Height zeta functions on generalized projective toric varieties (Q2918492)

Let \(V\) be a generalized projective toric variety in the sense of \textit{B. Sturmfels} [Proc. Symp. Pure Math. 62, Part 2, 437--449 (1997; Zbl 0914.14022)], defined over \(\mathbb Z\). Let \(T(x_0,\dots, x_n)\) be a homogeneous generalized polynomial (that is, exponents of the monomials can be arbitrary nonnegative real numbers) of degree \(d\) with positive coefficients and suppose that \(0\not\in T(\mathbb R^{n+ 1}_+\setminus \{0\})\). One defines a height NEWLINE\[NEWLINEH_G: \mathbb P^n(\mathbb Q)\to \mathbb R_+,\quad H_T: x\mapsto T(|x_0|,\dots,|x_n|)^{1/d},NEWLINE\]NEWLINE where \(x:=[x_0,\dots, x_n]\), \(x_i\in\mathbb Z\) for \(0\leq i\leq n\) and \((x_0,\dots, x_n)= (1)\). The corresponding height zeta-function is given by NEWLINE\[NEWLINEZ(H_T; U,s):= \sum_{x\in U} H_T(x)^{-s},NEWLINE\]NEWLINE where \(U\) is the maximal torus in \(V\). The author meromorphically continues the function \(s\mapsto Z(H_T; V, s)\) beyond the domain of convergence of its defining Dirichlet series and, as a corollary, obtains an asymptotic formula for the number of points of bounded height in \(U(\mathbb Q)\).NEWLINENEWLINEFor the entire collection see [Zbl 1242.11004].