New bounds on curve tangencies and orthogonalities

New bounds on curve tangencies and orthogonalities, Discrete Analysis 2016:18, 22 pp.

An important subfield of combinatorial geometry is that of incidence problems. Typically with such a problem one has two collections $A$ and $B$ of geometrical objects and some notion of incidence concerning them, and one wants to know how many incidences there can be. A fundamental theorem of this kind is the Szemerédi-Trotter theorem, which asserts that given $n$ points and $m$ lines in the plane, the number of incidences between them (that is, the number of pairs $(p,\ell)$ where $p$ is one of the points, $\ell$ is one of the lines, and $p$ is contained in $\ell$) is at most $C(n+m+(mn)^{2/3})$. Another important problem in the area is the joints problem, which is now a theorem of Larry Guth and Nets Katz. Given a set $L$ of lines in $\mathbb R^3$, define a joint to be a point $p\in\mathbb R^3$ such that there are three lines $\ell_1,\ell_2$ and $\ell_3$ that meet at $p$ and have linearly independent directions. Guth and Katz proved that $n$ lines can give rise to at most $O(n^{3/2})$ joints, which is best possible, as one can see by taking all axis parallel lines in $\mathbb R^3$ that intersect the grid $\{1,2,\dots,m\}^3$, with $m\sim\sqrt n$. This was part of the proof of their remarkable solution to the Erdős distance problem [2].

This paper is about incidences of a special kind between algebraic curves in the plane. Given a set $C$ of such curves, a tangency is a point $p$ in the plane and a line $\ell$ through $p$ such that at least two curves in $C$ go through $p$ in direction $l$. An orthogonality is a point $p$ and a line $l$ through $p$ such that at least one curve in $C$ goes through $p$ in the direction of $\ell$ and at least one curve in $C$ goes through $p$ in the direction orthogonal to $\ell$. (Throughout, we are assuming that the curves are not singular at $p$.)

The first main result of the paper roughly speaking states that a set $\mathcal L$ of $n$ algebraic curves of degree at most $D$ can give rise to at most $C_Dn^{3/2}$ tangencies. A more precise statement is that if you define the multiplicity of a tangency $(p,\ell)$ to be the number of curves in $\mathcal L$ that go through $p$ in direction $\ell$, then the sum of the multiplicities over all the tangencies is at most $C_Dn^{3/2}$. Like Guth and Katz’s proof of the joints problem, and Zeev Dvir’s famous solution of the finite-field Kakeya problem [1], the proof uses the polynomial method. They also prove the result in arbitrary fields, provided that $|\mathcal L|\leq c_D\chi^2$, where $\chi$ is the characteristic of the field (where this should be interpreted as meaning that there is no restriction if the characteristic is zero).

The theorem about orthogonalities is a little more complicated to state. One would like to say that $n$ plane curves of degree $D$ can give rise to at most $C_Dn^{3/2}$ orthogonalities, but that is false: for example, one can take $n/2$ parallel lines and another $n/2$ parallel lines that are orthogonal to the first ones. However, in a certain sense this is the only kind of counterexample. A precise statement can be found as Theorem 2 in the paper, but the rough idea is as follows. We say that $X$ is a family of curves if it can be defined in a polynomial manner. Then if $k$ is a field and $X$ is a family of curves in $k^2$ of degree at most $D$, then one of the following two situations occurs. Either every set $\mathcal L$ of $n$ curves from $X$ gives rise to at most $C_{D,X}n^{3/2}$ orthogonalities or for every $n\leq c_D\chi^2$ one can find $n$ curves in $X$ that give rise to $n^2(1/4-o_{D,X}(1))$ orthogonalities (where $o_{D,X}(1)$ denotes a term that tends to zero as $n$, and hence also the characteristic, tend to infinity while $D$ and the equations that define the family $X$ remain fixed).

Both results are proved by first transforming the set of curves into a set of curves in $k^3$ such that tangencies/orthogonalities in the original set correspond to intersections in the new set.

A simple example (given in the paper at the end of Section 1.3) shows that the condition that the curves should be algebraic of bounded degree is essential.

[1] Zeev Dvir, On the size of Kakeya sets in finite fields, JAMS 22 (2009), 1093–1097; preprint available online

[2] Larry Guth and Nets. Katz, On the Erdo ̋s distinct distance problem in the plane., Ann. of Math. 181 (2015), 155–190; or see arxiv:1011.4105