Computing the Dimension of Real Algebraic Sets

Abstract

Let V be the set of real common solutions to F = (f1, …, fs) in ℜ[x1, …;, xn] and D be the maximum total degree of the fi s. We design an algorithm which on input F computes the dimension of V. Letting L be the evaluation complexity of F and s=1, it runs using O∼ (L D n(d+3)+1) arithmetic operations in 𝒬 and at most Dn(d+1) isolations of real roots of polynomials of degree at most Dn. Our algorithm depends on the real geometry of V; its practical behavior is more governed by the number of topology changes in the fibers of some well-chosen maps. Hence, the above worst-case bounds are rarely reached in practice, the factor Dnd being in general much lower on practical examples. We report on an implementation showing its ability to solve problems which were out of reach of the state-of-the-art implementations.