TY - JOUR

T1 - On the solvability of bilinear equations in finite fields

AU - Shparlinski, Igor E.

N1 - Copyright 2008 Cambridge University Press. Article originally published in Glasgow Mathematical Journal, Volume 50, Issue 3, pp. 523-529. The original article can be found at http://dx.doi.org/10.1017/S0017089508004382

PY - 2008/9

Y1 - 2008/9

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=50849084466&partnerID=8YFLogxK

U2 - 10.1017/S0017089508004382

DO - 10.1017/S0017089508004382

M3 - Article

AN - SCOPUS:50849084466

VL - 50

SP - 523

EP - 529

JO - Glasgow Mathematical Journal

JF - Glasgow Mathematical Journal

SN - 0017-0895

IS - 3

ER -