On the bias of cubic polynomials

• David Kazhdan ^[1] ; Tamar Ziegler ^[1]
  1. [1] Hebrew University of Jerusalem

     Hebrew University of Jerusalem

     Israel

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 24, Nº. 1, 2018, págs. 511-520
• Idioma: inglés
• Enlaces
  • Texto completo (pdf)
• Resumen

  • 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 .