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