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- W2083832120 abstract "To any polynomial f∈ℂ[x 1,…,x n ]∖ℂ with f(0)=0 one associates the well-known motivic zeta function and its specialization to the level of Hodge polynomials. These zeta functions can be given very explicitly in terms of an embedded resolution of f −1{0} in 𝔸ℂ n . In this paper, where we work with polynomials in three variables, i.e., n=3, we find a geometric condition for having a nonzero contribution to the residue at a candidate pole. More precisely, for a given embedded resolution h we fix an exceptional surface E with h(E)={0}, which induces in a canonical way a candidate pole q of the motivic zeta function. Then we prove that, when the surface E is non-rational and we are in a generic situation, the maximality of the logarithmic Kodaira dimension of E ŝ implies the non-vanishing of the contribution of E to the residue at q. Here E ŝ denotes the part of E that doesn't belong to any other irreducible component of h −1(f −1{0}). The same result is already true on the level of Hodge polynomials." @default.
- W2083832120 created "2016-06-24" @default.
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- W2083832120 date "2003-10-01" @default.
- W2083832120 modified "2023-10-06" @default.
- W2083832120 title "Logarithmic Kodaira dimension and the poles of the Hodge and motivic zeta functions for surfaces" @default.
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- W2083832120 doi "https://doi.org/10.1007/s00229-003-0399-8" @default.
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