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Pairs of Bilinear Equations in a Finite Field

Published online by Cambridge University Press:  20 November 2018

A. Duane Porter
A. Duane Porter*
Affiliation:
University of Wyoming
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Let F = GF(g) be the finite field of q = pr elements, p arbitrary. We wish to consider the system of bilinear equations

1.1

where all coefficients are from F. The number of solutions in F of a single bilinear equation may be obtained from a theorem of John H. Hodges (3, Theorem 3) by properly defining the matrices U, V, A, B. In 1954, L. Carlitz (1) obtained, as a special case of his work on quadratic forms, the number of simultaneous solutions in F of (1.1) when all aj = 1 and p is odd. Carlitz considered the case p = 2 separately.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966 , pp. 561 - 565
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. Carlitz, L., Pairs of quadratic equations in a finite field., Amer. J. Math., 76 (1954), 137–153.Google Scholar
2. Cohen, E., Simultaneous pairs cf linear and quadratic equations in a Galois field, Can. J. Math., 9 (1957), 74–78.Google Scholar
3. Hodges, J. H., Representations by bilinear forms in a finite field, Duke Math. J., 22 (1955), 497–509.Google Scholar