# New bounds on curve tangencies and orthogonalities

Jordan S. Ellenberg Supported by NSF-DMS award 1402620, the John S. Guggenheim
Fellowship, and the Vilas Distinguished Achievement Professorship. Jozsef
Solymosi Supported by NSERC, ERC Advanced Research Grant AdG. 321104, and by
Hungarian National Research Grant NK 104183. Joshua Zahl Supported by a NSF
Postdoctoral Fellowship.

###### Abstract

We establish new bounds on the number of tangencies and orthogonal
intersections determined by an arrangement of curves. First, given a set of
$n$ algebraic plane curves, we show that there are $O(n^{3/2})$ points where
two or more curves are tangent. In particular, if no three curves are mutually
tangent at a common point, then there are $O(n^{3/2})$ curve-curve tangencies.
Second, given a family of algebraic plane curves and a set of $n$ curves from
this family, we show that either there are $O(n^{3/2})$ points where two or
more curves are orthogonal, or the family of curves has certain special
properties.

We obtain these bounds by transforming the arrangement of plane curves into an
arrangement of space curves so that tangency (or orthogonality) of the
original plane curves corresponds to intersection of space curves. We then
bound the number of intersections of the corresponding space curves. For the
case of curve-curve tangency, we use a polynomial method technique that is
reminiscent of Guth and Katz’s proof of the joints theorem. For the case of
orthogonal curve intersections, we employ a bound of Guth and the third author
to control the number of two-rich points in space curve arrangements.

title = New bounds on curve tangencies and orthogonalities, author = Jordan S.
Ellenberg, Jozsef Solymosi, and Joshua Zahl, plaintextauthor = Jordan S.
Ellenberg, Jozsef Solymosi, Joshua Zahl, plaintexttitle = New bounds on curve
tangencies and orthogonalities, copyrightauthor = J. Ellenberg, J. Solymosi,
J. Zahl, year=2016, number=18, received=1 April 2016, revised=28 October 2016,
published=4 November 2016, doi=10.19086/da990,

[classification=text]

##  1 Introduction

We will bound the number of tangencies and orthogonal intersections determined
by a set of $n$ algebraic plane curves. Bounding the maximal number of curve
tangencies plays an important role in combinatorial and computational
geometry. For the relevant works and an extended bibliography of the subject,
we refer to the classical paper of Agarwal et. al. [1] and the recent
publication of Pach et. al. [13]. The latter also deals in part with
tangencies between bounded degree algebraic curves, which is the main subject
of our paper.

###  1.1 Tangent curves

If $n$ curves are mutually tangent at a common point, then this would lead to
$\binom{n}{2}\sim n^{2}$ tangencies. To avoid this degenerate situation, we
could require that no three curves be mutually tangent at a common point.
Instead we will count a slightly different quantity.

######  Definition 1 (Directed points of tangency).

Let $k$ be a field and let $\mathcal{L}$ be a set of irreducible algebraic
curves in $k^{2}$. Let $p$ be a point in $k^{2}$ and let $\ell$ be a line
passing through $p$. We say that $(p,\ell)$ is a directed point of tangency
for $\mathcal{L}$ if there are at least two distinct curves in $\mathcal{L}$
that are smooth at $p$ and tangent to $\ell$ at $p$ (for readers unfamiliar
with algebraic geometry, the definition of a smooth point of a curve is given
in Section 2.3). Define $\mathcal{T}(\mathcal{L})$ to be the set of directed
points of tangency, and for each $(p,\ell)\in\mathcal{T}(\mathcal{L}),$ define
$\operatorname{mult}(p,\ell;\mathcal{L})$ to be the number of curves from
$\mathcal{L}$ that are smooth at $p$ and tangent to $\ell$ at $p$.

Our first main result is the following bound on curve tangencies.

######  Theorem 1.

Let $D\geq 1$ be an integer. Then there are constants $c_{D}>0$ (small) and
$C_{D}$ (large) so that the following holds. Let $k$ be a field, and let
$\mathcal{L}$ be a set of $n$ irreducible plane curves in $k^{2}$ of degree at
most $D$. Suppose that $n\leq c_{D}\operatorname{char}(k)^{2}$ (if
$\operatorname{char}(k)=0$ then we place no restrictions on $n$). Then

$\sum_{(p,\ell)\in\mathcal{T}(\mathcal{L})}\operatorname{mult}(p,\ell;\mathcal{L})\leq
C_{D}n^{3/2}.$ (1)

######  Corollary 1.

A set of $n$ (real or complex) plane algebraic curves of degree at most $D$,
no three of which pass through a common point, determine $O_{D}(n^{3/2})$
curve-curve tangencies.

We will prove Theorem 1 using the algebraic techniques first pioneered by Dvir
in [4] to solve the finite field Kakeya problem, and later used by Guth and
Katz in [8] to solve the joints problem. We will describe a process that
converts plane curves to space curves, so that curve-curve tangency in the
plane $k^{2}$ corresponds to curve-curve intersection in $k^{3}$. We will then
use a joints-like argument to show that not too many curve-curve intersections
can occur.

###  1.2 Orthogonal curves

We will also obtain a bound on the number of pairs of curves that can
intersect orthogonally. However, a problem immediately arises: it is possible
for $n$ lines to determine $n^{2}/4$ points where two lines intersect
orthogonally—just select a set of $n/2$ parallel lines and a second set of
$n/2$ parallel lines orthogonal to the first set. Similarly, it is possible to
find two sets of circles, each of cardinality $n/2$, so that each circle from
the first set intersects each circle from the second set orthogonally; such
arrangements are called Apollonian circles. These are two simple examples of
what are known as orthogonal trajectories.

In the examples above, $n$ curves can determine $\sim n^{2}$ orthogonal
intersections. We will establish a certain dichotomy: given a particular
family of curves (defined below), either every arrangement of curves from this
family determines very few orthogonal intersections, or it is possible to find
arrangements with a nearly maximal number of orthogonal intersections. This
will be stated precisely in Theorem 2 below. First however, we will need to
introduce several definitions.

The projective space $\mathbf{P}(k^{d})$ is the quotient of
$k^{d}\backslash\\{0\\}$ by the equivalence relation that identifies
$(w_{1},\ldots,w_{d})$ with $(w_{1}^{\prime},\ldots,w_{d}^{\prime})$ if there
is some non-zero $\lambda\in k$ such that $w_{i}=\lambda w_{i}^{\prime}$ for
each index $i$. We will often omit the brackets and write $\mathbf{P}k^{d}$ in
place of $\mathbf{P}(k^{d})$. A set $X\subset\mathbf{P}k^{d}$ is an (not
necessarily irreducible) algebraic variety if it can be written as the zero-
set of a collection of homogeneous polynomials in $d$ variables. If
$D\leq\operatorname{char}(k)$, then the set of algebraic curves in $k^{2}$ of
degree $\leq D$ are in one-to-one correspondence with the set of points in
$\mathbf{P}k^{\binom{D+2}{2}}$. We will frequently abuse notation and identify
these two sets. In particular, if $X\subset\mathbf{P}k^{\binom{D+2}{2}}$ is a
variety, then we will abuse notation and refer to $X$ as a family of degree
$\leq D$ curves.

######  Definition 2 (Directed points of orthogonality).

Let $k$ be a field and let $\mathcal{L}$ be a set of irreducible algebraic
curves in $k^{2}$. Let $p$ be a point in $k^{2}$, let $\ell$ be a line passing
through $p$, and let $\ell^{\perp}$ be the line passing through $p$ that is
orthogonal to $\ell$ (here the vector $(v_{1},v_{2})$ is orthogonal to the
vector $(v_{1}^{\prime},v_{2}^{\prime})$ if
$v_{1}v_{1}^{\prime}+v_{2}v_{2}^{\prime}=0$). We say that $(p,\ell)$ is a
directed point of orthogonality for $\mathcal{L}$ if there is a curve in
$\mathcal{L}$ that is smooth at $p$ and tangent to $\ell$ at $p$, and there is
a second curve in $\mathcal{L}$ that is smooth at $p$ and tangent to
$\ell^{\perp}$ at $p$.

######  Theorem 2.

Let $k$ be a field and let $X\subset\mathbf{P}k^{\binom{D+2}{2}}$ be a family
of degree $\leq D$ curves in $k^{2}$. Then exactly one of the following must
hold.

  * •

Every set of $n\leq c_{D}(\operatorname{char}k)^{2}$ curves from $X$
determines $O_{D,X}(n^{3/2})$ directed points of orthogonality.

  * •

Let $K$ be the algebraic closure of $k$, and let
$\bar{X}\subset\mathbf{P}K^{\binom{D+2}{2}}$ be the closure of $X$. Then for
each $1\leq n\leq c_{D,X}(\operatorname{char}k)^{2},$ we can find $n$ curves
in $\bar{X}$ that determine $\frac{n^{2}}{4}(1-o_{D,X}(1))$ directed points of
orthogonality111The expression $o_{D,X}(1)$ refers to a term that tends to 0
as $n$ (and thus $\operatorname{char}k$) tends to infinity, while the degree
$D$ and the equations defining the variety $X$ are kept fixed..

To prove Theorem 2, we will again convert plane curves to space curves so that
curve-curve orthogonality in the plane corresponds to curve-curve intersection
in $k^{3}$. We will then apply Theorem 1.2 from [9]. This theorem states that
for a given family of space curves, either (i) any set of curves from this
family determines few curve-curve intersections in $k^{3}$, or (ii) there are
many curves from the family that form two pairwise intersecting sets of
curves. Case (i) corresponds to few directed points of orthogonality, while
case (ii) allows us to construct arrangements of $n$ curves with
$\frac{n^{2}}{4}(1-o_{D,X}(1))$ directed points of orthogonality.

###  1.3 Previous work

In [3], Clarkson et. al. developed techniques to bound the number of
incidences between points and surfaces in $\mathbb{R}^{3}$. In [15, 16], Wolff
observed222The tangency bound for circles is not stated explicitly in [15,
16], but they are discussed in [14, Section 3], which is an expository paper
that discusses the results in [15, 16] that these techniques could be used to
bound the number of tangencies between circles in $\mathbb{R}^{2}$.
Specifically, $n$ circles in $\mathbb{R}^{2}$ determine $O(n^{3/2}\beta(n))$
directed points of tangency, where $\beta(n)$ is a very slowly growing
function. Some of the methods from [17] can also be used to slightly improve
this tangency bound to $O(n^{3/2})$. All of these methods, however, made
crucial use of two facts. First, circles in the plane have three degrees of
freedom—a circle can be described by three parameters. While these techniques
can be extended to general degree $D$ algebraic curves in $\mathbb{R}^{2}$,
the resulting exponent in the bound becomes worse as $D$ increases. Second,
these older techniques make crucial use of various “cuttings” or “polynomial
partitioning” theorems, and these results generally cannot be extended to
fields other than $\mathbb{R}$. Theorem 1 suffers from neither of these
constraints.

In [12], Megyesi and Szabó proved that if $k$ is a field whose characteristic
is not two, then $n$ conics in $k^{2}$ determine $O(n^{2-1/7633})$ directed
points of tangency. Theorem 1 improves this to $O(n^{3/2})$ in the special
case where $\operatorname{char}k=0$ or $n<C(\operatorname{char}k)^{2}$ for a
suitable (absolute) constant $C$.

In [11], Marcus and Tardos showed that any family of $n$ pseudocircles
determines $O(n^{3/2}\log n)$ tangencies. In the case where the pseudocircles
are defined by bounded-degree algebraic curves, Theorem 1 improves this bound.
This is also Problem 14 in Chapter 7.1 of Brass-Moser-Pach [2].

Purely combinatorial methods work well for bounding pseudocircle tangencies.
However, one can not expect general results for “pseudocubics” as the
following simple construction shows. In Figure 1 we present an arrangement of
curves in $\mathbb{R}^{2}$ where every two curves meet in two points and every
two curves are tangent; this results in $\binom{n}{2}$ directed points of
tangency. Thus the requirement that the curves be algebraic (and of controlled
degree) is crucial.

Figure 1: Pseudocubics

###  1.4 Proof sketch

In this section we will sketch the proof of Theorem 1 in the special case
where $k=\mathbb{R}$, the curves in $\mathcal{L}$ are circles, and no three
circles are tangent at a common point. Suppose that there are more than
$Cn^{3/2}$ tangencies. If $C$ is sufficiently large, we will obtain a
contradiction. To simply the proof sketch slightly, we will assume that each
circle is tangent to at least $(C/2)n^{1/2}$ other circles (we can always
reduce to this case by refining our collection of circles slightly). After
applying a rotation, we can assume that for every directed point of tangency
$(p,\ell)$, the line $\ell$ does not point in the $y$–direction.

For each circle $\gamma\in\mathcal{L}$, Define

$\beta[\gamma]=\Big{\\{}(x,y,z)\in\mathbb{R}^{3}\colon(x,y)\in\gamma,\
z=-\frac{x-x_{0}}{y-y_{0}}\Big{\\}},$

where $(x_{0},y_{0})$ is the center of $\gamma$. The reason for this
definition is as follows. Let $(x,y)\in\mathbb{R}^{2}$ and let $\ell$ be a
non-vertical line containing $(x,y)$. Let $z$ be the slope of $\ell$. Then
$\gamma$ is tangent to $\ell$ at the point $(x,y)$ if and only if
$(x,y,z)\in\beta[\gamma]$. In particular, the projection of $\beta[\gamma]$ to
the $xy$–plane is the set $\gamma\backslash\\{p_{1},p_{2}\\}$, where $p_{1}$
and $p_{2}$ are the two points where $\gamma$ is tangent to a vertical line.

Define $\mathcal{C}=\\{\beta[\gamma]\colon\gamma\in\mathcal{L}\\}.$ Observe
that $\gamma_{1}$ and $\gamma_{2}$ are tangent if and only if
$\beta[\gamma_{1}]\cap\beta[\gamma_{2}]\neq\emptyset$ (recall that no two
curves from $\mathcal{L}$ share a directed point of tangency $(p,\ell)$ where
$\ell$ is a vertical line). Next, suppose that
$(x,y,z)\in\beta[\gamma_{1}]\cap\beta[\gamma_{2}]$. If
$\gamma_{1}\neq\gamma_{2}$, then

$(0,0,1)\in\operatorname{span}\big{(}T_{(x,y,z)}\beta[\gamma_{1}],\
T_{(x,y,z)}\beta[\gamma_{2}]\big{)}.$ (2)

To see this, define $f_{i}(t)$ so that for all $t$ in a small neighborhood of
0, $(t+x,f_{i}(t))$ is a parameterization of $\gamma_{i}$ in a neighborhood of
$(x,y)$. Since $\gamma_{1}$ and $\gamma_{2}$ are tangent at $(x,y)$,
$\frac{df_{1}}{dt}(0)=\frac{df_{2}}{dt}(0)$. Since the circles $\gamma_{1}$
and $\gamma_{2}$ are distinct,
$\frac{d^{2}f_{1}}{dt^{2}}(0)\neq\frac{d^{2}f_{2}}{dt^{2}}(0)$. Next, observe
that in a neighborhood of $(x,y,z)$, the curve $\beta[\gamma_{i}]$ is
parameterized by $(t,f_{i}(t),\frac{df_{i}}{dt}(t))$. In particular, the
vector $\big{(}1,\frac{df_{i}}{dt}(0),\frac{d^{2}f_{i}}{dt^{2}}(0)\big{)}$ is
contained in the vector space $T_{(x,y,z)}\beta[\gamma_{i}]$. Thus $(0,0,1)$
is in the span of the vector

$\Big{(}1,\frac{df_{1}}{dt}(0),\frac{d^{2}f_{1}}{dt^{2}}(0)\Big{)}-\Big{(}1,\frac{df_{2}}{dt}(0),\frac{d^{2}f_{2}}{dt^{2}}(0)\Big{)},$

which is contained in
$\operatorname{span}\big{(}T_{(x,y,z)}\beta[\gamma_{1}],\
T_{(x,y,z)}\beta[\gamma_{2}]\big{)}.$ This establishes (2).

Next, let $P[x,y,z]$ be the non-zero polynomial of minimal degree that
vanishes on all of the curves in $\mathcal{C}$. In particular,
$\deg(P)=O(n^{1/2})$. By (2), we have that if $(x,y,z)$ is a point in
$\mathbb{R}^{3}$ where two curves from $\mathcal{C}$ intersect, then
$\partial_{z}P(x,y,z)=0$. Thus since each curve $\gamma\in\mathcal{L}$ is
tangent to at least $(C/2)n^{1/2}$ other curves from $\mathcal{L}$, and each
of these tangencies occurs at a distinct point of $\gamma$, we have that
$\partial_{z}P$ vanishes at $\geq(C/2)n^{1/2}$ points on each curve
$\beta\in\mathcal{C}$. If $C=O(1)$ is sufficiently large, then by Bézout’s
theorem we have that $\partial_{z}P$ vanishes on each curve from
$\mathcal{C}$. But since $P$ was the non-zero polynomial of minimal degree
that vanishes on all of the curves in $\mathcal{C}$, we conclude that
$\partial_{z}P=0$, i.e. $P(x,y,z)=Q(x,y)$ for some bivariate polynomial
$Q(x,y)$ of degree $O(n^{1/2})$. However, this immediately implies that each
of the $n$ circles in $\mathcal{L}$ must be contained in $Z(Q)$. This is
impossible, since $Q$ has degree $O(n^{1/2})$, while the algebraic curve
$\bigcup_{\gamma\in\Gamma}\gamma$ has degree $2n$. We conclude that
$\mathcal{L}$ determines fewer than $Cn^{3/2}$ tangencies.

When proving Theorem 1, we must overcome two difficulties that are absent in
the above proof sketch. First, since we are working over an arbitrary field,
we must make sense of what it means to parameterize $\gamma$ in a neighborhood
of $t$. Second, it need not be the case that
$\frac{d^{2}f_{1}}{dt^{2}}(0)\neq\frac{d^{2}f_{2}}{dt^{2}}(0)$; if the curves
are not circles, then it is possible for two curves to be tangent to order
higher than two. However, since the curves are algebraic of degree $\leq D$,
the derivatives of $f_{1}$ and $f_{2}$ are 0 must eventually differ. Sections
2–4 will develop the tools and techniques needed to make these ideas precise,
and then in Section 5 we will prove Theorem 1. Finally, in Section 6 we will
prove Theorem 2.

##  2 Algebraic geometry preliminaries

We will use several terms and tools from commutative algebra and algebraic
geometry. The results discussed in this section can be found in standard texts
on commutative algebra, such as [6]. In this section, all fields are
algebraically closed (in Section 3, we will explain why this assumption is
harmless), and all rings $R$ are commutative and contain a multiplicative
unit.

###  2.1 Power series and Bézout’s theorem

Define $k[x_{1},x_{2},\ldots,x_{d}]$ to be the ring of polynomials in $d$
variables that have coefficients in $k$ and define
$k[[x_{1},x_{2},\ldots,x_{d}]]$ to be the ring of formal power series in $d$
variables that have coefficients in $k$. Let $R$ be a commutative ring and let
$I\subset R$ be an ideal. We define the quotient $R/I$ to be the equivalence
class of objects $\\{\bar{a}\colon a\in R\\}$, where $\bar{a}=\bar{b}$ if
$a=b+r$ for some $r\in I$.

Note that there is a natural injection $k[x_{1},\ldots,x_{d}]\to
k[[x_{1},\ldots,x_{d}]]$. Let $I\subset k[x_{1},\ldots,x_{d}]$ be an ideal.
Abusing notation, we will identify $I$ with the ideal generated by its image
in $k[[x_{1},\ldots,x_{d}]].$ Now $k[[x_{1},\ldots,x_{d}]]/I$ is a $k$-vector
space; we denote its dimension, which may be finite or $\infty$, by
$\operatorname{dim}_{k}k[[x_{1},\ldots,x_{d}]]/I$.

######  Theorem 3 (Bézout).

Let $P,P^{\prime}\in k[x,y]$ be polynomials and let $(P,P^{\prime})\subset
k[x,y]$ be the ideal generated by $P$ and $P^{\prime}$. If
$\operatorname{dim}_{k}k[[x,y]]/(P,P^{\prime})>(\deg P)(\deg P^{\prime}),$
then $P$ and $P^{\prime}$ share a common factor.

Geometrically, one should think of the quantity
$\operatorname{dim}_{k}k[[x,y]]/(P,P^{\prime})$ as the multiplicity of the
intersection between the vanishing loci $\mathbf{Z}(P)$ and
$\mathbf{Z}(P^{\prime})$ at the point $(0,0)$; for instance, a “typical”
intersection between two curves will have multiplicity one, an intersection
point where two curves are tangent has multiplicity $2$, and an intersection
involving a higher degree of tangency will have still higher multiplicity.
Bézout’s theorem says that the total multiplicity of the intersection, summed
over all intersection points, between $\mathbf{Z}(P)$ and
$\mathbf{Z}(P^{\prime})$ is $(\deg P)(\deg P^{\prime})$, as long as $P$ and
$P^{\prime}$ have no common factor; this provides the claimed upper bound for
intersection multiplicity at the point $(0,0)$.

We will also use a second, less technical variant of Bézout’s theorem, which
states that if two varieties of degrees $D$ and $D^{\prime}$ intersect
properly (i.e. if the codimension of the intersection is the sum of the
codimensions of the varieties), then intersection is a variety of degree at
most $DD^{\prime}$.

###  2.2 Derivatives and tangent spaces

Let $P\in k[x_{1},\ldots,x_{d}]$ be a polynomial. We denote the (formal)
derivative of $P$ in the $x_{j}$ variable by $\partial_{x_{j}}P$. If $P/Q$ is
a rational function, we define the (formal) derivative $\partial_{x_{j}}(P/Q)$
using the Leibniz rule
$\partial_{x_{j}}(P/Q)=((\partial_{x_{j}}P)Q-P\partial_{x_{j}}Q)/Q^{2}$. If
$P\in k[[x_{1},\ldots,x_{d}]]$, then $\partial_{x_{j}}P$ will denote the
formal derivative of $P$. Note that we will never use Newton’s symbol $\prime$
to denote derivatives. By $\nabla P$ we mean the vector
$(\partial_{x_{1}}P,\ldots,\partial_{x_{d}}P)$.

Let $Z\subset k^{d}$ be a variety and let $p\in Z$. Let $\mathbf{I}(Z)$ be the
ideal of polynomials that vanish on $Z$. We define the the Zariski cotangent
space of $Z$ at $p$ to be the span of the vectors $\\{\nabla f(p)\colon
f\in\mathbf{I}(Z)\\}$. If $\mathbf{I}(Z)=(f_{1},\ldots,f_{\ell})$, then this
will be equal to the span of $\nabla f_{1}(p),\ldots\nabla f_{\ell}(p)$. The
Zariski tangent space $T_{p}(Z)$ is the dual of the Zariski cotangent space,
i.e.

$T_{p}(Z)=\Big{\\{}v\in
k^{d}\colon\sum_{i=1}^{d}\partial_{d_{x_{i}}}f(p)v_{i}=0\ \textrm{for all}\
f\in I(Z)\Big{\\}}.$

In particular, note that the Zariski tangent space is a vector space, i.e. if
$v_{1}$ and $v_{2}$ are vectors in $T_{p}(Z),$ then so is
$a_{1}v_{1}+a_{2}v_{2}$ for any $a_{1},a_{2}\in k$.

We can also compute the Zariski tangent space by working in the power series
ring. Suppose $p=0\in k^{d},$ and let $I\subset k[[x_{1},\ldots,x_{d}]]$ be
the image of $\mathbf{I}(Z)$ in $k[[x_{1},\ldots,x_{d}]]$. Then the Zariski
cotangent space $Z$ at $p=(0,0)$ is the span of the vectors $\\{\nabla
f(p)\colon f\in I\\}$, and the Zariski tangent space of $Z$ at $p=(0,0)$ is
given by

$\Big{\\{}v\in k^{d}\colon\sum_{i=1}^{d}\partial_{d_{x_{i}}}f(p)v_{i}=0\
\textrm{for all}\ f\in I\Big{\\}}.$

If $p\neq 0$, then we can apply a translation sending $p$ to the origin and
the above statement holds.

###  2.3 Plane curves

Let $P\in k[x,y]$ be a polynomial, and let $\zeta=\mathbf{Z}(P)=\\{(x,y)\in
k^{2}\colon P(x,y)=0\\}$. Sets of this form will be called algebraic curves.
If $P$ is irreducible, we say the curve $\zeta$ is irreducible as well. Let
$\zeta=\mathbf{Z}(P)$ be an irreducible curve. We say a point $(x,y)\in\zeta$
is singular if $\partial_{x}P(x,y)=\partial_{y}P(x,y)=0$ (or, in other words,
if the Zariski tangent space to $\zeta$ is two-dimensional.) If
$(x,y)\in\zeta$ is not a singular point, then we say it is a smooth point. In
this case, the tangent space to $\zeta$ at $(x,y)$ is one-dimensional, and
specifies the unique tangent direction there.

Let $\zeta$ be an irreducible curve. We say a set $X\subset\zeta$ is _Zariski
dense_ in $\zeta$ if $\zeta\backslash X$ is finite. We will sometimes make
statements like “a generic point on the irreducible curve $\zeta$ has the
following properties….” What this means is that all but a finite set of points
$\mathcal{Q}\subset\zeta$ have the properties in question; the set
$\mathcal{Q}$ depends only on $\zeta$ and the properties under consideration.
For example, a generic point on $\zeta\subset k^{2}$ is smooth. This is
because there is a finite set $\mathcal{Q}\subset\zeta$ of singular points,
and every point in $\zeta\backslash\mathcal{Q}$ is smooth.

###  2.4 Coordinates

We will mainly work in $k^{2}$ and $k^{3}$. We will use coordinates $(x,y)$ to
represent points in $k^{2}$, and $(x,y,z)$ to represent points in $k^{3}$.
Unless otherwise noted, $\pi$ will be the projection $\pi(x,y,z)=(x,y)$.
Sometimes we will restrict the domain of $\pi$ to a space curve in $k^{3}$, so
we will refer to a projection $\pi\colon\gamma\to\zeta$, and we will say that
“a generic fiber of this projection has the following properties….” This means
that the fiber of $\pi$ above a generic point of $\zeta$ has the properties in
question. In particular, the specified property holds for all but finitely
many fibers.

For simplicity, we will often work in coordinates. Thus it may appear that the
coordinate axes play a distinguished role. However, all of the quantities we
study are invariant under invertible affine transformations. Thus we will
sometimes apply a suitable transformation to our curve arrangement to ensure
that no coincidences involving the coordinate axes occur. In particular, we
will refer to a “generic linear transformation” in $k^{d}$ (generally $d=2$ or
$3$). What this means is that there is a Zariski-open subset
$X\subset\operatorname{GL}_{d}(k)$ that depends only on the curves from the
statement of Theorem 1 or 2, and we can select any linear transformation from
this set.

##  3 Plane and space curves

In this section we will establish notation and prove some results that are
useful for both Theorems 1 and 2. Later, the two proofs will diverge, and we
will handle the two theorems in separate sections. First, note that in
Theorems 1 and 2, we can assume without loss of generality that $k$ is
algebraically closed. Indeed, if $k$ is not closed then we can replace $k$ by
its algebraic closure.

###  3.1 Notation for plane and space curves

Let $k[x,y]_{\leq D}$ be the vector space of bivariate polynomials of degree
at most $D$; we can identify this vector space with $k^{\binom{D+2}{2}}$. For
each $P\in k[x,y]_{\leq D}$, let $\alpha_{P}\in k^{\binom{D+2}{2}}$ be the
corresponding point, and for each $\alpha\in k^{\binom{D+2}{2}}$, let
$P_{\alpha}$ be the corresponding polynomial. We say that $P\in k[x,y]_{\leq
D}$ is $x$– _monic_ if the coefficient of $x^{\ell}$ is 1, where
$\ell=\deg(P)$. Given a finite set of irreducible plane curves of degree $\leq
D<\operatorname{char}(k)$, we can always find an invertible affine
transformation $k^{2}\to k^{2}$ so that after applying the transformation to
the curves, each curve can be uniquely written as $\mathbf{Z}(P)$, where $P\in
k[x,y]_{\leq D}$ is $x$–monic and irreducible. If $\zeta\subset k^{2}$ is an
irreducible curve of degree $\leq D$ that can be written as the zero-set of an
irreducible $x$–monic polynomial, then define $P[\zeta]$ to be the
corresponding irreducible $x$–monic polynomial, and define
$\alpha_{\zeta}=\alpha_{P[\zeta]}$.

At several points we will construct space curves that capture relevant
properties from the plane curve arrangement. An important notion for an
arrangement of space curves is that of a _two-rich point_ , which we define
below.

######  Definition 3 (Two-rich point).

Let $\Gamma$ be a set of irreducible algebraic space curves. We say a point in
$k^{3}$ is _two-rich_ (with respect to $\Gamma$) if there are two distinct
curves from $\Gamma$ that contain the point.

###  3.2 Implicit differentiation from the formal viewpoint

Let $k$ be a field and let $P$ be an irreducible polynomial in $k[x,y]$ of
degree $D$. Let $\zeta=\mathbf{Z}(P)$ be the vanishing locus of $P$, so that
$\zeta$ is an algebraic curve in $k^{2}$. Let $(x_{0},y_{0})$ be a smooth
point of $\zeta$ where $\partial_{y}P\neq 0$, i.e. the tangent vector is not
vertical. Without loss of generality (for instance, by applying a translation)
we may assume $(x_{0},y_{0})=(0,0)$. However, we may sometimes refer to the
point $(x_{0},y_{0})$ in order to clarify the role played by this point.

Note first that $P$ can be written as

$P_{1}(x,y)+P_{2}(x,y)+\ldots+P_{D}(x,y)$

where $P_{i}$ is a homogeneous polynomial of degree $i$. We know there is no
constant term $P_{0}$ because $P$ vanishes at $(0,0)$. The fact that $\zeta$
is smooth at $(0,0)$ is equivalent to the nonvanishing of $P_{1}=ax+by$, and
the fact that the tangent vector is not vertical implies that $b\neq 0$. Thus
we can think of $P(x,y)$ as a polynomial $Q(y)$ with coefficients in $k[x]$,
such that $Q(0)\in k[x]$ vanishes modulo $x$, and $\partial_{y}Q(0)\in k[x]$
reduces to $b\neq 0$ mod $x$. We have already hypothesized that $Q$ is
irreducible in $k[x,y]$. However, $Q$ admits a factorization in the ring
$k[[x]][y]$. More precisely: by Hensel’s lemma (see e.g. [6, Thm 7.3]), there
is a unique power series $\phi_{(P;x_{0},y_{0})}(x)\in xk[[x]]$ such that
$y=\phi_{(P;x_{0},y_{0})}(x)$ is a root of $Q$.

######  Example 1.

If $P=x-2y-y^{2}=(1+x)-(1+y)^{2}$, then

$\phi_{(P;x_{0},y_{0})}(x)=\sqrt{1+x}-1=\frac{1}{2}x-\frac{1}{8}x^{2}+\frac{1}{16}x^{3}-\frac{5}{128}x^{4}+\ldots$

The other root of $Q$ is $y=2-\phi_{(P;x_{0},y_{0})}(x)$, which is a unit in
$k[[x]]$. We note that the denominators in the power series forbid us from
working in characteristic $2$; this is as it should be, since when
$\operatorname{char}k=2$ we have $P=x-y^{2}$, which has vertical slope at
$(0,0)$.

In other words, we can write

$P(x,y)=(y-\phi_{(P;x_{0},y_{0})}(x))R(x,y),$ (3)

where $R(x,y)$ is a power series in $k[[x,y]]$ with nonzero constant term.

We now define a sequence of rational functions $f_{1},f_{2},\ldots\in k(x,y)$
as follows. We first take

$f_{1}=(\partial_{x}P)/(\partial_{y}P).$

Write $\Delta$ for the linear differential operator defined by

$\Delta(f)=\partial_{x}f-f_{1}\partial_{y}f,$

and for all $i>1$, define

$f_{i}=\Delta f_{i-1}.$

In geometric terms, we may think of $f_{i}$ as the derivative of $f_{i-1}$
along the curve $\zeta$. Each $f_{i}$ is a rational function in $x$ and $y$
whose degree is bounded in terms of $i$ and $D$.

Define

$\hat{\mathcal{O}}=k[[x,y]]/P(x,y)=k[[x,y]]/(y-\phi_{(P;x_{0},y_{0})}(x)),$

where the second equality follows from the fact that $R(x,y)$ is a unit in
$k[[x,y]]$. $\hat{\mathcal{O}}$ is called the completed local ring of $\zeta$
at the point $(0,0)$. If $k=\mathbb{C}$, we may think of $\hat{\mathcal{O}}$
as the ring of germs of holomorphic functions at $(0,0)$. Note, though, that
we place no convergence condition on our power series; a series like $\sum
n!x^{n}y^{n}$ is a perfectly good element of $\mathbb{C}[[x,y]]$, though there
is no complex disc around $0$ in which this power series converges.

By the assumption that $\partial_{y}P$ doesn’t vanish at $(0,0)$, we can write
$f_{1}$ as an element of $k[[x,y]]$, and then project it to an element $g_{1}$
of the quotient $\hat{\mathcal{O}}$. We note that if $f=Ph$, for some function
$h\in k[[x,y]]$, then

$\Delta(f)=\partial_{x}(Ph)-f_{1}\partial_{y}(Ph)=(\partial_{x}P)h+P\partial_{x}h-\frac{\partial_{x}P}{\partial_{y}P}((\partial_{y}P)h+P\partial_{y}h)=P(\partial_{x}h-\frac{\partial_{x}P}{\partial_{y}P}\partial_{y}h)=P\Delta(h).$

In particular, this means that $\Delta$ preserves the principal ideal
$Pk[[x,y]]$, and so it descends to a well-defined linear operator (which we
continue to denote by $\Delta$) from $\hat{\mathcal{O}}$ to
$\hat{\mathcal{O}}$. In particular, if we define $g_{i}=\Delta g_{i-1}$ for
all $i>1$, then $g_{i}$ is the projection of $f_{i}$ to $\hat{\mathcal{O}}$.
For example, if $k=\mathbb{C}$ then we may think of $g_{i}$ as the germ of the
rational function $f_{i}$ in an infinitesimal neighborhood of $(0,0)$ on the
curve $\zeta$.

Recalling the factorization $P=(y-\phi_{(P;x_{0},y_{0})}(x))R$ in $k[[x,y]]$,
we have

$\partial_{y}P=(y-\phi_{(P;x_{0},y_{0})}(x))\partial_{y}R+R$

and

$\partial_{x}P=(y-\phi_{(P;x_{0},y_{0})}(x))\partial_{x}R-\partial_{x}\phi_{(P;x_{0},y_{0})}(x)R.$

Projection to $\hat{\mathcal{O}}$ sends $y-\phi_{(P;x_{0},y_{0})}(x)$ to $0$,
so

$g_{1}(x)=-\partial_{x}\phi_{(P;x_{0},y_{0})}(x).$

Note that $g_{1}$ depends on the polynomial $P$, the “base point”
$(x_{0},y_{0})$ and the value of $x$ at which it is evaluated. In practice, we
will always evaluate $g_{1}$ at $x=x_{0}$, but for now it will be helpful to
distinguish between $x$ and $x_{0}$. Now

$g_{2}(x)=\Delta
g_{1}(x)=\partial_{x}g_{1}(x)-f_{1}(x,y)\partial_{y}g_{1}(x)=-\partial_{x}^{2}\phi_{(P;x_{0},y_{0})}(x)$

and similarly,

$g_{i}(x)=-\partial_{x}^{i}\phi_{(P;x_{0},y_{0})}(x)$

for all positive integers $i$. In particular, setting $x=x_{0}$, we have

$g_{i}(x_{0})=-\partial_{x}^{i}\phi_{(P;x_{0},y_{0})}(x_{0}).$ (4)

To make the role of $P$ explicit, we will sometimes write $f_{i;P}$ for the
rational functions constructed above. When $P=P_{\alpha}$, we will write this
as $f_{i;P_{\alpha}}(x,y)$. If we want to refer to numerator and denominator
separately, we will call them $F_{i,\alpha}(x,y)$ and $G_{i,\alpha}(x,y)$, so

$f_{i;P_{\alpha}}(x,y)=\frac{F_{i,\alpha}(x,y)}{G_{i,\alpha}(x,y)},$ (5)

where the fraction is understood to be in lowest terms.

###  3.3 Space curves modeling plane curve tangency

In this section, for each pane curve $\zeta$ we will construct several space
curves that capture some of the relevant tangency information of $\zeta$.

Let $\alpha\in k^{\binom{D+2}{2}}$. Define

$\beta_{1}(\alpha)=\big{\\{}(x,y,z)\colon P_{\alpha}(x,y)=0,\
z\partial_{y}P_{\alpha}(x,y)-\partial_{x}P_{\alpha}(x,y)=0\big{\\}}.$ (6)

Then $\beta_{1}(\alpha)$ is an algebraic curve in $k^{3}$. If
$(x,y,z)\in\beta_{1}(\alpha)$ and if $\partial_{y}P_{\alpha}(x,y)\neq 0$, then
$z$ is the slope of the curve $\mathbf{Z}(P_{\alpha})$ at the point $(x,y)$.
Furthermore, for each $(x,y)\in\mathbf{Z}(P_{\alpha})$ for which
$\partial_{y}P_{\alpha}(x,y)\neq 0$, there is a unique $z\in k$ so that
$(x,y,z)\in\beta_{1}(\alpha)$. In particular, if $P_{\alpha}$ is irreducible
and if $\partial_{y}P_{\alpha}$ does not vanish identically on
$\mathbf{Z}(P_{\alpha})$, then the projection
$\pi\colon\beta_{1}(\alpha)\to\mathbf{Z}(P_{\alpha})$ is an isomorphism away
from the (finite) set of points on $\mathbf{Z}(P_{\alpha})$ where
$\partial_{y}P_{\alpha}$ vanishes. If $p=(x,y)$ is a point of
$\mathbf{Z}(P_{\alpha})$ where $\partial_{y}P_{\alpha}$ vanishes, there are
two situations to consider. If $\partial_{x}P_{\alpha}$ doesn’t vanish at $p$,
then the fiber of $\beta_{1}(\alpha)$ over $p$ is empty. If
$\partial_{x}P_{\alpha}(p)=0$, then $(x,y,z)$ lies in $\beta_{1}(\alpha)$ for
all $z$; in other words, the fiber is a vertical line. We conclude that
$\beta_{1}(\alpha)$ is the union of an irreducible curve
$\beta_{1}^{*}(\alpha)$ whose projection to the $xy$-plane is an open subset
of $\mathbf{Z}(P_{\alpha})$, and a finite set of vertical lines.

Note that $\beta_{1}^{*}(\alpha)$ is a component of the intersection of two
surfaces of degree $\leq D$, and thus has degree at most $D^{2}$.

More generally, for each $s$ we wish to define an algebraic space curve
$\beta_{s}^{*}(\alpha)$ so that for most points $(x,y,z)\in\beta_{s}(\alpha)$,
$z$ corresponds to the “$(s-1)$–st derivative” of the slope of
$\mathbf{Z}(P_{\alpha})$ at $(x,y)$. Let us now make this precise. Define

$\beta_{s}(\alpha)=\big{\\{}(x,y,z)\colon P_{\alpha}(x,y)=0,\
zG_{s,\alpha}(x,y)-F_{s,\alpha}(x,y)=0\big{\\}},$ (7)

Where $G_{s,\alpha}(x,y)$ and $F_{s,\alpha}(x,y)$ are defined in (5). Note
that the definition of $\beta_{1}(\alpha)$ from (6) agrees with the definition
from (7).

As before, if $P_{\alpha}$ is irreducible and if $\partial_{y}P_{\alpha}$ does
not vanish identically on $\mathbf{Z}(P_{\alpha})$, we can decompose
$\beta_{s}(\alpha)$ into the union of a finite set of vertical lines plus a
unique irreducible component $\beta_{s}^{*}(\alpha)$ that is not a vertical
line. This irreducible component will be of degree at most $O_{D,s}(1)$.
Finally, if $(x,y,z)\in\beta_{s}(\alpha)$ and if
$G_{s,\alpha}(x_{0},y_{0})\neq 0$, then
$z=F_{s,\alpha}(x_{0},y_{0})/G_{s,\alpha}(x_{0},y_{0})$.

######  Lemma 1.

Let $\alpha\in k^{\binom{D+2}{2}}$. Let
$(x_{0},y_{0},z_{0})\in\beta_{s}(\alpha)$, and let
$\phi=\phi_{(\alpha;x_{0},y_{0})}$ be the function from (3) associated to
$P_{\alpha}$ at the point $(x_{0},y_{0})$. Then

$\Big{(}1,\ \partial_{x}\phi(x_{0}),\ \partial_{x}^{s+1}\phi(x_{0})\Big{)}\in
T_{(x_{0},y_{0},z_{0})}\beta_{s}(\alpha).$ (8)

###### Proof.

After a translation, we can assume that $(x_{0},y_{0})=(0,0)$. Let
$\mathbf{I}(\beta_{s}(\alpha))$ be the ideal of polynomials in $k[x,y,z]$ that
vanish on $\beta_{s}(\alpha)$. Note that

$\mathbf{I}(\beta_{s}(\alpha))=\big{(}P_{\alpha}(x,y),\
zG_{s,\alpha}(x,y)-F_{s,\alpha}(x,y)\big{)}.$

In particular, the image of $\mathbf{I}(\beta_{s}(\alpha))$ in $k[[x,y,z]]$ is
$\Big{(}y-\phi(x),\ z-\partial_{x}^{s}\phi(x)\Big{)}.$ Thus, the Zariski
cotangent space of $\beta_{s}(\alpha)$ at $(0,0,0)$ is spanned by the vectors
$(-\partial_{x}\phi(0),1,0)$ and $(-\partial_{x}^{s+1}\phi(0),0,1)$ (these
vectors will be linearly independent if and only if $(0,0,0)$ is a smooth
point of $\beta_{s}(\alpha)$, but this fact is not relevant at the moment).
Thus $(1,\partial_{x}\phi(0),\partial_{x}^{s+1}\phi(0))$ lies in the Zariski
tangent space of $\beta_{s}(\alpha)$ at $(0,0,0).$ Undoing the translation to
the origin, we obtain (8).

∎

###  3.4 Space curves modeling plane curve orthogonality

Let $\alpha\in k^{\binom{D+2}{2}}$. Define

$\tilde{\beta}_{1}(\alpha)=\\{(x,y,z)\colon P_{\alpha}(x,y)=0,\
z\partial_{x}P_{\alpha}(x,y)+\partial_{y}P_{\alpha}(x,y)=0\\}.$ (9)

######  Remark 1.

Compare the definition of $\tilde{\beta}_{1}(\alpha)$ with the definition of
$\beta_{1}(\alpha)$ from (6)—if the following conditions are satisfied:

  * •

$\alpha\in k^{\binom{D+2}{2}}$,

  * •

$\operatorname{char}(k)=0$ or $D<\operatorname{char}(k)$,

  * •

$(x,y,z)\in\beta_{1}^{*}(\alpha)$,

  * •

$(x,y,\tilde{z})\in\tilde{\beta}_{1}^{*}(\alpha)$,

  * •

$\partial_{y}P_{\alpha}(x,y)\neq 0$,

  * •

$\partial_{x}P_{\alpha}(x,y)\neq 0$,

then $\tilde{z}=-1/z$.

Again, if $P_{\alpha}$ is irreducible and if $\partial_{x}P_{\alpha}$ does not
vanish identically on $\mathbf{Z}(P_{\alpha})$, then the projection
$\pi\colon\tilde{\beta}_{1}(\alpha)\to\mathbf{Z}(P_{\alpha})$ is an
isomorphism away from the (finite) set of points on $\mathbf{Z}(P_{\alpha})$
where $\partial_{x}P_{\alpha}$ vanishes. Again, $\tilde{\beta}_{1}(\alpha)$ is
the union of an irreducible curve $\tilde{\beta}_{1}^{*}(\alpha)$ whose
projection to the $xy$-plane is an open subset of $\mathbf{Z}(P_{\alpha})$,
and a finite set of vertical lines.

The key virtue of $\tilde{\beta}^{*}$ is that it translates curve-curve
orthogonality in the plane to curve-curve intersection in $k^{3}$. More
precisely, we have the following lemma, which follows from Remark 1:

######  Lemma 2.

Suppose $\operatorname{char}(k)=0$ or $D\leq\operatorname{char}(k)$. Let $P$
and $P^{\prime}$ be irreducible polynomials of degree $\leq D$ with
$\partial_{y}P$ (resp. $\partial_{x}P^{\prime}$) not vanishing identically on
$\mathbf{Z}(P)$ (resp. $\mathbf{Z}(P^{\prime})$), and let $(x,y)$ be a point
in $\mathbf{Z}(P)\cap\mathbf{Z}(P^{\prime})$ for which $\partial_{y}P\neq 0$
and $\partial_{x}P^{\prime}\neq 0.$ Then $\mathbf{Z}(P)$ and
$\mathbf{Z}(P^{\prime})$ intersect orthogonally at $(x,y)$ if and only if

$(x,y)\in\pi\big{(}\beta_{1}^{*}(\alpha_{P})\cap\tilde{\beta}_{1}^{*}(\alpha_{P^{\prime}})\big{)}.$
(10)

##  4 Plane curve tangency and space curve intersection

In this section we will prove several lemmas that relate tangencies of plane
curves with the intersections of the corresponding space curves. These results
will be helpful in proving Theorem 1.

######  Lemma 3.

Let $s\geq 1$. Let $\alpha,\alpha^{\prime}\in k^{\binom{D+2}{2}}$, with
$P_{\alpha}$ and $P_{\alpha^{\prime}}$ irreducible. Let
$(x_{0},y_{0},z_{0})\in\beta_{s}(\alpha)\cap\beta_{s}(\alpha^{\prime})$ and
suppose

$\begin{split}f_{1,P_{\alpha}}(x_{0},y_{0})&=f_{1,P_{\alpha}^{\prime}}(x_{0},y_{0}),\\\
f_{s+1,P_{\alpha}}(x_{0},y_{0})&\neq
f_{s+1,P_{\alpha}^{\prime}}(x_{0},y_{0}),\end{split}$ (11)

and the denominators of the above rational functions are not $0$. Then

$(0,0,1)\in\operatorname{span}\big{\\{}T_{(x_{0},y_{0},z_{0})}\beta_{s}(\alpha),\
T_{(x_{0},y_{0},z_{0})}\beta_{s}(\alpha^{\prime})\big{\\}}.$ (12)

###### Proof.

Define

$\begin{split}&m=f_{1,P_{\alpha}}(x_{0},y_{0})=f_{1,P_{\alpha}^{\prime}}(x_{0},y_{0}),\\\
&m_{s+1}=f_{s+1,P_{\alpha}}(x_{0},y_{0}),\\\
&m^{\prime}_{s+1}=f_{s+1,P_{\alpha}^{\prime}}(x_{0},y_{0}).\end{split}$

By (11), $m_{s+1}\neq m^{\prime}_{s+1}$. By Lemma 1,

$\begin{split}\operatorname{span}\big{\\{}T_{(x_{0},y_{0},z_{0})}\beta_{s}(\alpha),\
T_{(x_{0},y_{0},z_{0})}\beta_{s}(\alpha^{\prime})\big{\\}}&\supset\operatorname{span}\\{(1,m,m_{s+1}),(1,m,m^{\prime}_{s+1})\big{\\}}\\\
&=\operatorname{span}\\{(1,m,m_{s+1}),(0,0,m_{s+1}-m^{\prime}_{s+1})\\}\\\
&=\operatorname{span}\\{(1,m,m_{s+1}),(0,0,1)\\}.\qed\end{split}$

The following variant of Bézout’s theorem asserts that two disjoint
irreducible curves cannot be too tangent.

######  Lemma 4.

Let $P,P^{\prime}\in k[x,y]$ be polynomials of degree $\leq D$. Suppose that
either $\operatorname{char}(k)=0$ or $D^{2}+1<\operatorname{char}(k)$ and that
$P$ and $P^{\prime}$ are irreducible. Let
$(x_{0},y_{0})\in\mathbf{Z}(P)\cap\mathbf{Z}(P^{\prime})$. Suppose $P$ and
$P^{\prime}$ are smooth at $(x_{0},y_{0})$ and have non-vertical tangent.
Suppose furthermore that

$f_{i,P}(x_{0},y_{0})=f_{i,P^{\prime}}(x_{0},y_{0}),\qquad\quad
i=1,2,\ldots,D^{2}+1.$

Then $P^{\prime}=cP$ for some constant $c\in k$.

###### Proof.

Let $\phi_{P;x_{0},y_{0}}(x),\phi_{P^{\prime};x_{0},y_{0}}(x)\in xk[[x]]$ be
the power series defined in (3). Write

$\phi_{P,x_{0},y_{0}}(x)=a_{1}x+a_{2}x^{2}+\ldots,$

and

$\phi_{P^{\prime},x_{0},y_{0}}(x)=a^{\prime}_{1}x+a^{\prime}_{2}x^{2}+\ldots.$

Recall that from (4) and the surrounding discussion, we have that in the
complete local ring $\hat{\mathcal{O}}=k[[x,y]]/(P)$, the rational function
$f_{i,P}(x,y)$ is equal to $-\partial_{x}^{i}\phi_{P,x_{0},y_{0}}(x)$. Thus

$f_{i,P}(0,0)=-i!a_{i}.$

Similarly, $f_{i,P^{\prime}}(0,0)=-i!a_{i}$. By the hypothesis on
$\operatorname{char}k$, we know $i!\neq 0$ for $i\leq D^{2}+1$, so
$a_{i}=a^{\prime}_{i}$ for $i=1,2,\ldots,D^{2}+1$. In other words,
$\phi_{P,x_{0},y_{0}}(x)-\phi_{P^{\prime},x_{0},y_{0}}(x)$ is divisible by
$x^{D^{2}+1}$.

If $P$ and $P^{\prime}$ are irreducible and are not multiples of each other,
then by Bézout’s theorem (Theorem 3),

$\operatorname{dim}_{k}k[[x,y]]/(P,P^{\prime})\leq D^{2}.$

Now $P=(y-\phi_{P,x_{0},y_{0}}(x))R$ and
$P^{\prime}=(y-\phi_{P^{\prime},x_{0},y_{0}}(x))R^{\prime}$ where
$R,R^{\prime}$ are units in $k[[x,y]]$. So

$k[[x,y]]/(P,P^{\prime})=k[[x,y]](y-\phi_{P,x_{0},y_{0}}(x),y-\phi_{P^{\prime},x_{0},y_{0}}(x))=k[[x]]/(\phi_{P,x_{0},y_{0}}(x)-\phi_{P^{\prime},x_{0},y_{0}}(x)).$

We have shown above that
$\phi_{P,x_{0},y_{0}}(x)-\phi_{P^{\prime},x_{0},y_{0}}(x)$ lies in
$x^{D^{2}+1}k[[x]]$. We conclude that

$\operatorname{dim}_{k}k[[x,y]]/(P,P^{\prime})\geq D^{2}+1,$

which is a contradiction. Thus $P^{\prime}=cP$, as claimed. ∎

######  Remark 2.

We note that the condition on the characteristic of $k$ is not merely an
artifact of the method; not only Lemma 4 but the main theorem would fail to
hold without some such hypothesis. For instance, consider a family of $n$
curves of the form

$y=a_{i}x^{2}+b_{i}$

over $\bar{\mathbb{F}}_{2}$. These smooth, irreducible curves have horizontal
tangent at every point; in particular, each of the $n^{2}$ points of
intersection between these curves is a point of tangency.

##  5 Proof of Theorem 1

Let $\mathcal{L}$ be the set of curves from the statement of Theorem 1.
Suppose that

$\sum_{(p,\ell)\in\mathcal{T}(\mathcal{L})}\operatorname{mult}(p,\ell;\mathcal{L})>C_{D}n^{3/2}.$

If $C_{D}$ is sufficiently large, we will obtain a contradiction. After
applying a linear transformation to $k^{2}$, we can associate an irreducible
polynomial $P=P[\zeta]$ to each curve $\zeta\in\mathcal{L}$, as described in
Section 3.1, and since $\operatorname{char}(k)>D$, we can ensure that for each
curve $\zeta\in\mathcal{L}$, $\partial_{y}P[\zeta]$ does not vanish
identically on $\mathbf{Z}(P[\zeta])$.

First, note that for each $\zeta\in\mathcal{L}$ and each index $j\geq 1$,

$|\mathbf{Z}(P[\zeta])\cap\mathbf{Z}(G_{j,\alpha_{\zeta}})|=O_{j,D}(1),$ (13)

i.e. there are few points on $\zeta$ where the denominator of
$F_{j,\alpha_{P}}/G_{j,\alpha_{P}}$ from (5) vanishes. We say that $p\in\zeta$
is a _good_ point of $\zeta$ if $p$ is not in the set (13) for any
$j=1,\ldots,D^{2}$. Otherwise, $p$ is a _bad_ point of $\zeta$. Note that

$\sum_{\zeta\in\mathcal{L}}|\\{p\in\zeta\colon\ p\ \textrm{is a bad point of}\
\zeta\\}|=O_{D}(n).$ (14)

Define $\operatorname{mult}_{g}(p,\ell;\mathcal{L})$ to be the number of
curves from $\mathcal{L}$ that are smooth at $p$, tangent to $\ell$ at $p$,
and for which $p$ is a good point of the curve. Define
$\mathcal{T}^{\prime}\subset\mathcal{T}(\mathcal{L})$ to be the set of
directed points of tangency for which
$\operatorname{mult}_{g}(p,\ell;\mathcal{L})$ is larger than one.

By Lemma 4 and pigeonholing, we can find an index $1\leq j\leq D^{2}$ and a
set $\mathcal{T}^{\prime\prime}\subset\mathcal{T}^{\prime}$ of size
$|\mathcal{T}^{\prime\prime}|\geq(2D)^{-2}|\mathcal{T}^{\prime}|$ so that for
each $(p,\ell)\in\mathcal{T}^{\prime\prime}$, there exist curves $\zeta,\
\zeta^{\prime}$ in $\mathcal{C}$ so that $p$ is a good point for $\zeta$ and
$\zeta^{\prime}$, and (11) holds for this value of $j$ with
$\alpha=\alpha_{\zeta}$ and $\alpha^{\prime}=\alpha_{\zeta^{\prime}}$.

Define

$\Gamma=\\{\beta_{j}^{*}(\alpha_{\zeta})\colon\zeta\in\mathcal{L}\\}.$

If $\gamma\in\Gamma$, we define $\zeta(\gamma)$ to be the unique element of
$\mathcal{L}$ satisfying $\gamma=\beta_{j}^{*}(\alpha_{\zeta})$. If $p\in
k^{2}$ is a good point of $\zeta,\zeta^{\prime}\in\mathcal{L}$, and $(p,\ell)$
is a directed point of tangency for $\zeta$ and $\zeta^{\prime}$, then the
corresponding space curves $\beta_{j}^{*}(\alpha_{\zeta})$ and
$\beta_{j}^{*}(\alpha_{\zeta^{\prime}})$ intersect at the point $(x,y,z)$,
where $p=(x,y)$ and $z$ is the slope of $\ell$. We will abuse notation
slightly and identify directed points of tangency with points in $k^{3}$.

Consider the graph whose vertices are the curves from $\Gamma$, and two
vertices $\gamma,\gamma^{\prime}$ are adjacent if there is a directed point of
tangency $(p,\ell)\in\mathcal{T}^{\prime\prime}$ so that $p$ is a good point
for the curves $\zeta(\gamma)$ and $\zeta(\gamma^{\prime})$, and (11) holds
for $\zeta(\gamma)$ and $\zeta(\gamma^{\prime})$ with the value of $j$
specified above. This graph has $n$ vertices and at least
$(2D)^{-2}C_{D}n^{3/2}$ edges. Thus by [5, Lemma 2.8], we can find an induced
subgraph with $n^{\prime}\geq n^{1/2}$ vertices such that each vertex has
degree at least

$\frac{(2D)^{-2}C_{D}(n^{\prime})^{3/2}}{O(1)n^{\prime}}=\frac{C_{D}}{O_{D}(1)}(n^{\prime})^{1/2}.$

Let $\Gamma^{\prime}\subset\Gamma$ be the set of curves corresponding to the
vertices of the induced subgraph.

Define
$\mathcal{L}^{\prime}=\\{\zeta(\gamma)\colon\gamma\in\Gamma^{\prime}\\}$. Let
$R\in k[x,y,z]$ be a non-zero polynomial of minimal degree that vanishes on
all the curves in $\Gamma^{\prime}$.

######  Lemma 5.

$\deg R=O_{D}\big{(}(n^{\prime})^{1/2}\big{)}.$ (15)

###### Proof.

Recall that each curve in $\Gamma^{\prime}$ has degree $O_{D}(1)$. Let
$C_{D}^{\prime}$ be a constant to be chosen later, and let $\mathcal{P}$ be a
set of $C_{D}^{\prime}(n^{\prime})^{3/2}$ points, with
$C_{D}^{\prime}(n^{\prime})^{1/2}$ points on each curve of
$\mathcal{L}^{\prime}$. Let $S$ be a polynomial of degree $\leq
100(C_{D}^{\prime})^{1/3}(n^{\prime})^{1/2}$ that vanishes on the points of
$\mathcal{P}.$ By Bézout’s theorem, if $C_{D}^{\prime}$ is chosen sufficiently
large (depending only on $D$), then $S$ vanishes on every curve in
$\mathcal{L}^{\prime}.$ Since $R$ is the polynomial of lowest degree that
vanishes on the curves in $\mathcal{L}^{\prime}$, we have $\deg R\leq\deg S$,
which gives us (15). ∎

In particular, if the constant $c_{D}$ from the statement of Theorem 1 is
chosen sufficiently small, then

$(\deg\gamma)(\deg R)<\operatorname{char}(k)$ (16)

for every curve $\gamma\in\Gamma^{\prime}$.

######  Lemma 6.

If $c_{D}$ is sufficiently small and $C_{D}$ is sufficiently large (depending
only of $D$), then every curve in $\Gamma^{\prime}$ is contained in
$\mathbf{Z}(\partial_{z}R)$.

###### Proof.

Fix a curve $\gamma\in\Gamma^{\prime}$. There are at least
$\frac{C_{D}}{O_{D}(1)}(n^{\prime})^{1/2}$ distinct points $(x,y,z)\in\gamma$
so that the following holds

  * (i)

$(x,y,z)$ is a good point of $\gamma$.

  * (ii)

There is a second curve $\gamma^{\prime}\in\Gamma$ so that (11) holds at
$(x,y)$ (with the value of $j$ determined above) for $\alpha$ and
$\alpha^{\prime}$ corresponding to $\gamma$ and $\gamma^{\prime}$,
respectively.

  * (iii)

$(x,y,z)$ is a good point of $\gamma^{\prime}$.

Let $(x,y,z)$ be one such point and let $\gamma^{\prime}\in\Gamma^{\prime}$ be
the curve from Item (ii) above. By Lemma 3, the span of the tangent spaces of
$\beta_{k}^{*}(\alpha)$ and $\beta_{k}^{*}(\alpha^{\prime})$ at $(x,y,z)$
contains the vector $(0,0,1)$. This implies that $\partial_{z}R(x,y,z)=0$.
This means that there are at least $\frac{C_{D}}{O_{D}(1)}(n^{\prime})^{1/2}$
points on $\gamma$ for which $\partial_{z}R=0$. If $C_{D}$ is selected
sufficiently large (depending only on $D$), then the number of points on
$\gamma$ at which $\partial_{z}R$ vanishes is greater than
$(\deg\gamma)(\deg\partial_{z}R)$. Since $(\deg\gamma)(\deg
R)<\operatorname{char}(k)$, we can apply Bézout’s theorem to conclude that
$\gamma\subset\mathbf{Z}(\partial_{z}R)$. ∎

Since $R$ was a minimal degree non-zero polynomial whose zero-set contained
all the curves from $\Gamma^{\prime}$, Lemma 6 implies that $\partial_{z}R=0$.
In particular, if $x^{a}y^{b}z^{c}$ is a monomial appearing with nonzero
coefficient in $R$, we have $cx^{a}y^{b}z^{c-1}=0$, so $c$ is congruent to $0$
mod $\operatorname{char}(k)$; but $c\leq\deg R<\operatorname{char}(k)$, so
this implies that $c=0$. In other words, $R(x,y,z)=S(x,y)$ for some polynomial
$S\in k[x,y]$. However, each curve in $\mathcal{L}^{\prime}$ must be a
distinct irreducible component of $\mathbf{Z}(S)$, and this implies

$\deg(R)=\deg(S)\geq|\mathcal{L}^{\prime}|=n^{\prime}.$ (17)

If we select the constant $C_{D}$ in Theorem 1 sufficiently large, then this
contradicts (15). This completes the proof of Theorem 1.

##  6 Orthogonal curves

Before proving Theorem 2, we will introduce some additional tools and
terminology for dealing with space curves in $k^{3}$. An important tool will
be Proposition 4, which describes the structure of arrangements of space
curves that determine many two-rich points. In this section (as in previous
sections), we will always assume that $k$ is an algebraically closed field.

###  6.1 Constructible sets

A constructible set is a generalization of an algebraic variety. Algebraic
varieties are defined by finite sets of polynomial equalities; for example the
set $\\{w\in k^{d}\colon P_{1}(w)=0,\ldots,P_{s}(w)=0\\}$ is an algebraic
variety. (Affine) constructible sets are defined by finite sets of polynomial
equalities and non-equalities; for example the set

$\\{w\in k^{d}\colon P_{1}(w)=0,\ldots,P_{s}(w)=0,\ Q_{1}(w)\neq
0,\ldots,Q_{t}(w)\neq 0\\}$

is constructible. If $X$ and $Y$ are constructible subsets of $k^{d}$, then so
are $X\cup Y,\ X\cap Y$, and $X\backslash Y$. Similarly, (projective)
constructible sets are defined by finite sets of homogeneous polynomial
equalities and non-equalities. In particular, the set of irreducible degree
$\leq D$ plane curves in $k^{2}$ is a constructible subset of
$\mathbf{P}k^{\binom{D+2}{2}}.$

The degree of an algebraic variety is one measure of how “complicated” that
variety is. The analogous notation for constructible sets is called
complexity. See [10, Chapter 3] for an introduction to constructible sets, and
see [9] for further information about the complexity of constructible sets.

###  6.2 The Chow variety of curves

We will work with a parameter space of algebraic space curves called the Chow
variety. Normally, this is a projective variety that parameterizes projective
curves in $k^{3}$, but for our purposes it will be easier to work with a
slightly smaller object which is an affine variety that parameterizes (affine)
curves in $k^{3}$.

For each $D\geq 1$, let $\mathcal{C}_{D}$ be the (affine) Chow variety of
irreducible (affine) curves of degree $\leq D$ in $k^{3}$. $\mathcal{C}_{D}$
is a constructible set of complexity $O_{D}(1)$, and there is a constructible
set $\tilde{\mathcal{C}}_{D}\subset\mathcal{C}_{D}\times k^{3}$ of complexity
$O_{D}(1)$ that is equipped with two projections
$\pi_{1}\colon\tilde{\mathcal{C}}_{D}\to\mathcal{C}_{D}$ and
$\pi_{2}\colon\tilde{\mathcal{C}}_{D}\to k^{3}$ with the following properties:

  * •

For each $w\in\mathcal{C}_{D}$, $\pi_{2}\circ\pi^{-1}_{1}(w)=\gamma_{w}$ is an
irreducible algebraic curve in $k^{3}$.

  * •

For “every” irreducible algebraic curve $\gamma\subset k^{3}$ of degree $\leq
D$, there is a unique point $w_{\gamma}\in\mathcal{C}_{D}$ so that
$\gamma=\pi_{2}\circ\pi^{-1}_{1}(w_{\gamma})$.

We put the word “every” in quotes because when constructing the affine Chow
variety, we intersect the true Chow variety (which is a projective variety)
with the compliment of a generic hyperplane. Thus the above statement fails
for a small number of curves $\gamma\subset k^{3}$. However, since the
original problem we are interested in concerns a finite set of curves, this
subtlety will not affect us. For more information about the Chow variety, see
[10, Chapter 7] or [7]; both of these sources describe the projective version
of the Chow variety, and they only deal with the Chow variety of irreducible
curves of degree exactly $D$. For a precise construction of the affine Chow
variety $\mathcal{C}_{D}$ used here, see [9]. Henceforth, we will abuse
notation slightly and use the notation $\gamma\in\mathcal{C}_{D}$ to refer to
curves in the Chow variety.

###  6.3 Sets of curves with many rich points

Informally, Theorem 3.7 from [9] says that if an arrangement of space curves
(taken from a particular family of curves) determines many two-rich points,
then many of these curves must be contained in a low degree doubly-ruled
surface. In order to make this precise, we will need a definition.

######  Definition 4 (Doubly ruled surface).

Let $\mathcal{C}\subset\mathcal{C}_{D}$ be a constructible set. Let $Z\subset
k^{3}$ be an irreducible surface. We say that $Z$ is doubly ruled by curves
from $\mathcal{C}$ if there is a Zariski open set $O\subset Z$ so that for
every $w\in O$, there are at least two distinct curves from $\mathcal{C}$
containing $w$ and contained in $Z$.

######  Theorem 4 ([9] Theorem 3.7, special case).

Let $D\geq 1$ and let $\mathcal{C}\subset\mathcal{C}_{D}$ be a constructible
set of curves. Then there exist constants $c_{D,\mathcal{C}}>0$ (small) and
$C_{D,\mathcal{C}}$ (large) so that the following holds. Let
$\Gamma\subset\mathcal{C}$ be a finite set of curves of cardinality $n$, with
$n\leq c_{D,\mathcal{C}}\operatorname{char}(k)^{2}$ (if
$\operatorname{char}(k)=0$ then we place no restriction on $n$). Then at least
one of the following must hold:

  * •

The number of two-rich points is $\leq C_{D,\mathcal{C}}n^{3/2}$.

  * •

There is an irreducible surface $Z\subset k^{3}$ of degree $\leq 100D^{2}$
that is doubly ruled by curves from $\mathcal{C}$; this surface contains $\geq
c_{D,\mathcal{C}}n^{1/2}$ curves from $\Gamma$. Furthermore, for every $m\geq
1$, we can find two disjoint sets of curves, each of cardinality $m$, so that
every curve from the first set intersects every curve from the second set.

We will not require the full strength of Theorem 4. In particular, we will
never use the fact that the doubly ruled surface $Z$ (if it exists) has small
degree, nor will we use the fact that it contains $\Omega(n^{1/2})$ curves
from $\Gamma$.

###  6.4 Space curves respecting tangency conditions

######  Lemma 7.

Let $W\subset k^{\binom{D+2}{2}}$ be a constructible set of degree $\leq D$
curves. Then for each $D^{\prime}$, the sets

$\begin{split}&\mathcal{C}_{X,D^{\prime}}=\\{\gamma\in\mathcal{C}_{D^{\prime}}\colon\gamma=\beta_{1}^{*}(\alpha)\
\textrm{for some}\ \alpha\in W\\},\\\
&\tilde{\mathcal{C}}_{X,D^{\prime}}=\\{\gamma\in\mathcal{C}_{D^{\prime}}\colon\gamma=\tilde{\beta}_{1}^{*}(\alpha)\
\textrm{for some}\ \alpha\in W\\}\end{split}$

are constructible and have complexity $O_{W,D^{\prime}}(1)$.

###### Proof.

We will begin with $\mathcal{C}_{X,D^{\prime}}$. Define

$\begin{split}Y_{D^{\prime}}=&\big{\\{}(\gamma,(x,y,z),\alpha)\in\mathcal{C}_{D^{\prime}}\times
k^{3}\times W\colon(x,y,z)\in\gamma,\ P_{\alpha}(x,y)=0,\\\
&\qquad\qquad\qquad\qquad\qquad\partial_{y}P_{\alpha}(x,y)=0\ \textrm{OR}\
z\partial_{y}P_{\alpha}(x,y)-\partial_{x}P_{\alpha}(x,y)=0\big{\\}},\\\
Y_{D^{\prime}}^{\prime}=&\\{(\gamma,\alpha)\in\mathcal{C}_{D^{\prime}}\times
W\colon\gamma\ \textrm{is not a vertical line},\ (\gamma,(x,y,z),\alpha)\in
Y_{D^{\prime}}\ \textrm{for all}\ (x,y,z)\in\gamma\\}.\end{split}$

$Y_{D^{\prime}}$ is constructible of complexity $O_{D^{\prime},W}(1)$; the
condition $(x,y,z)\in\gamma$ can be written as
$(\gamma,(x,y,z))\in\tilde{\mathcal{C}}_{D^{\prime}}$, and the latter is a
constructible set of complexity $O_{D^{\prime}}(1)$. To verify that
$Y_{D^{\prime}}^{\prime}$ is constructible, observe that the set

$A=\\{(\gamma,(x,y,z),\alpha)\in\mathcal{C}_{D}\times k^{3}\times W\colon\
(x,y,z)\in\gamma,\ (\gamma,(x,y,z),\alpha)\notin Y_{D^{\prime}}\\}$ (18)

is constructible and has complexity $O_{D^{\prime},W}(1)$. Let
$\pi_{1}(\gamma,(x,y,z),\alpha)=(\gamma,\alpha)$ be the projection map. Then

$\pi_{1}(A)=\\{(\gamma,\alpha)\in\mathcal{C}_{D^{\prime}}\times W\colon\
\textrm{there exists}\ (x,y,z)\in\gamma,\ \textrm{such that}\
(\gamma,(x,y,z),\alpha)\notin Y_{D^{\prime}}\\}.$

By Chevalley’s theorem (see [10, Theorem 3.16]), $\pi_{1}(A)$ is also
constructible and has complexity $O_{D^{\prime},W}(1)$. Finally,

$Y_{D^{\prime}}^{\prime}=\big{(}\\{\gamma\in\mathcal{C}_{D^{\prime}}:\gamma\
\textrm{is not a vertical line}\\}\times W\big{)}\ \backslash\ \pi_{1}(A).$

Let $\pi_{2}\colon\mathcal{C}_{D^{\prime}}\times W\to\mathcal{C}_{D^{\prime}}$
be the projection map. Then
$\mathcal{C}_{D^{\prime},W}=\pi_{2}(Y_{D^{\prime}}^{\prime})$; this
establishes that $\mathcal{C}_{D^{\prime},W}$ is constructible and has
complexity $O_{D^{\prime},W}(1)$. An identical argument shows that
$\tilde{\mathcal{C}}_{W,D^{\prime}}$ is constructible and has complexity
$O_{W,D^{\prime}}(1)$. ∎

Define the set of monic irreducible polynomials,

$\operatorname{MI}_{D}=\\{\alpha\in k^{\binom{D+2}{2}}\colon P_{\alpha}\
\textrm{is irreducible and $x$--monic}\\}.$

Note that $\operatorname{MI}_{D}$ is a constructible set of complexity
$O_{D}(1)$, and there is an injection
$\varphi\colon\operatorname{MI}_{D}\to\mathbf{P}k^{\binom{D+2}{2}}$ that sends
the point $(w_{1},\ldots,w_{\binom{D+2}{2}})$ to the point
$[1:w_{1}:\ldots:w_{\binom{D+2}{2}}]$. We will be interested in
$\varphi^{-1}$.

Let $X\subset\mathbf{P}k^{\binom{D+2}{2}}$ be a family of curves and define

$\operatorname{Aff}(X)=\varphi^{-1}(\varphi(\operatorname{MI}_{D})\cap X).$
(19)

(This is the “affinization” of the projective set $X$). Then
$\operatorname{Aff}(X)\subset\operatorname{MI}_{D}$ is constructible and has
complexity $O_{D,X}(1)$. Note that for each $\alpha\in\operatorname{Aff}(X)$,
there is a unique element $\gamma\in\mathcal{C}_{\operatorname{Aff}(X),D^{2}}$
that satisfies $\gamma=\beta_{1}^{*}(\alpha)$. This is because
$\beta_{1}^{*}(\alpha)$ is an irreducible curve of degree at most $D^{2}$.
Similarly, there is a unique element
$\gamma\in\tilde{\mathcal{C}}_{\operatorname{Aff}(X),D^{2}}$ that satisfies
$\gamma=\tilde{\beta}_{1}^{*}(\alpha)$. We now have the necessary tools to
prove Theorem 2.

##  7 Proof of Theorem 2

###### Proof.

Let $\mathcal{L}$ be a set of $n$ irreducible curves in $X$. By applying a
suitable invertible linear transformation $T\colon k^{2}\to k^{2}$ if
necessary, we can guarantee that $w\in\varphi(\operatorname{MI}_{D})$ for each
$w\in\mathcal{L}$. The family $X$ is replaced by the family
$\\{T(\zeta)\colon\zeta\in X\\};$ this is also a family of curves of degree
$\leq D$. Abusing notation slightly, we will call this new family $X$ as well.

Let $X^{\prime}=\operatorname{Aff}(X)$, and let
$\mathcal{C}=\mathcal{C}_{X^{\prime},D^{2}}\cup\tilde{\mathcal{C}}_{X^{\prime},D^{2}}$.
This is a family of degree $\leq D^{2}$ curves of complexity $O_{D,X}(1)$.
Define

$\begin{split}&\Gamma=\\{\beta_{1}^{*}(\alpha)\colon\alpha\in\varphi^{-1}(\mathcal{L})\\},\\\
&\tilde{\Gamma}=\\{\tilde{\beta}_{1}^{*}(\alpha)\colon\alpha\in\varphi^{-1}(\mathcal{L})\\}.\end{split}$

$\Gamma$ and $\tilde{\Gamma}$ are sets of space curves that encode the slope
of the corresponding plane curves from $\mathcal{L}$. $\Gamma$ and
$\tilde{\Gamma}$ are finite subsets of $\mathcal{C}$. Let
$\Gamma_{0}=\Gamma\cup\tilde{\Gamma}.$

Apply Theorem 4 to $\Gamma_{0}$. Either there are $O_{X}(n^{3/2})$ two-rich
points determined by the arrangement of curves in $\Gamma_{0}$, or for any
$m\geq 1$, we can find a set $\Gamma^{*}\subset\mathcal{C}$ of cardinality $m$
that determines $\geq m^{2}/4$ two-rich points.

If the first option occurs, then by Lemma 2, the curves from $\mathcal{L}$
determine $O_{X}(n^{3/2})$ directed points of orthogonality, and we are done.

Suppose the second option occurs. Write
$\Gamma^{*}=(\Gamma^{*}\cap\mathcal{C}_{X^{\prime},D^{2}})\cup(\Gamma^{*}\cap\tilde{\mathcal{C}}_{X^{\prime},D^{2}})$.
Each two-rich point determined by
$\Gamma^{*}\cap\mathcal{C}_{X^{\prime},D^{2}}$ corresponds to a directed point
of tangency amongst the curves in

$\\{\mathbf{Z}(P_{\alpha})\colon\beta_{1}(\alpha)\in\Gamma^{*}\cap\mathcal{C}_{X^{\prime},D^{2}}\\}.$

By Theorem 1, there are $O_{D}(m^{3/2})$ such points. Similarly, each two-rich
point determined by $\Gamma^{*}\cap\tilde{\mathcal{C}}_{X^{\prime},D^{2}}$
corresponds to a directed point of tangency amongst the curves in

$\\{\mathbf{Z}(P_{\alpha})\colon\tilde{\beta}_{1}(\alpha)\in\Gamma^{*}\cap\tilde{\mathcal{C}}_{X^{\prime},D^{2}}\\}.$

Again, by Theorem 1, there are $O_{D}(m^{3/2})$ such points.

Define

$\mathcal{L}_{m}=\\{\mathbf{Z}(P_{\alpha})\colon\beta_{1}(\alpha)\in\Gamma^{*}\cap\mathcal{C}_{X^{\prime},D^{2}}\\}\cup\\{\mathbf{Z}(P_{\alpha})\colon\tilde{\beta}_{1}(\alpha)\in\Gamma^{*}\cap\tilde{\mathcal{C}}_{X^{\prime},D^{2}}\\}.$

There are $\Omega_{X}(m^{2})$ points that are hit by at least one curve from
$\Gamma^{*}\cap\mathcal{C}_{X^{\prime},D^{2}}$ and at least one curve from
$\Gamma^{*}\cap\tilde{\mathcal{C}}_{X^{\prime},D^{2}}$. By Lemma 2, each of
these points correspond to a distinct directed point of orthogonality from the
curves in the arrangement $\mathcal{L}_{m}$. Therefore we have constructed an
arrangement of $m$ curves from the family $X$ that determine
$(m/2)^{2}(1-o_{X}(1))$ directed points of orthogonality. ∎

##  8 Generalizations and further directions

###  8.1 Improved bounds

It is natural to ask whether Theorem 1 is sharp. Remark 2 shows that if $k$
has characteristic two, then it is impossible to obtain nontrivial bounds on
the number of directed tangencies. If $n\sim(\operatorname{char}(k))^{2}$,
then Theorem 1 is best possible. For example, let $\mathcal{L}$ be the set of
all unit circles in $\mathbb{F}_{p}^{2}$; then
$|\mathcal{L}|=|\mathbb{F}_{p}^{2}|\sim(\operatorname{char}(\mathbb{F}_{p}))^{2}$,
and for each point $p\in\mathbb{F}_{p}^{2}$ and each line $\ell$ passing
through $p$, there are exactly two unit circles that are smooth at $p$ and
tagent to $\ell$ at $p$. Thus no three curves from $\mathcal{L}$ are tangent
at a common point, and $\mathcal{L}$ determines $\sim
p^{3}=|\mathcal{L}|^{3/2}$ tangencies, so the bound from Theorem 1 is attained
(technically, Theorem 1 only applies if $|\mathcal{L}|$ is less than
$c_{D}(\operatorname{char}(\mathbb{F}_{p}))^{2}$, (here $D=2$) but we can meet
this requirement by randomly selecting a subset of $\mathcal{L}$, where each
circle is chosen with probability $\sim c_{D}$).

However, if $|\mathcal{L}|$ is much smaller than
$(\operatorname{char}(k))^{2}$, then we do not know if the bound from Theorem
1 is sharp. For example, if $k=\mathbb{R}$ or $k=\mathbb{C}$, the best
construction we are aware of attains a lower bound of $\Omega(n^{4/3})$. In
brief, the construction is as follows. Let $\mathcal{P}$ be a set of $n$
points in $\mathbb{R}^{2}$ and let $\mathcal{L}_{0}$ be a set of $n$ lines in
$\mathbb{R}^{2}$ that determine $\sim n^{4/3}$ point-line incidences. Let
$\mathcal{L}_{1}$ be the set of $n$ unit circles centered at the points of
$\mathcal{P},$ and let $\mathcal{L}_{2}$ be the set of $n$ lines obtained by
translating each line $\ell\in\mathcal{L}_{1}$ one unit in the direction
$\ell^{\perp}$. Then if $(p,\ell)$ is a point-line incidence from the original
collection of points and lines, the corresponding circle and line will be
tangent. Let $\mathcal{L}=\mathcal{L}_{1}\cup\mathcal{L}_{2}$; this is a set
of $2n$ irreducible curves of degree $\leq 2$ that determines $\sim n^{4/3}$
tangencies. By applying an inversion transform centered at a point that does
not lie on any of the circles or lines, one can transform $\mathcal{L}$ into a
set of $2n$ circles that determine $\sim n^{4/3}$ tangencies. See Chapter 7.1
of [2] for further details. The same example can be realized in
$\mathbb{C}^{2}$ by taking the complexification of the curves from
$\mathcal{L}.$

###  8.2 Higher-order tangency

We believe that an analogue of Theorem 1 should hold for higher-order
tangencies. As the order of tangency increases, the number of such tangencies
should decrease. Heuristically, the number of “$j$–th order tangencies”
spanned by an arrangement of $n$ curves should be $O(n^{(j+2)/(j+1)})$. We
will make this notion more precise below.

######  Definition 5 (Jet Bundle).

Let $k$ be a field and let $1\leq s<\operatorname{char}(k)$ (if
$\operatorname{char}(k)=0$, we only require $s\geq 1$). We define the bundle
of one-dimensional $s$–jets in $k^{2}$ to be the set of equivalence classes

$\mathcal{J}_{s}=\\{\overline{(p,\tau)}\colon p\in\ k^{2},\ \tau\ \textrm{is a
curve through $p$ that is smooth at}\ p\\},$

where $\overline{(p,\tau)}=\overline{(p^{\prime},\tau^{\prime})}$ if
$p=p^{\prime}$ and $\tau$ and $\tau^{\prime}$ intersect at $p$ with
multiplicity $\geq s$. If $s=1$, then $\mathcal{J}_{1}$ can be identified with
the set of tuples $(p,\ell)$, where $p\in k^{2}$ and $\ell$ is a line passing
through $p$.

######  Definition 6 (Directed points of higher order tangency).

Let $k$ be a field and let $1\leq s<\operatorname{char}(k)$ (if
$\operatorname{char}(k)=0$, we only require $s\geq 1$). Let $\mathcal{L}$ be a
set of irreducible algebraic curves in $k^{2}$. Let
$\overline{(p,\tau)}\in\mathcal{J}_{s}$. We say that $\overline{(p,\tau)}$ is
a directed point of $s$–th order tangency for $\mathcal{L}$ if there are at
least two distinct curves in $\mathcal{L}$ that are smooth at $p$ and that
intersect $\tau$ at $p$ with multiplicity $\geq s$ (crucially, this does not
depend on the choice of representative $\tau$ from the equivalence class
$\overline{(p,\tau)}$).

Define $\mathcal{T}_{s}(\mathcal{L})$ be the set of directed points of $s$–th
order tangency, and for each $\overline{(p,\tau)}\in\mathcal{T}(\mathcal{L}),$
define $\operatorname{mult}_{s}(\overline{(p,\tau)};\mathcal{L})$ to be the
number of curves from $\mathcal{L}$ that are smooth at $p$ and intersect
$\tau$ at $p$ with multiplicity $\geq s$.

The natural analogue of the space curves described in Section 3 would be
curves in $\mathbb{R}^{d+s}$. This leads us to conjecture the following bound
on the number of higher-order tangencies.

######  Conjecture 1.

Fix $D,s\geq 1$. Let $\mathcal{L}$ be a set of $n$ irreducible algebraic
curves of degree $\leq D$, with $n\leq c_{D,s}\operatorname{char}(k)^{s+1}$.
Then

$\sum_{(p,\gamma)\in\mathcal{T}_{j}(\mathcal{L})}\operatorname{mult}_{s}(\overline{(p,\tau)};\mathcal{L})\leq
C_{D,s}n^{(s+2)/(s+1)}.$ (20)

###  8.3 Higher-dimensional hypersurfaces

We conjecture than an analogue of Theorem 1 should be true for hypersurfaces
in higher dimension.

######  Definition 7 (Higher dimensional directed points of tangency).

Let $k$ be a field and let $\mathcal{S}$ be a set of irreducible hypersurfaces
in $k^{d}$. Let $p$ be a point in $k^{d}$ and let $H$ be a hyperplane
containing $p$. We say that $(p,H)$ is a directed point of tangency for
$\mathcal{S}$ if there are at least two distinct hypersurfaces in
$\mathcal{S}$ that are smooth at $\zeta$ and tangent to $H$ at $\zeta$. Define
$\mathcal{T}(\mathcal{S})$ be the set of directed points of tangency, and for
each $(p,H)\in\mathcal{T}(\mathcal{S}),$ define
$\operatorname{mult}(p,H;\mathcal{S})$ to be the number of hypersurfaces from
$\mathcal{S}$ that are smooth at $p$ and tangent to $H$ at $p$.

The natural analogue of the space curves described in Section 3 would be $d-1$
dimensional varieties in $\mathbb{R}^{2d-1}$. This leads us to conjecture the
following higher-dimensional analogue of Theorem 1.

######  Conjecture 2.

Let $\mathcal{S}$ be a set of $n$ irreducible hypersurfaces in $k^{d}$ of
degree at most $D$, with $n\leq c_{D}\operatorname{char}(k)^{d}$. Then

$\sum_{(p,H)\in\mathcal{T}(\mathcal{S})}\operatorname{mult}(p,H;\mathcal{S})\leq
C_{D}n^{(2d-1)/d}.$ (21)

## References

  * [1] Pankaj K. Agarwal, Eran Nevo, János Pach, Rom Pinchasi, Micha Sharir, and Shakhar Smorodinsky, _Lenses in arrangements of pseudo-circles and their applications_ , J. ACM 51 (2004), no. 2, 139–186.
  * [2] Peter Brass, William Moser, and János Pach, _Research problems in discrete geometry_ , Springer, New York, 2005.
  * [3] Kenneth L. Clarkson, Herbert Edelsbrunner, Leonidas J. Guibas, Micha Sharir, and Emo Welzl, _Combinatorial complexity bounds for arrangements of curves and spheres_ , Discrete Comput. Geom. 5 (1990), no. 2, 99–160.
  * [4] Zeev Dvir, _On the size of Kakeya sets in finite fields_ , JAMS 22 (2009), 1093–1097.
  * [5] Zeev Dvir and Sivakanth Gopi, _On the number of rich lines in truly high dimensional sets_ , Proceedings of the 31st International Symposium on Computational Geometry, 2015, pp. 584–598.
  * [6] David Eisenbud, _Commutative algebra: with a view toward algebraic geometry_ , vol. 150, Springer Science & Business Media, 2013.
  * [7] Mark Green and Ian Morrison, _The equations defining Chow varieties_ , Duke Math. J 53 (1986), 733–747.
  * [8] Larry Guth and Nets. Katz, _On the Erdős distinct distance problem in the plane._ , Ann. of Math. 181 (2015), 155–190.
  * [9] Larry Guth and Joshua Zahl, _Algebraic curves, rich points, and doubly-ruled surfaces_ , arXiv:1503.02173 (2015).
  * [10] Joe Harris, _Algebraic geometry: A first course_ , Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1995.
  * [11] Adam Marcus and Gábor Tardos, _On topological graphs without self-intersecting 4-cycles_ , Proc. Graph Drawings, 2005.
  * [12] Gábor Megyesi and Endre Szabó, _On the tacnodes of configurations of conics in the projective plane_ , Math. Ann. 305 (1996), 693–703.
  * [13] János Pach, Natan Rubin, and Gábor Tardos, _On the Richter-Thomassen conjecture about pairwise intersecting closed curves_ , SODA, 2015, pp. 1506–1516.
  * [14] T Wolff, _Recent work connected with the kakeya problem_ , Prospects in mathematics, 1999, pp. 129–162.
  * [15] Thomas Wolff, _A Kakeya type problem for circles_ , Am. J. Math 119 (1997), no. 5, 985–1026.
  * [16] , _Local smoothing type estimates on $L$p for large $p$_, GAFA 10 (2000), no. 5, 1237–1288.
  * [17] Joshua Zahl, _An improved bound on the number of point-surface incidences in three dimensions._ , Contrib. Discrete Math 8 (2013), no. 1, 100–121.

[JSE] Jordan S. Ellenberg
University of Wisconsin, Madison
Madison, WI
ellenbermathwiscedu [JS] Jozsef Solymosi
University of British Colombia
Vancouver, BC
solymosimathubcca [JZ] Joshua Zahl
University of British Colombia
Vancouver, BC
jzahlmathubcca


# Real-World Chaos-Based Cryptography Using Synchronised Chua Chaotic Circuits
††thanks: Funded by DFG, under Projects 440182124 and 439892735 of SPP 2253,
and by BMBF under Joint Project 16KISK034 for the 6G-RIC.

Emiliia Nazarenko1, Nikolaos Athanasios Anagnostopoulos12, Stavros G.
Stavrinides3,
Nico Mexis1, Florian Frank1, Tolga Arul12, Stefan Katzenbeisser1  1Faculty of
Computer Science and Mathematics, University of Passau, Innstraße 43, 94032
Passau, Germany
Emails: {nazare02, anagno02, mexis01, frank55, arul01<EMAIL_ADDRESS>passau.de
2Computer Science Department, Technical University of Darmstadt,
Hochschulstraße 10, 64289 Darmstadt, Germany
Emails: {anagnostopoulos<EMAIL_ADDRESS>3School of
Science and Technology, International Hellenic University, Thermi Campus,
57001 Thessaloniki, Greece
Email<EMAIL_ADDRESS>
###### Abstract

This work presents the hardware demonstrator of a secure encryption system
based on synchronised Chua chaotic circuits. In particular, the presented
encryption system comprises two Chua circuits that are synchronised using a
dedicated bidirectional synchronisation line. One of them forms part of the
transmitter, while the other of the receiver. Both circuits are tuned to
operate in a chaotic mode. The output (chaotic) signal of the first circuit
(transmitter) is digitised and then combined with the message to be encrypted,
through an XOR gate. The second Chua circuit (receiver) is used for the
decryption; the output chaotic signal of this circuit is similarly digitised
and combined with the encrypted message to retrieve the original message. Our
hardware demonstrator proves that this method can be used in order to provide
extremely lightweight real-world, chaos-based cryptographic solutions.

###### Index Terms:

chaos, Chua circuit, stream encryption, security

## Introduction

The rapid increase in the number of electronic devices in everyday use leads
to an unlimited growth of vulnerable communication that must be protected from
possible attacks. Having in focus the growing Internet of Things ecosystem, as
well as edge computing, secure data transmission appears as a very important
part. One of the strongest tools for mitigating attacks on electronic
communications, is cryptography.

In our work, we focus on securing the transmitted information through
symmetric stream cryptography that is based on the chaotic signal produced by
two synchronised Chua chaotic circuits [1], one of which forms part of the
transmitter and the other of the receiver. The produced chaotic signal acts as
a random number stream that is shared by the synchronised chaotic circuits.
Thus, the message can be encrypted at the transmitter by being XORed with a
digitised form of the aforementioned chaotic signal, and decrypted at the
receiver by XORing the encrypted message stream with the digitised form of the
same chaotic signal.

## Description of the Hardware Demonstrator

A simplified architecture of the proposed synchronized chaotic encryption-
decryption system is shown in Figure 1. A proof-of-concept circuit of this
system has been designed and appears in Figure 2, which demonstrates the full
circuitry. The relevant characteristics and values of the components used for
implementing the circuits are listed in Table I.

Figure 1: The overall architecture of the encryption-decryption system.

Figure 2: Schematic diagram of the overall implemented system.

The Chua circuits that we are using, are based on the ones examined by Kennedy
[2]. In our approach, two channels are utilized, one for achieving
synchronisation and another for transmitting information. The signal produced
by the chaotic circuits is digitised and processed according to the method
described in [3, 4]. After being processed, this signal is used to encrypt a
message, by performing an XOR operation. In order to perform a successful
decoding, the transmitter and receiver are synchronised using a dedicated
bidirectional synchronisation line.

In our demo, the message is generated by a wave generator with the following
parameters: frequency $6$ kHz, amplitude $2.5$ Vpp, offset $+1.25$ V, phase
$0.0^{\circ}$, and duty cycle $50\%$. The encrypted message (coming from the
XOR) is transmitted to the receiver, which in its turn decodes it, using the
XOR operation and its own synchronized chaotic signal, which is processed in a
similar way as the chaotic signal of the transmitter. The decrypted message is
then cleared by an RC low-pass filter. A real-world implementation of the
system is shown in Figure 3.

TABLE I: Components of the Chua Circuit and Their Characteristics Component | Characteristics
---|---
Inductors $L,L2$ | 18 mH, 10% tolerance
Capacitors $C1,C21$ | 100 nF, 5% tolerance
Capacitors $C2,C22$ | 10 nF, 5% tolerance
Variable resistors $Rv,R2v$ | 1555 $\Omega$
Resistors $R1,R21$ | 3.3 k$\Omega$, 5% tolerance
Resistors $R2,R22$ | 22 k$\Omega$, 5% tolerance
Resistors $R3,R23$ | 2.2 k$\Omega$, 5% tolerance
Resistors $R4,R24$ | 220 $\Omega$, 5% tolerance
OpAmps | TL082ACP, TL081DIP
Batteries | 2 V, 9 V
Capacitor $Cfil$ | 7 nF, 5% tolerance
Resistor $Rfil$ | 1 k$\Omega$, 5% tolerance

Figure 3: The implementation of the proposed system.

In Figure 4a, the attractor produced by the transmitter’s circuit is
illustrated. The chaotic mode of operation of the Chua circuit is evident. The
transmitter and the receiver are reliably chaotic-synchronised, as
demonstrated by the synchronization phase portrait appearing in Figure 4b,
based on the voltages on capacitors C1 and C21. The synchronization quality
(perfect in this case) is responsible for the system’s ability to provide
efficient decryption. Finally, Figure 5 presents the initial message (magenta)
encrypted in the transmitter, the encrypted message (blue) that is
transmitted, and the decrypted message (yellow) at the receiver’s output. It
is rather evident that the initial message and the decrypted one match.

The circuitry shown in Figure 3, together with the relevant power supply
(batteries), oscilloscope, and wave generator, form the main part of our
hardware demonstrator setup.

Figure 4: (a) Phase portrait of the transmitter Chua circuit, coming from the
voltages on the capacitors C1 and C2. (b) Synchronisation phase portrait
between the chaotic signals of the transmitter and the receiver.

Figure 5: Initial message (magenta), encrypted message (blue) and decrypted
message (yellow). The coincidence between the initial and the decrypted
message is evident.

## Conclusion

Concluding, one may support that our hardware demonstrator proves that the
proposed encryption-decryption system, which is based on two synchronized Chua
chaotic circuits, can provide an efficient, lightweight, and practical
solution for real-world cryptographic applications.

## References

  * [1] N. A. Anagnostopoulos, “Stability and robustness of three non-linear cryptographic systems under external noise conditions,” B.Sc. Thesis, Aristotle University, Thessaloniki, Greece, 2010.
  * [2] M. P. Kennedy, “Robust OP amp realization of Chua’s circuit,” _Frequenz_ , vol. 46, no. 3-4, pp. 66–80, 1992.
  * [3] A. Miliou, S. Stavrinides, A. Valaristos, and A. Anagnostopoulos, “Nonlinear electronic circuit, Part II: synchronization in a chaotic MODEM scheme,” _Nonlinear Analysis: Theory, Methods & Applications_, vol. 71, 12 2009.
  * [4] S. Stavrinides, A. Anagnostopoulos, A. Miliou, A. Valaristos, L. Magafas, K. Kosmatopoulos, and S. Papaioannou, “Digital chaotic synchronized communication system,” _Journal of Engineering Science and Technology Review_ , vol. 2, 06 2009.