Integrability at infinity of negative powers of polynomials in the plane and its application to convergence of Dirichlet series
Volume 266 / 2022
Studia Mathematica 266 (2022), 55-79
MSC: Primary 42B20, 30B50; Secondary 26D10, 14P99.
DOI: 10.4064/sm210211-19-11
Published online: 2 May 2022
Abstract
Let $f: \mathbb {R}^2 \rightarrow \mathbb {R}$ be a polynomial function in two variables of the form $$ f(x,y) = a_0y^D + a_1(x)y^{D-1} + \cdots + a_D(x),$$ where $D = \deg f$ is the degree of $f$. Assume that $f(x,y) \not = 0$ for all $(x,y) \in \mathbb {R}^2$. We study the set of all positive $\varepsilon $ for which $$\int _{\mathbb {R}^2}|f(x,y)|^{-\varepsilon }\,dx\,dy \lt \infty .$$ We provide some conditions under which this set can be expressed in natural ways using the Newton polygon of $f$. As a consequence, we consider Dirichlet series associated with polynomials in two variables. We describe the domain of convergence in terms of Newton polygons of polynomials defining the series.
