Expanding Polynomials: A Generalization of the Elekes-Rónyai Theorem to d Variables

Orit E. Raz*, Zvi Shem-Tov

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

We prove the following statement. Let f ∈ ℝ[x1,…,xd], for some d ≥ 3, and assume that f depends non-trivially in each of x1,…, xd. Then one of the following holds.(i)For every finite sets A1,…, Ad ⊂ℝ, each of size n, we have (Formula presented.) with constant of proportionality that depends on deg f.(ii)f is of one of the forms (Formula presented.) or (Formula presented.) for some univariate real polynomials h(x), pi(x),…,pd(x). This generalizes the results from [2,5,7], which treat the cases d = 2 and d = 3.

Original languageAmerican English
Pages (from-to)721-748
Number of pages28
JournalCombinatorica
Volume40
Issue number5
DOIs
StatePublished - Nov 2020
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2020, János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg.

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