Periods and the asymptotics of a diophantine problem. I (Q1197462)

Let \(P\) and \(\varphi\) be two polynomials of two variables with positive coefficients. One is interested in the asymptotic behaviour (as \(x\to \infty\)) of the sum \(N_ \varphi(x)=\sum_{m\in\mathbb{N}^ 2,P(m)\leq x}\varphi(m)\), where \(\mathbb{N}\) denotes the set of natural numbers. The Dirichlet series \(D_ P(s,\varphi)=\sum_{m\in\mathbb{N}^ 2}(\varphi(m)/P(m)^ s)\) is known to define a meromorphic function of \(s\) with poles \(p_ 0(\varphi)>p_ 1(\varphi)>\dots\) of order \(\leq 2\). Assuming \(p_ 0(\varphi)\) is a simple pole, the author proves that the dominant term in the asymptotic formula for \(N_ \varphi(x)\) is, in a sense, a cohomological invariant for ``generic'' polynomials \(\varphi\). He requires, in particular, that the first poles of \(D_ P(s,\varphi)\) and \(I_ P(s,\varphi)\) and their residues coincide, where \(I_ P(s,\varphi)\) is the meromorphic function defined by the equation \[ I_ P(s,\varphi)=\int_{[1,\infty)^ 2}P(z_ 1,z_ 2)^{-s} \varphi(z_ 1,z_ 2) dz_ 1 dz _ 2. \] The bulk of the paper is devoted to a geometric investigation of this integral, much too complicated to be described here. [For part II see the review below].