On a~hypothesis on Poincar\'e series

G. I. Gusev

Matematičeskie zametki, Tome 14 (1973) no. 3, pp. 453-463.

Résumé

Let $F(x_1,\dots,x_m)$ ($m\ge1$) be a polynomial with integral $p$-adic coefficients, and let $N_\alpha$, be the number of solutions of the congruence $F(x_1,\dots,x_m)\equiv0\pmod{p^\alpha}$ proof is given that the Poincaré series $\Phi(t)=\sum_{\alpha=0}^\infty N_\alpha t^\alpha$ is rational for a class of isometrically-equivalent polynomials of $m$ variables ($m\ge2$) containing a form of degree $n\ge2$ of two variables.