Gröbner basis computation of Drazin inverses with multivariate rational function entries

Abstract

In this paper we show how to apply Gröbner bases to compute the Drazin inverse of a matrix with multivariate rational functions as entries. When the coefficients of the rational functions depend on parameters, we give sufficient conditions for the Drazin inverse to specialize properly. In addition, we extend the method to weighted Drazin inverses. We present an empirical analysis that shows a good timing performance of the method.

The Rees Algebra of a monomial plane parametrization

“…Here, m p (C) denotes the multiplicity of p with respect to C. This definition is not precise at all, and we refer the reader to [CA00, Sections 4.1 and 4.8] for a correct statement of this concept. Adjoint curves are of importance in computational algebra due to their use in the inverse of the implicitization problem, see for instance [SWP08] and the references therein.…”

Computation of Moore–Penrose generalized inverses of matrices with meromorphic function entries

“…In [23], elimination theory techniques are applied to compute the Drazin inverse of matrices with rational function entries. For the case of the Moore-Penrose inverse of a matrix one may try to proceed similarly.…”

“…Also, contributions to the case of polynomial matrices can be found [11], [12], [13], [26]. In addition, the symbolic treatment of the problem of computing generalized inverses have been addressed in [19], [23], [24], [25]. Some of the papers mentioned above provide characterizations to ensure the existence of the Moore-Penrose inverse of a given matrix with entries over different types of rings or over a field.…”

Moving Curve Ideals of Rational Plane Parametrizations

“…This assumption is not a big one, due to the fact that generic parametrizations are generically injective, and moreover, thanks to Luröth's theorem (see [vdW66]), every parametrization φ as in (1) can be factorized as φ = φ • P, with φ : P 1 → P 2 generically injective, and P : P 1 → P 1 being a map defined by a pair of coprime homogeneous polynomial both of them having degree β. One can then regard φ as a "reparametrization" of C, and there are very efficient algorithms to deal with this problem, see for instance [SWP08].…”

“…But not all algebraic curve is rational, and moreover, if the curve has degree 3 or more, a generic algebraic curve will not be rational. Being rational or not is actually a geometric property of the curve,and one should not expect to detect it from the form of the defining polynomial, see [SWP08] for algorithms to decide whether a given polynomial G(X 0 , X 1 , X 2 ) defines a rational curve or not. For instance, the Folium of Descartes (see Figure 3) is a rational curve with parametrization (t 0 : t 1 ) → (t 3 0 + t 3 1 : 3t 2 0 t 1 : 3t 0 t 2 1 ), and implicit equation given by the polynomial F (X 0 , X 1 , X 2 ) = X 3 1 + X 3 2 − 3X 0 X 1 X 2 .…”