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Matematicheskie Zametki, 1973, Volume 14, Issue 3, Pages 453–463 (Mi mzm7275)  

On a hypothesis on Poincaré series

G. I. Gusev

Saratov State University
Abstract: 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.
Received: 04.07.1972
English version:
Mathematical Notes, 1973, Volume 14, Issue 3, Pages 817–822
DOI: https://doi.org/10.1007/BF01147462
Bibliographic databases:
UDC: 511
Language: Russian
Citation: G. I. Gusev, “On a hypothesis on Poincaré series”, Mat. Zametki, 14:3 (1973), 453–463; Math. Notes, 14:3 (1973), 817–822
Citation in format AMSBIB
\Bibitem{Gus73}
\by G.~I.~Gusev
\paper On a~hypothesis on Poincar\'e series
\jour Mat. Zametki
\yr 1973
\vol 14
\issue 3
\pages 453--463
\mathnet{http://mi.mathnet.ru/mzm7275}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=330111}
\zmath{https://zbmath.org/?q=an:0288.10009}
\transl
\jour Math. Notes
\yr 1973
\vol 14
\issue 3
\pages 817--822
\crossref{https://doi.org/10.1007/BF01147462}
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