Full Rank Representation of Real Algebraic Sets and Applications

Abstract

We introduce the notion of the full rank representation of a real algebraic set, which represents it as the projection of a union of real algebraic manifolds VR(Fi) of Rm, m ≥ n, such that the rank of the Jacobian matrix of each Fi at any point of VR(Fi) is the same as the number of polynomials in F:i. By introducing an auxiliary variable, we show that a squarefree regular chain T can be transformed to a new regular chain C having various nice properties, such as the Jacobian matrix of C attains full rank at any point of VR(C). Based on a symbolic triangular decomposition approach and a numerical critical point technique, we present a hybrid algorithm to compute a full rank representation. As an application, we show that such a representation allows to better visualize plane and space curves with singularities. Effectiveness of this approach is also demonstrated by computing witness points of polynomial systems having rank-deficient Jacobian matrices.