Congruences, Trees, and -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 | References | Similar Articles | Additional Information

Abstract: Let be a polynomial in one variable with integer coefficients, and a prime. A solution of the congruence may branch out into several solutions modulo , or it may be extended to just one solution, or it may not extend to any solution. Again, a solution modulo may or may not be extendable to solutions modulo , etc. In this way one obtains the ``solution tree'' of congruences modulo for . We will deal with the following questions: What is the structure of such solution trees? How many ``isomorphism classes'' are there of trees when ranges through polynomials of bounded degree and height? We will also give bounds for the number of solutions of congruences in terms of and the degree of .

**1.**S. Lang,*Algebra*, 2nd ed., Addison-Wesley, 1984. MR**86j:00003****2.**C. L. Stewart,*On the number of solutions of polynomial congruences and Thue equations*, J. Amer. Math. Soc. 4 (1991), 793-835. MR**92j:11032**

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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