A Dimension Series for Multivariate Splines.
In Computer Algebra, Subresultant Theory provides a powerful method to construct algorithms solving problems for polynomials in one variable in an optimal way. So, using this method we can compute the greatest common divisor of two polynomials in one variable with integer coefficients avoiding the exponential growth of the coefficients that will appear if we use the Euclidean Algorithm.In this note, generalizing a forgotten construction appearing in [Hab], we extend the Subresultant Theory to the...
The paper introduces the calculation of a greatest common divisor of two univariate polynomials. Euclid’s algorithm can be easily simulated by the reduction of the Sylvester matrix to an upper triangular form. This is performed by using - transformation and -factorization methods. Both procedures are described and numerically compared. Computations are performed in the floating point environment.
In this work, we propose a new method to find monic irreducible polynomials with integer coefficients, only real roots, and span less than 4. The main idea is to reduce the search of such polynomials to the solution of Integer Linear Programming problems. In this frame, the coefficients of the polynomials we are looking for are the integer unknowns. We give inequality constraints specified by the properties that the polynomials should have, such as the typical distribution of their roots. These...
The main purpose of this note is to show how Sturm-Habicht Sequence can be generalized to the multivariate case and used to compute the number of real solutions of a polynomial system of equations with a finite number of complex solutions. Using the same techniques, some formulae counting the number of real solutions of such polynomial systems of equations inside n-dimensional rectangles or triangles in the plane are presented.
In 2000 A. Alesina and M. Galuzzi presented Vincent’s theorem “from a modern point of view” along with two new bisection methods derived from it, B and C. Their profound understanding of Vincent’s theorem is responsible for simplicity — the characteristic property of these two methods. In this paper we compare the performance of these two new bisection methods — i.e. the time they take, as well as the number of intervals they examine in order to isolate the real roots of polynomials — against that...
We are concerned with solving polynomial equations over rings. More precisely, given a commutative domain A with 1 and a polynomial equation antn + ...+ a0 = 0 with coefficients ai in A, our problem is to find its roots in A.We show that when A = B[x] is a polynomial ring, our problem can be reduced to solving a finite sequence of polynomial equations over B. As an application of this reduction, we obtain a finite algorithm for solving a polynomial equation over A when A is F[x1, ..., xN] or F(x1,...
Dado un polinomio f perteneciente a K[x], determinar si existen otros dos g y h de grado mayor que uno tales que f(x) = g(h(x)) = g o h, y, en caso de que existan, encontrarlos, es conocido como problema de descomposición para polinomios. Cuando dicha descomposición existe, problemas como la evaluación de f en un punto o la resolución de la ecuación f = 0 se pueden resolver de manera más simple. La generalización del problema de la descomposición al caso de funciones racionales es sin duda un problema...
À l’aide du Nullstellensatz effectif, on trouve des bornes inférieure et supérieure explicites des valeurs critiques non nulles d’un polynôme, en termes des coefficients de celui-ci.