New bounds on curve tangencies and orthogonalities

- Mathematics, University of Wisconsin, Madison (WI)
- More about Jordan S. Ellenberg

- Mathematics, University of British Columbia (BC)
- More about Jozsef Solymosi

- Mathematics, Massachusetts Institute of Technology (MA)
- More about Joshua Zahl

*Discrete Analysis*, November. https://doi.org/10.19086/da.990.

### Editorial introduction

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,ℓ) where p is one of the points, ℓ is one of the lines, and p is contained in ℓ) 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 R3, define a *joint* to be a point p∈R3 such that there are three lines ℓ1,ℓ2 and ℓ3 that meet at p and have linearly independent directions. Guth and Katz proved that n lines can give rise to at most O(n3/2) joints, which is best possible, as one can see by taking all axis parallel lines in R3 that intersect the grid {1,2,…,m}3, with m∼√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 ℓ 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 ℓ and at least one curve in C goes through p in the direction orthogonal to ℓ. (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 L of n algebraic curves of degree at most D can give rise to at most CDn3/2 tangencies. A more precise statement is that if you define the *multiplicity* of a tangency (p,ℓ) to be the number of curves in L that go through p in direction ℓ, then the sum of the multiplicities over all the tangencies is at most CDn3/2. 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 |L|≤cDχ2, where χ 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 CDn3/2 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 k2 of degree at most D, then one of the following two situations occurs. Either every set L of n curves from X gives rise to at most CD,Xn3/2 orthogonalities or for every n≤cDχ2 one can find n curves in X that give rise to n2(1/4−oD,X(1)) orthogonalities (where oD,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 k3 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