2013/09/09 by Pierre-Jean Spaenlehauer, Spaenlehauer, Pierre-Jean
Computer Science · Engineering · #Advanced Numerical Analysis Techniques #Coding theory and cryptography #FOS: Computer and information sciences #Polynomial and algebraic computation #Symbolic Computation (cs.SC)
paper · doi:10.48550/arxiv.1309.2138
openalex publication_date 2013/09/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Computing the critical points of a polynomial function q∈\mathbb Q[X1,…,Xn] restricted to the vanishing locus V⊂\mathbb Rn of polynomials f1,…, fp∈\mathbb Q[X1,…, Xn] is of first importance in several applications in optimization and in real algebraic geometry. These points are solutions of a highly structured system of multivariate polynomial equations involving maximal minors of a Jacobian matrix. We investigate the complexity of solving this problem by using Gröbner basis algorithms under genericity assumptions on the coefficients of the input polynomials. The main results refine known complexity bounds (which depend on the maximum D=max(deg(f1),…,deg(fp),deg(q))) to bounds which depend on the list of degrees (deg(f1),…,deg(fp),deg(q)): we prove that the Gröbner basis computation can be performed in δO(log(A)/log(G)) arithmetic operations in \mathbb Q, where δ is the algebraic degree of the ideal vanishing on the critical points, and A and G are the arithmetic and geometric average of a multiset constructed from the sequence of degrees. As a by-product, we prove that solving such generic optimization problems with Gröbner bases requires at most DO(n) arithmetic operations in \mathbb Q, which meets the best known complexity bound for this problem. Finally, we illustrate these complexity results with experiments, giving evidence that these bounds are relevant for applications.