Resumen de On the bias of cubic polynomials

David Kazhdan, Tamar Ziegler

• Let V be a vector space over a finite field k=Fq of dimension N. For a polynomial P:V→k we define the bias b~1(P) to be b~1(P)=|∑v∈Vψ(P(v))|qN where ψ:k→C⋆ is a non-trivial additive character. A. Bhowmick and S. Lovett proved that for any d≥1 and c>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=∑ri=1QiRi 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 .