Tight Bounds for Chevalley--warning--ax--katz Type Estimates, with Improved Applications
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چکیده
In 1935, C. Chevalley proved a conjecture by E. Artin: if F ðX1; . . . ; XmÞ is a homogeneous polynomial of total degree d < m over a 7nite 7eld Fq having q 1⁄4 p elements, then F has a non-trivial zero. Chevalley showed the result to hold even when the homogeneity hypothesis is replaced by the weaker requirement that F ðX1; . . . ; XmÞ has no constant term. Almost immediately, E. Warning showed that the characteristic of the 7eld divides the number of zeros. A proof of the Chevalley-Warning theorem can be found in [2] and [9]. In [2], Ax followed Dwork’s ideas [4] and developed a p-adic technique which revealed previously unseen solutions. He proved that if b is equal to dm=de 1, where dae is the smallest integer larger than or equal to a, then the number of zeros of F is divisible by q. Ax’s results were based on an estimate of the p-divisibility of the exponential sum associated to F . Using the principle of inclusion and exclusion, Ax extended his result to a system of polynomials. Let FiðX1; . . . ; XmÞ; for i 1⁄4 1; . . . ; r; be a collection of polynomials over Fq of degree di respectively. Set 1⁄4 m Pr i1⁄41 di Pr i1⁄41 di :
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تاریخ انتشار 2004