Good reduction of Puiseux series and applications

We have designed a new symbolic–numeric strategy for computing efficiently and accurately floating point Puiseux series defined by a bivariate polynomial over an algebraic number field. In essence, computations modulo a well-chosen prime number pp are used to obtain the exact information needed to guide floating point computations. In this paper, we detail the symbolic part of our algorithm. First of all, we study modular reduction of Puiseux series and give a good reduction criterion for ensuring that the information required by the numerical part is preserved. To establish our results, we introduce a simple modification of classical Newton polygons, that we call “generic Newton polygons”, which turns out to be very convenient. Finally, we estimate the size of good primes obtained with deterministic and probabilistic strategies. Some of these results were announced without proof at ISSAC’08.

► Generic Newton polygons, a better tool for computing Puiseux series of algebraic curves.
► A reduction criterion for preserving properties of the curve modulo a prime number.
► Bounds for the size of a good prime for deterministic and probabilistic algorithms.
► The symbolic part of a symbolic–numeric method for computing Puiseux series and invariants.