Practical and Theoretical Issues for the Computation of Generalized Critical Values of a Polynomial Mapping

Let f ∈ Q[X1, . . . , Xn] be a polynomial of degree D. Computing the set of generalized critical values of the mapping e f : x ∈ C → f(x) ∈ C (i.e. {c ∈ C | ∃(xk)k∈N f(xk) → c and ||xk||.||dxkf || → 0 when k → ∞}) is an important step in algorithms computing sampling points in semi-algebraic sets defined by a single inequality. A previous algorithm allows us to compute the set of generalized critical values of e f . This one is based on the computation of the critical locus of a projection on a plane P . This plane P must be chosen such that some global properness properties of some projections are satisfied. These properties, which are generically satisfied, are difficult to check in practice. Moreover, choosing randomly the plane P induces a growth of the coefficients appearing in the computations. We provide here a new certified algorithm computing the set of generalized critical values of e f . This one is still based on the computation of the critical locus on a plane P . The certification process consists here in checking that this critical locus has dimension 1 (which is easy to check in practice), without any assumption of global properness. Moreover, this allows us to limit the growth of coefficients appearing in the computations by choosing a plane P defined by sparse equations. Additionally, we prove that the degree of this critical curve is bounded by (D−1)n−1−d where d is the sum of the degrees of the positive dimensional components of the ideal 〈 ∂f ∂X1 , . . . , ∂f ∂Xn 〉. We also provide complexity estimates on the number of arithmetic operations performed by a probabilistic version of our algorithm. Practical experiments at the end of the paper show the relevance and the importance of these results which improve significantly in practice previous contributions.