On the computation of the topology of plane curves

International audienceLet P be a square free bivariate polynomial of degree at most d and with integer coefficients of bit size at most t. We give a deterministic algorithm for the computation of the topology of the real algebraic curve definit by P, i.e. a straight-line planar graph isotopic to the curve. Our main result is an algorithm for the computation of the local topology in a neighbourhood of each of the singular and critical points of the projection wrt the X axis in $\tilde{O} (d^6 t)$ bit operations where $\tilde{O}$ means that we ignore logarithmic factors in $d$ and $t$. Combined to state of the art sub-algorithms used for computing a Cylindrical Algebraic Decomposition, this result avoids a generic shear and gives a deterministic algorithm for the computation of the topology of the curve in $\tilde{O} (d^6 t + d^7)$ bit operations

Computing the number of roots of the determinant of a polynomial matrix in a región of the complex plane

RESUMEN: El cálculo de las raíces del determinante de una matriz polinomial en una variable es un problema recurrente en Álgebra Computacional. Es bien conocido que su cálculo mediante el desarrollo de dicho determinante trae consigo una explosión combinatoria que produce polinomios de alto grado con coeficientes difíciles de manejar en la práctica. En el trabajo de H. Dym y D. Volok se demuestra como calcular el número de raíces del determinante de dicha matriz polinomial N(λ) en una región del plano complejo en términos de la signatura de una matriz numérica X que es construida a partir de los coeficientes de la matriz N(λ). Puesto que los algoritmos de aproximación de las raíces de una ecuación, en muchas ocasiones, parten de técnicas que calculan el número de raíces en una región determinada y van haciendo esta más pequeña para aproximar cada una de las raíces, nos proponemos aquí entender la demostración del Teorema de H. Dym y D. Volok antes mencionado con el fin de que pueda ser utilizada en el contexto de la separación de las raíces del determinante de una matriz polinomial.ABSTRACT: Computing the roots of the determinant of a polynomial matrix in one variable is a recurrent problem in Symbolic Computation. It is well known that performing this task through the determinant expansion results in a combinatorial explosion that produces high-degree polynomials with huge coe cients that are di cult to handle in practice. In their work, H. Dym and D. Volok prove how to compute the number of zeros of the determinant of the matrix polynomial N(λ) inside a region of the complex plane in terms of the signature of a numerical matrix X that is constructed from the coe cients of N(λ). Since typically, approximation algorithms for finding the roots of an equation often start from techniques that compute the number of roots in a given region and make successive reductions of this region in order to approximate each root, our purpose here is to understand the proof of H. Dym and D. Volok Theorem in order to be used in the context of the separation of the roots of the determinant of the considered polynomial matrix.Máster en Matemáticas y Computació