Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9939-1983-0687622-8
MathSciNet review: 687622
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] Z. I. Borevich and I. R. Shafarevich, Number theory, Academic Press, New York, 1966. MR 0195803 (33:4001)
  • [2] J. Igusa, Complex powers and asymptotic expansions. II, J. Reine Angew. Math 278/279 (1975). 307-321. MR 0404215 (53:8018)
  • [3] -, Some observations on higher degree characters, Amer. J. Math. 99 (1977), 393-417. MR 0441933 (56:324)
  • [4] -, Lectures on forms of higher degree, Tata Institute Notes, Springer-Verlag, Berlin and New York, 1978. MR 546292 (80m:10020)
  • [5] D. Knuth, The art of computer programming, Vol. 1: Fundamental algorithms, 2nd ed., Addison-Wesley, Reading, Mass., 1973. MR 0378456 (51:14624)
  • [6] D. Meuser, On the rationality of certain generating functions, Math. Ann. 256 (1981), 303-310. MR 626951 (83g:12015)
  • [7] E. Stevenson, The rationality of the Poincaré series of a diagonal form, Acta Arith. (to appear).

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 11T99, 11E45, 11E76

Retrieve articles in all journals with MSC: 11T99, 11E45, 11E76


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0687622-8
Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society