On the solvability of bilinear equations in finite fields

Abstract

We consider the equation ab + cd = λ, a ε A, b ε B, c ε c, d ε D over a finite field 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 Hart and Iosevich, who, in particular, prove that if $ A # B # C # D ≥ C q3 ,$ for some absolute constant C > 0, then above equation has a solution for any q*. Here we show that using bounds of multiplicative character sums allows us to extend the class of sets which satisfy this property. © 2008 Glasgow Mathematical Journal Trust.