61 research outputs found
Critical Points and Gr\"obner Bases: the Unmixed Case
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 be polynomials in \Q[x_1,..., x_n] of degree
, V\subset\C^n be their complex variety and be the projection map
. The critical points of the restriction of
to are defined by the vanishing of and some maximal minors
of the Jacobian matrix associated to . Such a system is
algebraically structured: the ideal it generates is the sum of a determinantal
ideal and the ideal generated by . 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
, the complexity is polynomial in the generic number of critical
points, i.e. . More particularly, in the
quadratic case D=2, the complexity of such a Gr\"obner basis computation is
polynomial in the number of variables and exponential in . We also give
experimental evidence supporting these theoretical results.Comment: 17 page
Separation of periods of quartic surfaces
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
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
Let be closed (possibly singular) subschemes of a smooth
projective toric variety . We show how to compute the Segre class
as a class in the Chow group of . 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 . 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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