Schwartz-Zippel bounds for two-dimensional products

Abstract

We prove bounds on intersections of algebraic varieties in C 4 with Cartesian products of finite sets from C 2 , and we point out connections with several classic theorems from combinatorial geometry. Consider an algebraic variety X ⊂ C 4 of degree d, such that not all polynomials that vanish on X are of the formwhere G, H, K, L are polynomials and G and K are not constant. Let P, Q ⊂ C 2 be finite sets of size n. If X has dimension one or two, then we prove |X ∩ (P × Q)| = O d (n), while if X has dimension three, then |X ∩ (P × Q)| = O d,ε (n 4/3+ε ) for any ε > 0. Both bounds are best possible in this generality (except for the ε).These bounds can be viewed as different generalizations of the Schwartz-Zippel lemma, where we replace a product of "one-dimensional" finite subsets of C by a product of "twodimensional" finite subsets of C 2 . The bound for three-dimensional varieties generalizes the Szemerédi-Trotter theorem. A key ingredient in our proofs is a two-dimensional version of a special case of Alon's combinatorial Nullstellensatz.As corollaries of our two bounds, we obtain bounds on the number of repeated and distinct values of polynomials and polynomial maps of pairs of points in C 2 , with a characterization of those maps for which no good bounds hold. These results generalize known bounds on repeated and distinct Euclidean distances.

The Elekes–Szabó Theorem in four dimensions

“…Theorem 1.1 replaces this combinatorial condition with an algebraic condition on F . An incidence bound with a similar (but different) algebraic condition was obtained in [12], for point sets and curve sets that do not have this Cartesian product structure.…”

An o-minimal Szemerédi–Trotter theorem

“…Remark 4. The formulation of Theorem 3 is inspired by the main results and the proofs in [15,18,17]. In these papers one is interested in a bound on the cardinality of a set of the form V ∩ (P × Q), where V ⊂ R 4 is an algebraic variety of fixed degree, and each of P, Q is a finite subset of R 2 (of arbitrarily large cardinality).…”

A Survey of Elekes-Rónyai-Type Problems

“…The study of such questions was initiated by Nassajian Mojarrad et al [33]. Even analogues of the Schwartz-Zippel lemma become more complicated: The easiest case would be a bound of the form |X ∩ (P × Q)| for a variety X ⊂ C 4 and finite sets P, Q ⊂ C 2 , but there are varieties for which no non-trivial bound holds.…”

“…We say that a variety X ⊂ C 4 is Cartesian if every polynomial vanishing on X is Cartesian with the same G, K. For a Cartesian variety X, we can have |X ∩ (P × Q)| = |P ||Q|, since we can take P ⊂ Z(G) and Q ⊂ Z(K) to get P × Q ⊂ X. It is proved in [33] that if X is not Cartesian, then one can obtain non-trivial upper bounds on |X ∩ (P × Q)|. As a consequence, we obtain the following lower bound on the number of distinct values of a polynomial map F = (F 1 , F 2 ) : C 2 × C 2 → C 2 .…”