On the solvability of bilinear equations in finite fields

Igor E. Shparlinski

doi:10.1017/S0017089508004382

On the solvability of bilinear equations in finite fields

Igor E. Shparlinski

School of Computing

Research output: Contribution to journal › Article › peer-review

26 Citations (Scopus)

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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.

Original language	English
Pages (from-to)	523-529
Number of pages	7
Journal	Glasgow Mathematical Journal
Volume	50
Issue number	3
DOIs	https://doi.org/10.1017/S0017089508004382
Publication status	Published - Sept 2008

Bibliographical note

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

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title = "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.",

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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.

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On the solvability of bilinear equations in finite fields

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