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Newton polyhedra and solutions of congruences. (English) Zbl 0601.12023

Diophantine analysis, Proc. Number Theory Sect. Aust. Math. Soc. Conv., Univ. New South Wales 1985, Lond. Math. Soc. Lect. Note Ser. 109, 67-82 (1986).
[For the entire collection see Zbl 0583.00005.]
The classical Newton polygon for functions in one variable over a field with \(p\)-adic valuation is generalized to a Newton polyhedron for functions of two variables. From the polyhedron corresponding to a polynomial f information about the valuations of (the infinitude of) zeros of \(f\) can be obtained. For instance, the minimum value of the valuation of a zero of \(f\) can be obtained. The authors state a conjecture about the occurrence of a common zero of two polynomials, which may be deduced from correspondences between the Newton polyhedra of both polynomials. A special case of this conjecture is proved by the authors.
Reviewer: F.J.van der Linden

MSC:

11S05 Polynomials
12J25 Non-Archimedean valued fields

Citations:

Zbl 0583.00005