MATHEMATICS OF COMPUTATION Volume 70, Number 233, Pages 329–335 S 0025-5718(00)00934-0 Article electronically published on July 10, 2000 A CONDITION NUMBER THEOREM FOR UNDERDETERMINED POLYNOMIAL SYSTEMS J ´ ER ˆ OME D ´ EGOT Abstract. The condition number of a numerical problem measures the sen- sitivity of the answer to small changes in the input. In their study of the complexity of B´ ezout’s theorem, M. Shub and S. Smale prove that the condi- tion number of a polynomial system is equal to the inverse of the distance from this polynomial system to the nearest ill-conditioned one. Here we explain how this result can be extended to underdetermined systems of polynomials (that is with less equations than unknowns). 1. Introduction The condition number of a numerical problem measures the sensitivity of the answer to small changes in the input. We call the problem ill-posed if its condition number is infinite. For a given problem, a condition number theorem asserts that the condition number μ is equal to the inverse of the distance of that problem to the set Σ of ill-posed ones. A first example of such a theorem is due to Eckart and Young [4] about the problem of matrix inversion. See Demmel [3] and Dedieu [2] for a general study concerning condition number theorems for various numerical problems like: matrix inversion, computing eigenvalues and eigenvectors, finding zeroes of polynomials, pole assignment in linear control, ... In the case of polynomial systems with the same number of equations as un- knowns, Shub and Smale [6] have proved a condition number theorem. We will give a direct and elementary proof of this theorem, using various properties of Bombieri’s scalar product. This allows us to extend their result to the case of underdetermined polynomial systems (that is with less equations than unknowns). 2. Background Here, we deal only with homogeneous polynomials. Extensions to the affine case are generally trivial, fixing the first variable to 1. Let H d denote the linear space of all homogeneous polynomial systems P = (P 1 ,...,P m ): C n+1 → C m , where each P i is a homogeneous polynomial of n + 1- variables, of degree d i and d =(d 1 ,...,d m ). Remark 1. We denote by n + 1 the number of variable, because the first variable z 0 is added to homogenize an ordinary affine polynomial system C n → C m . Received by the editor August 13, 1996. 2000 Mathematics Subject Classification. Primary 65H10. c 2000 American Mathematical Society 329