On the bias of cubic polynomials

David Kazhdan; Tamar Ziegler

doi:10.1007/s00029-017-0358-y

On the bias of cubic polynomials

David Kazhdan^*, Tamar Ziegler

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Let V be a vector space over a finite field k= F_q of dimension N. For a polynomial P: V→ k we define the biasb~ ₁(P) to be (Formula presented.) Nwhere ψ: k→ C^⋆ is a non-trivial additive character. A. Bhowmick and S. Lovett proved that for any d≥ 1 and cCloseSPigtSPi 0 there exists r= r(d, c) such that any polynomial P of degree d with b~ ₁(P) ≥ c can be written as a sum P=∑i=1rQiRi where Q_i, R_i: V→ k are non constant polynomials. We show the validity of a modified version of the converse statement for the case d= 3.

Original language	English
Pages (from-to)	511-520
Number of pages	10
Journal	Selecta Mathematica, New Series
Volume	24
Issue number	1
DOIs	https://doi.org/10.1007/s00029-017-0358-y
State	Published - 1 Mar 2018

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Publisher Copyright:
© 2017, Springer International Publishing AG.

Keywords

11T55

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10.1007/s00029-017-0358-y

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AU - Kazhdan, David

AU - Ziegler, Tamar

PY - 2018/3/1

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N2 - Let V be a vector space over a finite field k= Fq of dimension N. For a polynomial P: V→ k we define the biasb~ 1(P) to be (Formula presented.) Nwhere ψ: k→ C⋆ is a non-trivial additive character. A. Bhowmick and S. Lovett proved that for any d≥ 1 and cCloseSPigtSPi 0 there exists r= r(d, c) such that any polynomial P of degree d with b~ 1(P) ≥ c can be written as a sum P=∑i=1rQiRi where Qi, Ri: V→ k are non constant polynomials. We show the validity of a modified version of the converse statement for the case d= 3.

AB - Let V be a vector space over a finite field k= Fq of dimension N. For a polynomial P: V→ k we define the biasb~ 1(P) to be (Formula presented.) Nwhere ψ: k→ C⋆ is a non-trivial additive character. A. Bhowmick and S. Lovett proved that for any d≥ 1 and cCloseSPigtSPi 0 there exists r= r(d, c) such that any polynomial P of degree d with b~ 1(P) ≥ c can be written as a sum P=∑i=1rQiRi where Qi, Ri: V→ k are non constant polynomials. We show the validity of a modified version of the converse statement for the case d= 3.

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JO - Selecta Mathematica, New Series

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On the bias of cubic polynomials

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