| Popis: |
The Chevalley–Warning Theorem is a result on the solution set of a system of polynomial equations f 1 , … , f r in n variables over a finite field F q in the low degree case d ≔ ∑ j = 1 r deg ( f j ) n . In this note we reformulate that result in terms of fibers of the associated polynomial map and, following Heath-Brown, show that something weaker continues to hold when d = n . This result invites a search for homogeneous degree n polynomials in n variables over F q for which the associated polynomial function F q n → F q is not surjective, and we exhibit several families of such polynomials. |
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