If polynomials over a finite field have total degree sum strictly smaller than their number of variables, then the number of their common zeros is divisible by the field characteristic.
Every sequence of elements of , where is prime, contains terms whose sum is zero. A proof applies the Chevalley-Warning theorem to the cardinality and two coordinate sums and then double-counts zero-sum subsequences.
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The Chevalley–Warning theorem is a result in algebraic geometry and number theory that concerns the existence of rational points on certain types of algebraic varieties. More specifically, it deals with the solutions of systems of polynomial equations over finite fields.