## Critical Points and Gröbner Bases: the Unmixed Case (2012)

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@MISC{Faugère12criticalpoints,

author = {Jean-charles Faugère and Mohab Safey El Din and Pierre-jean Spaenlehauer},

title = {Critical Points and Gröbner Bases: the Unmixed Case},

year = {2012}

}

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### Abstract

We consider the problem of computing critical points of the restriction of a polynomial map to an algebraic variety. This is of first importance since the global minimum of such a map is reached at a critical point. Thus, these points appear naturally in non-convex polynomial optimization which occurs in a wide range of scientific applications (control theory, chemistry, economics,...). Critical points also play a central role in recent algorithms ofeffectiverealalgebraicgeometry. Experimentally, it has been observed that Gröbner basis algorithms are efficient to compute such points. Therefore, recent software based on the so-called Critical Point Method are built on Gröbner bases engines. Let f1,...,fp be polynomials in Q[x1,...,xn] of degree D, V ⊂ C n be their complex variety and π1 be the projection map (x1,...,xn) ↦ → x1. Thecriticalpointsoftherestrictionofπ1to V are defined by the vanishing of f1,...,fp and some maximal minors of the Jacobian matrix associated to f1,...,fp. Suchasystemisalgebraicallystructured:theidealitgenerates is the sum of a determinantal ideal and the ideal generated by f1,...,fp. We provide the first complexity estimates on the computation of Gröbner bases of such systems defining critical points. We prove that under genericity assumptions on f1,...,fp, thecomplexityis polynomial in the generic number of critical points, i.e. D p (D − 1) n−p () n−1.Moreparticularly,inthe p−1 quadratic case D =2,thecomplexityofsuchaGröbnerbasiscomputationispolynomial in the number of variables n and exponential in p. We also give experimental evidence supporting these theoretical results.