On the distance to the zero set of a homogeneous polynomial

In this paper we prove simple estimates relating the value of a complex homogeneous polynomial at a point to the distance of the point to the zero set of the polynomial, and also the distance to the zero set along a projective line to the distance in projective space. Our motivation for doing this was to give a quantitative aspect to the algorithm of W. Zulehner and to relate it to the algorithms of J. Renegar, J. Canny, and M. Kim. © 1989.