• Ultrametric root counting, Vol 36, No.4

Avendaño, Martín, Math Department, Texas A&M University, College Station, TX 77843 (avendano@math.tamu.edu), and Ibrahim, Ashraf, Math Department, Texas A&M University, College Station, TX 77843 (aibrahim@math.tamu.edu). 
Ultrametric root counting, pp. 1011-1022.
ABSTRACT. Let K be a complete non-archimedian field with respect to a discrete valuation, f be a polynomial with coefficients in K and non-zero discriminant, A the valuation ring of K, and M the maximal ideal of A. The first main result of this paper is a reformulation of Hensel's Lemma that connects the number of roots of f with the number of roots of its reduction modulo a power of M. We then define a condition - regularity - that yields a simple method to compute the exact number of roots of f in K. In particular, we show that regularity implies that the number of roots of f equals the sum of the numbers of roots of certain binomials derived from the Newton polygon.

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Ultrametric root counting, Vol 36, No.4

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