Finding at least one point in each connected component of a real algebraicset defined by a single equation

Deciding efficiently the emptiness of a real algebraic set defined by a sin gle equation is a fundamental problem of computational real algebraic geome try. We propose an algorithm for this test. We find, when the algebraic set is non empty, at least one point on each semi-algebraically connected comp onent. The problem is reduced to deciding the existence of real critical po ints of the distance function and computing them. (C) 2000 Academic Press.