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This article considers the problem of solving a system of $n$ real polynomialequations in $n+1$ variables. We propose an algorithm based on Newton's methodand subdivision for this problem. Our algorithm is intended only fornondegenerate cases, in which case the solution is a 1-dimensional curve. Ourfirst main contribution is a definition of a condition number measuringreciprocal distance to degeneracy that can distinguish poor and wellconditioned instances of this problem. (Degenerate problems would be infinitelyill conditioned in our framework.) Our second contribution, which is the mainnovelty of our algorithm, is an analysis showing that its running time isbounded in terms of the condition number of the problem instance as well as $n$and the polynomial degrees.

A condition number analysis of an algorithm for solving a system of polynomial equations with one degree of freedom