Computations in the neighbourhood of algebraic singularities

It is known that finite precision versus exact computation is a crucial issue only when the computation takes place in the neighbourhood of a singularity. In such a situation, it is essential to know the distance to singularity. Many attention has been dedicated to the relationship between the distance to singularity δ and the condition number K of the problem under study. The well-known Turing theorem states that, for a linear system Ax = b the distance to singularity, in a normwise measure, is the reciprocal of the normwise condition number ||A −1 |||A|| In this Paper, we examine the possibility of extending this theorem for nonlinear problems in the neighbourhood of algebraic singularities. After reviewing the literature on that topic ([Demmel 1987, 1990], [Shub and Smale 1992]), we propose and check on the computer a conjecture which makes more explicit Demmel's bounds on the distance to singularity.