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Proceedings of the American Mathematical Society

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Numbers of solutions of congruences: Poincaré series for strongly nondegenerate forms

Author: Jay R. Goldman
Journal: Proc. Amer. Math. Soc. 87 (1983), 586-590
MSC: Primary 11T99; Secondary 11E45, 11E76
MathSciNet review: 687622
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Abstract: Let $ p$ be a fixed prime, $ f({x_1}, \ldots ,{x_k})$ a polynomial over $ {{\mathbf{Z}}_p}$, the $ p$-adic integers. $ {c_n}$ the number of solutions of $ f = 0$ over $ {\mathbf{Z}}/{p^n}{\mathbf{Z}}$ and $ {P_f}(t) = \sum\nolimits_{i = 0}^\infty {{c_i}{t^i}} $ the Poincaré series. A general approach to computing $ {c_n}$ and $ {P_f}(t)$ is given and explicit formulas are derived for strongly nondegenerate forms.

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Article copyright: © Copyright 1983 American Mathematical Society