Electronic Resource
Springer
Mathematical notes
17 (1975), S. 142-147
ISSN:
1573-8876
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract Let Nα denote the number of solutions to the congruence F(xi,..., xm) ≡ 0 (mod pα) for a polynomial F(xi,..., xm) with integral p-adic coefficients. We examine the series $$\varphi (t) = \sum\nolimits_{\alpha = 0}^\infty {N_{\alpha ^{t^\alpha } } } $$ . called the Poincaré series for the polynomial F. In this work we prove the rationality of the series ϕ(t) for a class of isometrically equivalent polynomials of m variables, m ≥ 2, containing the sum of two forms ϕn(x, y) + ϕn+1(x, y) respectively of degrees n and n+1, n ≥ 2. In particular the Poincaré series for any third degree polynomial F3(x, y) (over the set of unknowns) with integral p-adic coefficients is a rational function of t.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01161870
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