions of bounded variation §7.4
  * splitting §2.2.2, §3.2.1
  * stable point §5.6.1
  * $\operatorname{Sym}(d)$ §5.4.1, Notation
  * system of the roots Notation
  * theorem
    * Abhyankar–Jung Theorem 2.11
    * Bronshtein’s Theorem 3.1, Theorem 3.11, Theorem 3.2
    * Hermite–Sylvester version of Sturm’s Theorem 8.3
    * Rellich’s Theorem 2.1, Theorem 2.2
    * slice §5.3.3
  * Tschirnhausen form §2.2.1
  * Tschirnhausen transformation §2.2.1
  * $\operatorname{U}_{n}(\mathbb{C})$ Notation
  * unordered tuple §4.2.1, Notation
  * Vandermonde manifold §8
  * Vandermonde mapping §8
  * $V{/\\!\\!/}G$ §5.2, §5.3.1, §5.6
  * Wasserstein distance §6.2
    * inner §6.2
  * weak Lebesgue space §4.1, §7.2
  * Weyl group §5.2.1, §5.6.1
  * $W^{k,p}$ §7.3