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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Congruences, Trees, and $p$-Adic Integers


Authors: Wolfgang M. Schmidt and C. L. Stewart
Journal: Trans. Amer. Math. Soc. 349 (1997), 605-639
MSC (1991): Primary 11A12, 11S05
DOI: https://doi.org/10.1090/S0002-9947-97-01547-X
MathSciNet review: 1340185
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Abstract: Let $f$ be a polynomial in one variable with integer coefficients, and $p$ a prime. A solution of the congruence $f(x) \equiv 0 (\text {mod} \,p)$ may branch out into several solutions modulo $p^{2}$, or it may be extended to just one solution, or it may not extend to any solution. Again, a solution modulo $p^{2}$ may or may not be extendable to solutions modulo $p^{3}$, etc. In this way one obtains the ``solution tree'' $T = T(f)$ of congruences modulo $p^{\lambda }$ for $\lambda = 1,2,\ldots $. We will deal with the following questions: What is the structure of such solution trees? How many ``isomorphism classes'' are there of trees $T(f)$ when $f$ ranges through polynomials of bounded degree and height? We will also give bounds for the number of solutions of congruences $f(x) \equiv 0 (\text {mod} \,p^{\lambda })$ in terms of $p, \lambda $ and the degree of $f$.


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Additional Information

Wolfgang M. Schmidt
Affiliation: Department of Mathematics, University of Colorado, Boulder, Colorado 80309
Email: schmidt@euclid.Colorado.edu

C. L. Stewart
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Email: cstewart@watserv1.uwaterloo.ca

DOI: https://doi.org/10.1090/S0002-9947-97-01547-X
Received by editor(s): August 30, 1994
Additional Notes: The first author was supported in part by NSF grant DMS–9108581.
The second author was supported in part by Grant A3528 from the Natural Sciences and Engineering Research Council of Canada.
Article copyright: © Copyright 1997 American Mathematical Society