Geodesic
Diagonal Equations Over Large Finite Fields
Canadian journal of mathematics, Tome 36 (1984) no. 2, pp. 249-262

Voir la notice de l'article provenant de la source Cambridge University Press

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 d 1,..., 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. 
Small, Charles. Diagonal Equations Over Large Finite Fields. Canadian journal of mathematics, Tome 36 (1984) no. 2, pp. 249-262. doi: 10.4153/CJM-1984-016-6
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