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Abstract
l. Borevich & l. R. Shafarevich conjectured the rationality of the Poincaré series ∑n≥0CnUn where C0= 1 y Cn (≥1) denotes the number ofsolutions ofthe reduction modulo ℓr, ℓe a rational prime, of a polynomial H(t) € Z,[t], t= (t1…,t).This conjecture was settled in the affrrmative by J. lgusa, using Hironalca's deep resolution of singularities teorem. Later on, J. Denef produced a new proof of this result, essentially using the fact that Q, admits elimination of quantifiers, avoiding thus Hironalca's result. The same conjecture for characteristic > O is still an open problem, and none of the techniques used in characteristic O seem to help in characteristic > O. In this short note we prove the conjecture in characteristic > O for sorne special cases of polynomials, using elementary methods.
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References
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