TY - GEN
T1 - Polynomials Vanishing on Cartesian Products
T2 - 31st International Symposium on Computational Geometry, SoCG 2015
AU - Raz, Orit E.
AU - Sharir, Micha
AU - De Zeeuw, Frank
PY - 2015/6/1
Y1 - 2015/6/1
N2 - Let F ε ℂ[x, y, z] be a constant-degree polynomial, and let A,B,C ⊆ ℂ with |A| = |B| = |C| = n. We show that F vanishes on at most O(n11/6) points of the Cartesian product A×B ×C (where the constant of proportionality depends polynomially on the degree of F), unless F has a special group-related form. This improves a theorem of Elekes and Szabó [2], and generalizes a result of Raz, Sharir, and Solymosi [9]. The same statement holds over R. When A,B,C have different sizes, a similar statement holds, with a more involved bound replacing O(n11/6). This result provides a unified tool for improving bounds in various Erdos-type problems in combinatorial geometry, and we discuss several applications of this kind.
AB - Let F ε ℂ[x, y, z] be a constant-degree polynomial, and let A,B,C ⊆ ℂ with |A| = |B| = |C| = n. We show that F vanishes on at most O(n11/6) points of the Cartesian product A×B ×C (where the constant of proportionality depends polynomially on the degree of F), unless F has a special group-related form. This improves a theorem of Elekes and Szabó [2], and generalizes a result of Raz, Sharir, and Solymosi [9]. The same statement holds over R. When A,B,C have different sizes, a similar statement holds, with a more involved bound replacing O(n11/6). This result provides a unified tool for improving bounds in various Erdos-type problems in combinatorial geometry, and we discuss several applications of this kind.
KW - Combinatorial geometry
KW - Incidences
KW - Polynomials
UR - http://www.scopus.com/inward/record.url?scp=84958166190&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.SOCG.2015.522
DO - 10.4230/LIPIcs.SOCG.2015.522
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AN - SCOPUS:84958166190
T3 - Leibniz International Proceedings in Informatics, LIPIcs
SP - 522
EP - 536
BT - 31st International Symposium on Computational Geometry, SoCG 2015
A2 - Pach, Janos
A2 - Pach, Janos
A2 - Arge, Lars
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Y2 - 22 June 2015 through 25 June 2015
ER -