Section: New Results
Fundamental algorithms and structured polynomial systems
The Berlekamp–Massey–Sakata algorithm and the Scalar-FGLM algorithm both compute the ideal of relations of a multidimensional linear recurrent sequence. Whenever quering a single sequence element is prohibitive, the bottleneck of these algorithms becomes the computation of all the needed sequence terms. As such, having adaptive variants of these algorithms, reducing the number of sequence queries, becomes mandatory. A native adaptive variant of the Scalar-FGLM algorithm was presented by its authors, the so-called Adaptive Scalar-FGLM algorithm. In [3], our first contribution is to make the Berlekamp–Massey–Sakata algorithm more efficient by making it adaptive to avoid some useless relation test-ings. This variant allows us to divide by four in dimension 2 and by seven in dimension 3 the number of basic operations performed on some sequence family. Then, we compare the two adaptive algorithms. We show that their behaviors differ in a way that it is not possible to tweak one of the algorithms in order to mimic exactly the behavior of the other. We detail precisely the differences and the similarities of both algorithms and conclude that in general the Adaptive Scalar-FGLM algorithm needs fewer queries and performs fewer basic operations than the Adaptive Berlekamp–Massey–Sakata algorithm. We also show that these variants are always more efficient than the original algorithms.
The problem of finding
Gröbner bases is one the most powerful tools in algorithmic non-linear
algebra. Their computation is an intrinsically hard problem with a complexity
at least single exponential in the number of variables. However, in most of
the cases, the polynomial systems coming from applications have some kind of
structure. For example , several problems in computer-aided design, robotics,
vision, biology , kinematics, cryptography, and optimization involve sparse
systems where the input polynomials have a few non-zero terms. In
[16], our approach to exploit sparsity is to embed the
systems in a semigroup algebra and to compute Gröbner bases over this algebra.
Up to now, the algorithms that follow this approach benefit from the sparsity
only in the case where all the polynomials have the same sparsity structure,
that is the same Newton polytope. We introduce the first algorithm that
overcomes this restriction. Under regularity assumptions, it performs no
redundant computations. Further, we extend this algorithm to compute Gröbner
basis in the standard algebra and solve sparse polynomials systems over the
torus
In [10], we consider the problem of approximating numerically the moments and the supports of measures which are invariant with respect to the dynamics of continuous- and discrete-time polynomial systems, under semialgebraic set constraints. First, we address the problem of approximating the density and hence the support of an invariant measure which is absolutely continuous with respect to the Lebesgue measure. Then, we focus on the approximation of the support of an invariant measure which is singular with respect to the Lebesgue measure. Each problem is handled through an appropriate reformulation into a linear optimization problem over measures, solved in practice with two hierarchies of finite-dimensional semidefinite moment-sum-of-square relaxations, also called Lasserre hierarchies. Under specific assumptions, the first Lasserre hierarchy allows to approximate the moments of an absolutely continuous invariant measure as close as desired and to extract a sequence of polynomials converging weakly to the density of this measure. The second Lasserre hierarchy allows to approximate as close as desired in the Hausdorff metric the support of a singular invariant measure with the level sets of the Christoffel polynomials associated to the moment matrices of this measure. We also present some application examples together with numerical results for several dynamical systems admitting either absolutely continuous or singular invariant measures.