TY - GEN

T1 - Polynomials vanishing on grids

T2 - 30th Annual Symposium on Computational Geometry, SoCG 2014

AU - Raz, Orit E.

AU - Sharir, Micha

AU - Solymosi, József

PY - 2014

Y1 - 2014

N2 - In this paper we characterize real bivariate polynomials which have a small range over large Cartesian products. We show that for every constant-degree bivariate real polynomial f, either |f(A;B)| = Ω(n4/3), for every pair of finite sets A,B,⊂ ℝ, with |A| = |B| = n (where the constant of proportionality depends on deg f), or else f must be of one of the special forms f(u, v) = h(φ (u)·ψ (v)), or f(u, v) = h(ψ (u) · ψ (v)), for some univariate polynomials φ ψ ; h over ℝ. This significantly improves a result of Elekes and Rónyai [7]. Our results are cast in a more general form, in which we give an upper bound for the number of zeros of z = f(x, y) on a triple Cartesian product A × B × C, when the sizes |A|, |B|, |C| need not be /the same; the upper bound is O(n11/6) when |A| = |B| = |C| = n, where the constant of proportionality depends on deg f, unless f has one of the aforementioned special forms. This result provides a unified tool for improving bounds in various Erdo″s-type problems in geometry and additive combinatorics. Several applications of our results to problems of these kinds are presented. For example, we show that the number of distinct distances between n points lying on a constant-degree parametric algebraic curve which does not contain a line, in any dimension, is (n4/3), extending the result of Pach and de Zeeuw [19] and improving the bound of Charalambides [2], for the special case where the curve under consideration has a polynomial parameterization. We also derive improved lower bounds for several variants of the sum-product problem in additive combinatorics.

AB - In this paper we characterize real bivariate polynomials which have a small range over large Cartesian products. We show that for every constant-degree bivariate real polynomial f, either |f(A;B)| = Ω(n4/3), for every pair of finite sets A,B,⊂ ℝ, with |A| = |B| = n (where the constant of proportionality depends on deg f), or else f must be of one of the special forms f(u, v) = h(φ (u)·ψ (v)), or f(u, v) = h(ψ (u) · ψ (v)), for some univariate polynomials φ ψ ; h over ℝ. This significantly improves a result of Elekes and Rónyai [7]. Our results are cast in a more general form, in which we give an upper bound for the number of zeros of z = f(x, y) on a triple Cartesian product A × B × C, when the sizes |A|, |B|, |C| need not be /the same; the upper bound is O(n11/6) when |A| = |B| = |C| = n, where the constant of proportionality depends on deg f, unless f has one of the aforementioned special forms. This result provides a unified tool for improving bounds in various Erdo″s-type problems in geometry and additive combinatorics. Several applications of our results to problems of these kinds are presented. For example, we show that the number of distinct distances between n points lying on a constant-degree parametric algebraic curve which does not contain a line, in any dimension, is (n4/3), extending the result of Pach and de Zeeuw [19] and improving the bound of Charalambides [2], for the special case where the curve under consideration has a polynomial parameterization. We also derive improved lower bounds for several variants of the sum-product problem in additive combinatorics.

KW - Combinatorial geometry

KW - Incidences

KW - Polynomials

UR - http://www.scopus.com/inward/record.url?scp=84904430726&partnerID=8YFLogxK

U2 - 10.1145/2582112.2582150

DO - 10.1145/2582112.2582150

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AN - SCOPUS:84904430726

SN - 9781450325943

T3 - Proceedings of the Annual Symposium on Computational Geometry

SP - 251

EP - 260

BT - Proceedings of the 30th Annual Symposium on Computational Geometry, SoCG 2014

PB - Association for Computing Machinery

Y2 - 8 June 2014 through 11 June 2014

ER -