On the Complexity of Computing Critical Points with Gröbner Bases

Computing the critical points of a polynomial function $q\in\mathbb{Q}[X_1,\ldots,X_n]$ restricted to the vanishing locus $V\subset\mathbb{R}^n$ of polynomials $f_1,\ldots, f_p\in\mathbb{Q}[X_1,\ldots, X_n]$ 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 Grobner basis algorithms under genericity assumptions on the coefficients of the input polynomials. The main results refine known complexity bounds (which depend on the maximum $D=\max(\deg(f_1),\ldots,\deg(f_p),\deg(q))$) to bounds which depend on the list $(\deg(f_1),\ldots,\deg(f_p),\deg(q))$: we prove that the Grobner basis computation can be performed in $\delta^{O(\log(A)/\log(G))}$ arithmetic operations in $\mathbb{Q}$, where $\delta$ is the algebraic degree of the ideal vanishing on the critical poin...