Article contents
Diagonal Equations Over Large Finite Fields
Published online by Cambridge University Press: 20 November 2018
Extract
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
- Copyright
- Copyright © Canadian Mathematical Society 1984
References
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20190725034118487-0188:S0008414X00021647:S0008414X00021647_inline1.gif?pub-status=live)
- 10
- Cited by