Sonia L. Rueda

Approximate parametrization of plane algebraic curves by linear systems of curves

It is well known that an irreducible algebraic curve is rational (i.e. parametric) if and only if its genus is zero. In this paper, given a tolerance @e>0 and an @e-irreducible algebraic affine plane curve C of proper degree d, we introduce the notion of @e-rationality, and we provide an algorithm to parametrize approximately affine @e-rational plane curves by means of linear systems of (d-2)-degree curves. The algorithm outputs a rational parametrization of a rational curve C@? of degree d which has the same points at infinity as C. Moreover, although we do not provide a theoretical analysis, our empirical analysis shows that C@? and C are close in practice.

Linear complete differential resultants and the implicitization of linear DPPEs

The linear complete differential resultant of a finite set of linear ordinary differential polynomials is defined. We study the computation by linear complete differential resultants of the implicit equation of a system of n linear differential polynomial parametric equations in n-1 differential parameters. We give necessary conditions to ensure properness of the system of differential polynomial parametric equations.

An algorithm to parametrize approximately space curves

This is the author’s version of a work that was accepted for publication in Journal of Symbolic Computation. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Symbolic Computation vol. 56 pp. 80-106 (2013). DOI: 10.1016/j.jsc.2013.04.002

Finite dimensional representations of invariant differential operators

Let k be an algebraically closed field of characteristic 0, Y = k r × (k × ) s , and let G be an algebraic torus acting diagonally on the ring of algebraic differential operators V(Y). We give necessary and sufficient conditions for D(V) G to have enough simple finite dimensional representations, in the sense that the intersection of the kernels of all the simple finite dimensional representations is zero. As an application we show that if K → GL(V) is a representation of a reductive group K and if zero is not a weight of a maximal torus of K on V, then D(V) K has enough finite dimensional representations. We also construct examples of FCR-algebras with any integer GK dimension ≥ 3.

On the performance of the approximate parametrization algorithm for curves

In Perez-Diaz et al. (2009) [5], the authors present an algorithm to parametrize approximately @e-rational curves, and they show that the Hausdorff distance, w.r.t. the Euclidean distance, between the input and output curves is finite. In this paper, we analyze this distance for a family of curves randomly generated and we empirically find a reasonable upper bound of the Hausdorff distance between each input and output curve of the family.

On the Computation of Differential Resultants

The definition of the differential resultant of a set of ordinary differential polynomials is reviewed and its computation via determinants is revisited, using a modern language. This computation is also extended to differential homogeneous resultants of homogeneous ordinary differential polynomials. A numeric example is included and an example of the application of elimination theory to biological modelling is revisited, in terms of differential resultants.

Rational Hausdorff divisors: A new approach to the approximate parametrization of curves

In this paper, we introduce the notion of rational Hausdorff divisor, we analyze the dimension and irreducibility of its associated linear system of curves, and we prove that all irreducible real curves belonging to the linear system are rational and are at finite Hausdorff distance among them. As a consequence, we provide a projective linear subspace where all (irreducible) elements are solutions of the approximate parametrization problem for a given algebraic plane curve. Furthermore, we identify the linear system with a plane curve that is shown to be rational and we develop algorithms to parametrize it analyzing its fields of parametrization. Therefore, we present a generic answer to the approximate parametrization problem. In addition, we introduce the notion of Hausdorff curve, and we prove that every irreducible Hausdorff curve can always be parametrized with a generic rational parametrization having coefficients depending on as many parameters as the degree of the input curve.

Differential elimination by differential specialization of Sylvester style matrices

Differential resultant formulas are defined, for a system P of n ordinary Laurent differential polynomials in n−1 differential variables. These are determinants of coefficient matrices of an extended system of polynomials obtained from P through derivations and multiplications by Laurent monomials. To start, through derivations, a system ps(P) of L polynomials in L− 1 algebraic variables is obtained, which is non sparse in the order of derivation. This enables the use of existing formulas for the computation of algebraic resultants, of the multivariate sparse algebraic polynomials in ps(P), to obtain polynomials in the differential elimination ideal generated by P. The formulas obtained are multiples of the sparse differential resultant defined by Li, Yuan and Gao, and provide order and degree bounds in terms of mixed volumes in the generic case.

An approximate algorithm to parametrize algebraic curves

We provide approximate algorithms to solve the approximate parametrization problem for planar and spacial real algebraic curves. The solution we give to this problem is a global one, under the requirement that the Hausdorff distance between input and output curves is finite. A theoretical reasoning has also been developed to describe linear systems of curves whose elements are solutions to the approximate problem for plane curves. Methods to estimate and certify that the Hausdorff distance between the input and output curves is small (with respect to tolerance) are also given.

Linear differential implicitization and differential resultants

Given a system P of n linear ordinary differential polynomial parametric equations (linear DPPEs) in n-1 differential parameters, we proved in [2] that if nonzero a differential resultant gives the implicit equation of P. Differential resultants often vanish under specialization, which prevented us from giving an implicitization algorithm in [2]. Motivated by Cannys method and its generalizations we consider now a linear perturbation of P and use it to give an algorithm to decide if the dimension of the implicit ideal of P is n-1 and in the affirmative case obtain the implicit equation of P. This poster presentation will contain this development together with examples illustrating the results. An extended version of this work can be found in [1].