2020/08/26 by Kien Huu Nguyen, Nguyen, Kien Huu, Willem Veys +1
Mathematics · #Algebraic Geometry and Number Theory #Advanced Mathematical Identities #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.2008.11637
Let f be a polynomial in n variables over some number field and Z a\nsubscheme of affine n-space. The notion of motivic oscillation index of f\nat Z was initiated by Cluckers (2008) and Cluckers-Musta ct va-Nguyen\n(2019). In this paper we elaborate on this notion and raise several questions.\nThe first one is stability under base field extension; this question is linked\nto a deep understanding of the density of non-archimedean local fields over\nwhich Igusa's local zeta functions of f has a pole with given real part. The\nsecond one is around Igusa's conjecture for exponential sums with bounds in\nterms of the motivic oscillation index. Thirdly, we wonder if the above\nquestions only depend on the analytic isomorphism class of singularities. By\nusing various techniques as the GAGA theorem, resolution of singularities and\nmodel theory, we can answer the third question up to a base field extension.\nNext, by using a transfer principle between non-archimedean local fields of\ncharacteristic zero and positive characteristic, we can link all three\nquestions with a conjecture on weights of \ℓ-adic cohomology groups of\nArtin-Schreier sheaves associated to jet polynomials. This way, we can answer\nall questions positively if f is a polynomial of Thom-Sebastiani type with\nnon-rational singularities. As a consequence, we prove Igusa's conjecture for\narbitrary polynomials in three variables and polynomials with singularities of\nADE type. In an appendix, we answer affirmatively a recent question of\nCluckers-Musta ct va-Nguyen (2019) on poles of twisted Igusa's local zeta\nfunctions of maximal order.\n