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Diagonal Equations Over Large Finite Fields

Published online by Cambridge University Press:  20 November 2018

Charles Small
Charles Small*
Affiliation:
Queen's University, Kingston, Ontario
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We consider polynomials of the form

with non-zero coefficients ai in a finite field F. For any finite extension field K ⊇ F, let fk:Kn → K be the mapping defined by f. We say f is universal over K if fK is surjective, and f is isotropic over K if fK has a non-trivial “kernel“; the latter means fK(X) = 0 for some 0 ≠ x ∊ Kn.

We show (Theorem 1) that f is universal over K provided |K| (the cardinality of K) is larger than a certain explicit bound given in terms of the exponents d1,…, dn. The analogous fact for isotropy is Theorem 2.

It should be noted that in studying diagonal equations

we fix both the number of variables n and the exponents di, and ask how large the field must be to guarantee a solution.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 36 , Issue 2 , 01 April 1984 , pp. 249 - 262
Copyright
Copyright © Canadian Mathematical Society 1984

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