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On the degree of a local zeta function. (English) Zbl 0627.14020
Let $$K$$ be a number field, let $$R$$ be its ring of integers, and let $$\{$$ $$v\}$$ be the set of finite places of $$K$$. For a finite place v of K, denote by $$|.|_ v$$ the associated absolute value on $$K$$, denote by $$K_ v$$ the completion of K with respect to $$|.|_ v,$$ denote by $$R_ v$$ the ring of integers of $$K_ v$$, and denote by $$k_ v$$ the residue field of $$R_ v$$. Then $$k_ v$$ is a finite field with $$q_ v$$ elements. For a homogeneous polynomial $$f(x)=f(x_ 1,...,x_ n)\in K[x_ 1,...,K_ n]$$ of degree $$m$$ and a finite $$place\quad v,$$ consider $$Z(t)=\int_{R^ n_ v}| f(x)|^ s_ v| dx|_ v\quad$$ where $$s\in {\mathbb{C}}$$ with $$Re(s)>0$$ and $$t=q_ v^{- s}$$. Due to Igusa, $$Z(t)$$ is a rational function of t. Writing $$Z(t)=P(t)/Q(t)$$, we define $$\deg Z(t) = \deg P(t)-\deg Q(t).$$
The author proves that if f is non-degenerate with respect to its Newton polyhedron then the degree $$\deg(Z(t))=-m$$ for almost all finite places v of K. This establishes a conjecture of Igusa for “generic” homogeneous polynomials.
Reviewer: W.Lütkebohmert
##### MSC:
 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 11S40 Zeta functions and $$L$$-functions
##### Keywords:
degree of zeta function
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##### References:
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