Igusa's local zeta functions of semiquasihomogeneous polynomials (Q2716135)

The author considers Igusa's local zeta function NEWLINE\[NEWLINE Z(f,s) = \int_{{\mathcal O}^n_K} |f(x)|^s_K |\text{ d} x|, \quad s\in{\mathbb C}, \Re(s)>0, NEWLINE\]NEWLINE associated to \(f\in K[x]\), \(x=(x_1, \dots,x_n)\), with \(K\) a non-Archimedean local field and \({\mathcal O}_K\) the ring of integers of \(K\). In the main result of the paper he proves that if \(F(x) \in K[x]\) is a semiquasihomogeneous polynomial whose homogeneous part \(f(x)\) has weight \(d\) and exponents \(\alpha_1\), \dots, \(\alpha_n\), then NEWLINE\[NEWLINE Z(F,s) = { {L(q^{-s})} \over {(1-q^{-1}q^{-s})(1-q^{-|\alpha|}q^{-ds})} } , NEWLINE\]NEWLINE where \(\alpha = (\alpha_1, \dots,\alpha_n)\) and the polynomial \(L(q^{-s})\) can be computed effectively. The proof is based on Igusa's stationary phase formula [cf. \textit{J.-I. Igusa}, in Algebraic geometry and its applications, Conf. Purdue Univ. 1990, Springer-Verlag, 175-194 (1994; Zbl 0904.11035)] and Néron \(\pi\)-desingularization.