Good reduction of puiseux series and complexity of the Newton-Puiseux algorithm over finite fields

In [12], we sketched a numeric-symbolic method to compute Puiseux series with floating point coefficients. In this paper, we address the symbolic part of our algorithm. We study the reduction of Puiseux series coefficients modulo a prime ideal and prove a good reduction criterion sufficient to preserve the required information, namely Newton polygon trees. We introduce a convenient modification of Newton polygons that greatly simplifies proofs and statements of our results. Finally, we improve complexity bounds for Puiseux series calculations over finite fields, and estimate the bit-complexity of polygon tree computation.

[1]  Joachim von zur Gathen,et al.  Modern Computer Algebra , 1998 .

[2]  William Fulton,et al.  Hurwitz Schemes and Irreducibility of Moduli of Algebraic Curves , 1969 .

[3]  Adrien Poteaux,et al.  Computing monodromy groups defined by plane algebraic curves , 2007, SNC '07.

[4]  Victor Shoup,et al.  A computational introduction to number theory and algebra , 2005 .

[5]  Adrien Poteaux,et al.  Towards a Symbolic-Numeric Method to Compute Puiseux Series: The Modular Part , 2008, ArXiv.

[6]  Victor Shoup.  A Computational Introduction to Number Theory and Algebra: Finite fields , 2005 .

[7]  Bernard Deconinck,et al.  Computing the Abel map , 2008 .

[8]  H. T. Kung,et al.  All Algebraic Functions Can Be Computed Fast , 1978, JACM.

[9]  D. Duval.  Rational Puiseux expansions , 1989 .

[10]  Henri Cohen,et al.  A course in computational algebraic number theory , 1993, Graduate texts in mathematics.

[11]  Bernard Deconinck,et al.  Computing Riemann matrices of algebraic curves , 2001 .

[12]  Bernard Dwork,et al.  On natural radii of $p$-adic convergence , 1979 .

[13]  C. Chevalley,et al.  Introduction to the theory of algebraic functions of one variable , 1951 .

[14]  Barry M. Trager,et al.  Integration of algebraic functions , 1984 .

[15]  J. W. Bruce,et al.  LE PROBLÈME DES MODULES POUR LES BRANCHES PLANES , 1988 .

[16]  Mark van Hoeij,et al.  An Algorithm for Computing an Integral Basis in an Algebraic Function Field , 1994, J. Symb. Comput..

[17]  P. G. Walsh,et al.  A polynomial-time complexity bound for the computation of the singular part of a Puiseux expansion of an algebraic function , 2000, Math. Comput..

[18]  Martin Eichler,et al.  Introduction to the Theory of Algebraic Numbers and Functions , 1966 .