The Elekes–Szabó Theorem in four dimensions

Raz, Orit E.; Sharir, Micha; Zeeuw, Frank de

doi:10.1007/s11856-018-1728-7

Cited by 11 publications

(12 citation statements)

References 24 publications

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“…With this lemma, we can prove that there are not many points for which C q is not a curve, unless F is Cartesian. A very similar fact is proved in [17].…”

Section: Curves and Dual Curvessupporting

confidence: 78%

Schwartz-Zippel bounds for two-dimensional products

Mojarrad¹,

Pham²,

Valculescu³

et al. 2018

Discrete Anal

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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.

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“…With this lemma, we can prove that there are not many points for which C q is not a curve, unless F is Cartesian. A very similar fact is proved in [17].…”

Section: Curves and Dual Curvessupporting

confidence: 78%

Schwartz-Zippel bounds for two-dimensional products

Mojarrad¹,

Pham²,

Valculescu³

et al. 2018

Discrete Anal

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“…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).…”

Section: Main Resultmentioning

confidence: 99%

An o-minimal Szemerédi–Trotter theorem

Basu

Raz

2017

The Quarterly Journal of Mathematics

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“…Analogously, one would expect that the bound in Theorem 4.1 can be improved on, unless the variety is of some special type. The only other case that has so far been studied is D = 4 and dim(X) = 3, for which Raz, Sharir, and De Zeeuw [42] proved the following. The proof of Theorem 4.2 mostly uses the same techniques as that of Theorem 2.1, and the setup turns out to be simpler.…”

Section: Other Variants 41 Longer One-dimensional Productsmentioning

confidence: 99%

A Survey of Elekes-Rónyai-Type Problems

Zeeuw

2018

Bolyai Society Mathematical Studies

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We give an overview of recent progress around a problem introduced by Elekes and Rónyai. The prototype problem is to show that a polynomial f ∈ R[x, y] has a large image on a Cartesian product A×B ⊂ R 2 , unless f has a group-related special form. We discuss this problem and a number of variants and generalizations. This includes the Elekes-Szabó problem, which generalizes the Elekes-Rónyai problem to a question about an upper bound on the intersection of an algebraic surface with a Cartesian product, and curve variants, where we ask the same questions for Cartesian products of finite subsets of algebraic curves.These problems lie at the crossroads of combinatorics, algebra, and geometry: They ask combinatorial questions about algebraic objects, whose answers turn out to have applications to geometric questions involving basic objects like distances, lines, and circles, as well as to sum-product-type questions from additive combinatorics. As part of a recent surge of algebraic techniques in combinatorial geometry, a number of quantitative and qualitative steps have been made within this framework. Nevertheless, many tantalizing open questions remain.

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The Elekes–Szabó Theorem in four dimensions

Cited by 11 publications

References 24 publications

Schwartz-Zippel bounds for two-dimensional products

Schwartz-Zippel bounds for two-dimensional products

An o-minimal Szemerédi–Trotter theorem

A Survey of Elekes-Rónyai-Type Problems

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