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Improvements of the Chevalley-Warning and the Ax-Katz theorems. (English) Zbl 0824.11017

Let \(k\) be a finite field with \(q= p^ f\) elements, and let \(F_ i\) \((1\leq i\leq r)\) be polynomials over \(k\) having \(N\) common zeros in \(k\). The theorems of Chevalley-Warning and Ax-Katz show that \(q^ \mu\mid N\), where \(\mu\) is the smallest integer which is at least as large as \[ {{n- \sum\deg (F_ i)} \over {\max \deg (F_ i)}}. \] This form of the result is due to N. Katz [Am. J. Math. 93, 485-499 (1971; Zbl 0237.12012)]. The present paper establishes an alternative form of the result, which is sometimes stronger, in which \(p^ \lambda \mid N\), where \(\lambda\) is the smallest integer which is at least as large as \[ f \Biggl\{ {{n- \sum w(F_ i)} \over {\max w(F_ i)}} \Biggr\}. \] To define the weight \(w(F)\) one writes \(\sigma (d)\) for the sum of the digits in the expansion of \(d\) in base \(p\). Then \(w\) of a monomial \(\prod x_ i^{d_ i}\) is defined as the sum of \(\sigma (d_ i)\) and \(W(F)\) is defined as the maximum of the weights of the monomials appearing in \(F\). The result is deduced from Katz’ theorem by the principle of restriction of scalars.

MSC:

11D79 Congruences in many variables
11T55 Arithmetic theory of polynomial rings over finite fields

Citations:

Zbl 0237.12012
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