相关论文: Puiseux power series solutions for systems of equa…

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…

We consider discrete best approximation problems in the setting of tropical algebra, which is concerned with the theory and application of algebraic systems with idempotent operations. Given a set of input--output pairs of an unknown…

We show how to solve explicitly an equation satisfied by a real function belonging to certain general quasianalytic classes. Examples of the classes under consideration are the collection of convergent generalised power series, a class of…

We establish general weak majorization inequalities, relating the leading exponents of the eigenvalues of matrices or matrix polynomials over the field of Puiseux series with the tropical analogues of eigenvalues. We also show that these…

We introduce new discrete best approximation problems, formulated and solved in the framework of tropical algebra, which deals with semirings and semifields with idempotent addition. Given a set of samples, each consisting of the input and…

Tropical Differential Algebraic Geometry considers difficult or even intractable problems in Differential Equations and tries to extract information on their solutions from a restricted structure of the input. The Fundamental Theorem of…

The problem of solving tropical linear systems, a natural problem of tropical mathematics, has already proven to be very interesting from the algorithmic point of view: it is known to be in $NP\cap coNP$ but no polynomial time algorithm is…

A large family of linear, usually overdetermined, systems of partial differential equations that admit a multiplication of solutions, i.e, a bi-linear and commutative mapping on the solution space, is studied. This family of PDE's contains…

Let $Y=\{f(x,y)=0\}$ be the germ of an irreducible plane curve. We present an algorithm to obtain polynomials, whose valuations coincide with the semigroup generators of $Y$. These polynomials are obtained sequentially, adding terms to the…

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…

A novel method of summation for power series is developed. The method is based on the self-similar approximation theory. The trick employed is in transforming, first, a series expansion into a product expansion and in applying the…

In this work we present a power series method for solving ordinary and partial differential equations. To demonstrate our method we solve a system of ordinary differential equations describing the movement of a random walker on a…

Reciprocal space methods for solving Poisson's equation for finite charge distributions are investigated. Improvements to previous proposals are presented, and their performance is compared in the context of a real-space density functional…

Already for bivariate tropical polynomials, factorization is an NP-Complete problem. In this paper, we give an efficient algorithm for factorization and rational factorization of a rich class of tropical polynomials in $n$ variables.…

A relationship between Puiseux series satisfying an ordinary differential equation corresponding to a polynomial dynamical system and degrees of irreducible invariant algebraic curves is studied. A bound on the degrees of irreducible…

The goal in this preprint is to give an efficient algorithm to compute Puiseux expansions at cusps of X0(N). It is based on a relation with a hypergeometric function that holds for any N.

The well-known Newton-Puiseux Theorem states that each real branch of a planar real analytic curve admits a Puiseux expansion. We generalize this result to characteristic orbit of an isolated singularity of a planar real analytic vector…

We illustrate an efficient new method for handling polynomial systems with degenerate solution sets. In particular, a corollary of our techniques is a new algorithm to find an isolated point in every excess component of the zero set (over…

Though Mahler equations have been introduced nearly one century ago, the study of their solutions is still a fruitful topic for research. In particular, the Galois theory of Mahler equations has been the subject of many recent papers.…

We introduce an efficient way, called Newton algorithm, to study arbitrary ideals in C[[x,y]], using a finite succession of Newton polygons. We codify most of the data of the algorithm in a useful combinatorial object, the Newton tree. For…