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

Orit E. Raz; Zvi Shem-Tov

doi:10.1007/s00493-020-4041-0

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

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Abstract

We prove the following statement. Let f ∈ ℝ[x₁,…,x_d], for some d ≥ 3, and assume that f depends non-trivially in each of x₁,…, x_d. Then one of the following holds.(i)For every finite sets A₁,…, A_d ⊂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), p_i(x),…,p_d(x). This generalizes the results from [2,5,7], which treat the cases d = 2 and d = 3.

Original language	English
Pages (from-to)	721-748
Number of pages	28
Journal	Combinatorica
Volume	40
Issue number	5
DOIs	https://doi.org/10.1007/s00493-020-4041-0
State	Published - Nov 2020
Externally published	Yes

All Science Journal Classification (ASJC) codes

Discrete Mathematics and Combinatorics
Computational Mathematics

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10.1007/s00493-020-4041-0

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@article{28b7a782e45840908750479a98001bca,

title = "Expanding Polynomials: A Generalization of the Elekes-R{\'o}nyai Theorem to d Variables",

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.",

author = "Raz, {Orit E.} and Zvi Shem-Tov",

note = "Publisher Copyright: {\textcopyright} 2020, J{\'a}nos Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg.",

year = "2020",

month = nov,

doi = "10.1007/s00493-020-4041-0",

language = "الإنجليزيّة",

volume = "40",

pages = "721--748",

journal = "Combinatorica",

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publisher = "Janos Bolyai Mathematical Society",

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TY - JOUR

T1 - Expanding Polynomials

T2 - A Generalization of the Elekes-Rónyai Theorem to d Variables

AU - Raz, Orit E.

AU - Shem-Tov, Zvi

PY - 2020/11

Y1 - 2020/11

N2 - 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.

AB - 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.

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U2 - 10.1007/s00493-020-4041-0

DO - 10.1007/s00493-020-4041-0

M3 - مقالة

SN - 0209-9683

VL - 40

SP - 721

EP - 748

JO - Combinatorica

JF - Combinatorica

IS - 5

ER -

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

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