61 research outputs found

    Critical Points and Gr\"obner Bases: the Unmixed Case

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    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 of effective real algebraic geometry. Experimentally, it has been observed that Gr\"obner basis algorithms are efficient to compute such points. Therefore, recent software based on the so-called Critical Point Method are built on Gr\"obner bases engines. Let f1,...,fpf_1,..., f_p be polynomials in \Q[x_1,..., x_n] of degree DD, V\subset\C^n be their complex variety and π1\pi_1 be the projection map (x1,..,xn)x1(x_1,.., x_n)\mapsto x_1. The critical points of the restriction of π1\pi_1 to VV are defined by the vanishing of f1,...,fpf_1,..., f_p and some maximal minors of the Jacobian matrix associated to f1,...,fpf_1,..., f_p. Such a system is algebraically structured: the ideal it generates is the sum of a determinantal ideal and the ideal generated by f1,...,fpf_1,..., f_p. We provide the first complexity estimates on the computation of Gr\"obner bases of such systems defining critical points. We prove that under genericity assumptions on f1,...,fpf_1,..., f_p, the complexity is polynomial in the generic number of critical points, i.e. Dp(D1)np(n1p1)D^p(D-1)^{n-p}{{n-1}\choose{p-1}}. More particularly, in the quadratic case D=2, the complexity of such a Gr\"obner basis computation is polynomial in the number of variables nn and exponential in pp. We also give experimental evidence supporting these theoretical results.Comment: 17 page

    Separation of periods of quartic surfaces

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    The periods of a quartic surface X are complex numbers obtained by integrating a holomorphic 2-form over 2-cycles in X. For any quartic X defined over the algebraic numbers and any 2-cycle G in X, we give a computable positive number c(G) such that the associated period A satisfies either |A| > c(G) or A = 0. This makes it possible in principle to certify a part of the numerical computation of the Picard group of quartics and to study the Diophantine properties of periods of quartics

    On the complexity of computing real radicals of polynomial systems

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    International audienceLet f= (f1, ..., fs) be a sequence of polynomials in Q[X1,...,Xn] of maximal degree D and V⊂ Cn be the algebraic set defined by f and r be its dimension. The real radical re associated to f is the largest ideal which defines the real trace of V . When V is smooth, we show that re , has a finite set of generators with degrees bounded by V. Moreover, we present a probabilistic algorithm of complexity (snDn )O(1) to compute the minimal primes of re . When V is not smooth, we give a probabilistic algorithm of complexity sO(1) (nD)O(nr2r) to compute rational parametrizations for all irreducible components of the real algebraic set V ∩ Rn. Experiments are given to show the efficiency of our approaches

    Segre Class Computation and Practical Applications

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    Let XYX \subset Y be closed (possibly singular) subschemes of a smooth projective toric variety TT. We show how to compute the Segre class s(X,Y)s(X,Y) as a class in the Chow group of TT. Building on this, we give effective methods to compute intersection products in projective varieties, to determine algebraic multiplicity without working in local rings, and to test pairwise containment of subvarieties of TT. Our methods may be implemented without using Groebner bases; in particular any algorithm to compute the number of solutions of a zero-dimensional polynomial system may be used
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