TY - JOUR

T1 - The Elekes–Szabó Theorem in four dimensions

AU - Raz, Orit E.

AU - Sharir, Micha

AU - de Zeeuw, Frank

N1 - Publisher Copyright:
© 2018, Hebrew University of Jerusalem.

PY - 2018/8/1

Y1 - 2018/8/1

N2 - Let F ∈ C[x, y, s, t] be an irreducible constant-degree polynomial, and let A,B,C,D ⊂ C be finite sets of size n. We show that F vanishes on at most O(n8/3) points of the Cartesian product A × B × C × D, unless F has a special group-related form. A similar statement holds for A,B,C,D of unequal sizes, with a suitably modified bound on the number of zeros. This is a four-dimensional extension of our recent improved analysis of the original Elekes–Szabó theorem in three dimensions. We give three applications: an expansion bound for three-variable real polynomials that do not have a special form, a bound on the number of coplanar quadruples on a space curve that is neither planar nor quartic, and a bound on the number of four-point circles on a plane curve that has degree at least five.

AB - Let F ∈ C[x, y, s, t] be an irreducible constant-degree polynomial, and let A,B,C,D ⊂ C be finite sets of size n. We show that F vanishes on at most O(n8/3) points of the Cartesian product A × B × C × D, unless F has a special group-related form. A similar statement holds for A,B,C,D of unequal sizes, with a suitably modified bound on the number of zeros. This is a four-dimensional extension of our recent improved analysis of the original Elekes–Szabó theorem in three dimensions. We give three applications: an expansion bound for three-variable real polynomials that do not have a special form, a bound on the number of coplanar quadruples on a space curve that is neither planar nor quartic, and a bound on the number of four-point circles on a plane curve that has degree at least five.

UR - http://www.scopus.com/inward/record.url?scp=85049139647&partnerID=8YFLogxK

U2 - 10.1007/s11856-018-1728-7

DO - 10.1007/s11856-018-1728-7

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AN - SCOPUS:85049139647

SN - 0021-2172

VL - 227

SP - 663

EP - 690

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

IS - 2

ER -