Related papers: The Newton-Puiseux algorithm and effective algebra…
We have designed a new symbolic-numeric strategy to compute 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 $p$…
In this paper, an explanation of the Newton-Peiseux algorithm is given. This explanation is supplemented with well-worked and explained examples of how to use the algorithm to find fractional power series expansions for all branches of a…
We deal with the algebraicity of a Puiseux series in terms of the properties of its coefficients. We show that the algebraicity of a Puiseux series for given bounded degree is determined by a finite number of explicit polynomial formulae.…
In this paper we study systems of autonomous algebraic ODEs in several differential indeterminates. We develop a notion of algebraic dimension of such systems by considering them as algebraic systems. Afterwards we apply differential…
Algebraic power series are formal power series which satisfy a univariate polynomial equation over the polynomial ring in n variables. This relation determines the series only up to conjugacy. Via the Artin-Mazur theorem and the implicit…
We give an algorithm to compute term by term multivariate Puiseux series expansions of series arising as local parametrizations of zeroes of systems of algebraic equations at singular points. The algorithm is an extension of Newton's method…
We deal with the algebraicity of an iterated Puiseux series in several variables in terms of the properties of its coefficients. Our aim is to generalize to several variables the results from [HM15]. We show that the algebraicity of such a…
In this paper, we study the algebraic, rational and formal Puiseux series solutions of certain type of systems of autonomous ordinary differential equations. More precisely, we deal with systems which associated algebraic set is of…
A constructive version of Newton-Puiseux theorem for computing the Puiseux expansions of algebraic curves is presented. The proof is based on a classical proof by Abhyankar. Algebraic numbers are evaluated dynamically; hence the base field…
In this paper we outline an algorithmic approach to compute Puiseux series expansions for algebraic surfaces. The series expansions originate at the intersection of the surface with as many coordinate planes as the dimension of the surface.…
We survey and unify recent results on the existence of accurate algorithms for evaluating multivariate polynomials, and more generally for accurate numerical linear algebra with structured matrices. By "accurate" we mean that the computed…
Computing the real solutions to a system of polynomial equations is a challenging problem, particularly verifying that all solutions have been computed. We describe an approach that combines numerical algebraic geometry and sums of squares…
With the aid of Hensel Lemma, we refine the 2-adic Newton polygon algorithm proposed by Magron, Koprowski, and Vaccon at ISSAC 2023 to express computationally a given positive univariate polynomial with rational coefficients as a sum of…
We develop a theory of arithmetic Newton polygons of higher order, that provides the factorization of a separable polynomial over a $p$-adic field, together with relevant arithmetic information about the fields generated by the irreducible…
We use Newton's method to find all roots of several polynomials in one complex variable of degree up to and exceeding one million and show that the method, applied to appropriately chosen starting points, can be turned into an algorithm…
We develop an iterative method to calculate the roots of arbitrary polynomials over the field of Puiseux series including non-separable ones. The method works by transforming the polynomial and its roots into a special form and then…
We develop a collection of numerical algorithms which connect ideas from polyhedral geometry and algebraic geometry. The first algorithm we develop functions as a numerical oracle for the Newton polytope of a hypersurface and is based on…
Formal Laurent-Puiseux series are important in many branches of mathematics. This paper presents a {\it Mathematica} implementation of algorithms developed by the author for converting between certain classes of functions and their…
There exist several methods for computing exact solutions of algebraic differential equations. Most of the methods, however, do not ensure existence and uniqueness of the solutions and might fail after several steps, or are restricted to…
The representation of polynomials by arithmetic circuits evaluating them is an alternative data structure which allowed considerable progress in polynomial equation solving in the last fifteen years. We present a circuit based computation…