0 Fe b 20 06 THE MODULO p AND p 2 CASES OF THE IGUSA CONJECTURE ON EXPONENTIAL SUMS AND THE MOTIVIC OSCILLATION INDEX
نویسنده
چکیده
— We prove the modulo p and modulo p 2 cases of the Igusa conjecture on exponential sums. This conjecture predicts specific uniform bounds in the homogeneous polynomial case of exponential sums modulo p m when p and m vary. We introduce the motivic oscillation index of a polynomial and prove the possibly stronger, analogue bounds for m = 1, 2 using this index instead of the bounds of the conjecture. The modulo p 2 case of our bounds holds for all polynomials; the modulo p case holds for homogeneous polynomials and under extra conditions also for non-homogeneous polynomials. We prove natural lower bounds for the motivic oscillation index by using results of Segers. We also prove that, for p big enough, Igusa's local zeta function has a nontrivial pole when one can expect so. This implies that the mo-tivic oscillation index is nontrivial whenever one can expect so. Finally, we introduce a new invariant of a polynomial f , the flaw of f , which we esteem important for a better understanding of Igusa's conjecture.
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