Poincar\'e series

G. I. Gusev

Matematičeskie zametki, Tome 17 (1975) no. 2, pp. 245-254.

Résumé

Let $N_\alpha$ denote the number of solutions to the congruence $F(x_i,\dots,x_m)\equiv\pmod{p^\alpha}$ for a polynomial $F(x_i,\dots,x_m)$ with integral $p$-adic coefficients. We examine the series $\varphi(t)=\sum_{\alpha=0}^\infty N_\alpha t^\alpha$ called the Poincaré series for the polynomial $F$. In this work we prove the rationality of the series $\varphi(t)$ for a class of isometrically equivalent polynomials of $m$ variables, $m\ge2$, containing the sum of two forms $\varphi_n(x,y)+\varphi_{n+1}(x,y)$ respectively of degrees $n$ and $n+1$, $n\ge2$. In particular the Poincaré series for any third degree polynomial $F_3(x,y)$ (over the set of unknowns) with integral $p$-adic coefficients is a rational function of $t$.