TY - JOUR
T1 - Polynomials vanishing on cartesian products
T2 - The elekes-szabó theorem revisited
AU - Raz, Orit E.
AU - Sharir, Micha
AU - Zeeuw, Frank De
N1 - Publisher Copyright: © 2016.
PY - 2016
Y1 - 2016
N2 - Let F e C[x,y,z] be a constant-degree polynomial, and let A,B,C ⊂ C be finite sets of size n. We show that F vanishes on at most O(n11/6) points of the Cartesian product A × B × C, unless F has a special group-related form. This improves a theorem of Elekes and Szabó and generalizes a result of Raz, Sharir, and Solymosi. The same statement holds over C, and a similar statement holds when A, B, C have different sizes (with a more involved bound replacing O.(n11/6)). This result provides a unified tool for improving bounds in various Erdös-type problems in combinatorial geometry, and we discuss several applications of this kind.
AB - Let F e C[x,y,z] be a constant-degree polynomial, and let A,B,C ⊂ C be finite sets of size n. We show that F vanishes on at most O(n11/6) points of the Cartesian product A × B × C, unless F has a special group-related form. This improves a theorem of Elekes and Szabó and generalizes a result of Raz, Sharir, and Solymosi. The same statement holds over C, and a similar statement holds when A, B, C have different sizes (with a more involved bound replacing O.(n11/6)). This result provides a unified tool for improving bounds in various Erdös-type problems in combinatorial geometry, and we discuss several applications of this kind.
UR - http://www.scopus.com/inward/record.url?scp=85002779002&partnerID=8YFLogxK
U2 - https://doi.org/10.1215/00127094-3674103
DO - https://doi.org/10.1215/00127094-3674103
M3 - مقالة
SN - 0012-7094
VL - 165
SP - 3517
EP - 3566
JO - Duke Mathematical Journal
JF - Duke Mathematical Journal
IS - 18
ER -