On The Solvability of Bilinear Equations in Finite Fields

Details On The Solvability of Bilinear Equations in Finite Fields On The Solvability of Bilinear Equations in Finite Fields

2007-08-16

arxiv.org/abs/0708.2130v2

Mathematics

Number Theory

Scientific paper

We consider the equation $$ ab + cd = \lambda, \qquad a\in A, b \in B, c\in C, d \in D, $$ over a finite field $F_q$ of $q$ elements, with variables from arbitrary sets $ A, B, C, D \subseteq 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 $$ #A #B #C #D \gg q^3, $$ then above equation has a solution for any $\lambda \in F_q^*$. 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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