Abstract
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
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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References
Deligne, P.: La conjecture de Weil. II. Inst. Hautes Études Sci. Publ. Math. 52, 137–252 (1980)
Bhowmick, A., Lovett, S.: Bias vs structure of polynomials in large fields, and applications in effective algebraic geometry and coding theory. Arxiv:1506.02047
Laumon, G.: Transformation de Fourier, constantes d’quations fonctionnelles et conjecture de Weil. Inst. Hautes Études Sci. Publ. Math. 65, 131–210 (1987)
Laumon, G.: Exponential sums and l-adic cohomology: a survey. Isr. J. Math. 120(part A), 225–257 (2000)
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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). https://doi.org/10.1007/s00029-017-0358-y
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DOI: https://doi.org/10.1007/s00029-017-0358-y