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On the bias of cubic polynomials

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Let V be a vector space over a finite field \(k=\mathbb {F}_q\) of dimension N. For a polynomial \(P:V\rightarrow k\) we define the bias \(\tilde{b}_1(P)\) to be

$$\begin{aligned} \tilde{b}_1(P)=\frac{|\sum _{v\in V}\psi (P(v))|}{q^N} \end{aligned}$$

where \(\psi :k\rightarrow {\mathbb {C}}^\star \) is a non-trivial additive character. A. Bhowmick and S. Lovett proved that for any \(d\ge 1\) and \(c>0\) there exists \(r=r(d,c)\) such that any polynomial P of degree d with \(\tilde{b}_1(P)\ge c\) can be written as a sum \(P=\sum _{i=1}^rQ_iR_i\) where \(Q_i,R_i:V\rightarrow k\) 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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Correspondence to David Kazhdan.

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The Tamar Ziegler is supported by ERC Grant ErgComNum 682150.

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Kazhdan, D., Ziegler, T. On the bias of cubic polynomials. Sel. Math. New Ser. 24, 511–520 (2018).

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