On The Solvability of Bilinear Equations in Finite Fields

2007/08/16 by Igor E. Shparlinski, Shparlinski, Igor E. · 3 citations
Computer Science · Mathematics · #11L40 #11T30 #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.0708.2130

openalex publication_date 2007/08/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the equation ab + cd = λ, a∈ A, b ∈ B, c∈ C, d ∈ D, over a finite field F_q of q elements, with variables from arbitrary sets A, B, C, D ⊆ F_q. The question of solvability of such and more general equations has recently been considered by D. Hart and A. Iosevich, who, in particular, proved that if #A #B #C #D ≫ q³, then above equation has a solution for any λ∈ F_q^*. Here we show that using bounds of multiplicative character sums allows us to extend the class of sets which satisfy this property.

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