A polynomial-time complexity bound for the computation of the singular part of a Puiseux expansion of an algebraic function

Walsh, P.

doi:10.1090/S0025-5718-00-01246-1

A polynomial-time complexity bound for the computation of the singular part of a Puiseux expansion of an algebraic function
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Math. Comp. 69 (2000), 1167-1182 Request permission

Abstract:

In this paper we present a refined version of the Newton polygon process to compute the Puiseux expansions of an algebraic function defined over the rational function field. We determine an upper bound for the bit-complexity of computing the singular part of a Puiseux expansion by this algorithm, and use a recent quantitative version of Eisenstein’s theorem on power series expansions of algebraic functions to show that this computational complexity is polynomial in the degrees and the logarithm of the height of the polynomial defining the algebraic function.

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