A condition number theorem for underdetermined polynomial systems

The condition number of a numerical problem measures the sensitivity of the answer to small changes in the input. In their study of the complexity of Bézout's theorem, M. Shub and S. Smale prove that the condition 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).