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
SN - 0017-0895
VL - 50
SP - 523
EP - 529
JO - Glasgow Mathematical Journal
JF - Glasgow Mathematical Journal
IS - 3
ER -