6 research outputs found
Elimination for generic sparse polynomial systems
We present a new probabilistic symbolic algorithm that, given a variety
defined in an n-dimensional affine space by a generic sparse system with fixed
supports, computes the Zariski closure of its projection to an l-dimensional
coordinate affine space with l < n. The complexity of the algorithm depends
polynomially on combinatorial invariants associated to the supports.Comment: 22 page
Homotopy techniques for solving sparse column support determinantal polynomial systems
Let be a field of characteristic zero with
its algebraic closure. Given a sequence of polynomials
and a
polynomial matrix , with , we are interested in determining the isolated
points of , the algebraic set of points in
at which all polynomials in and all
-minors of vanish, under the assumption .
Such polynomial systems arise in a variety of applications including for
example polynomial optimization and computational geometry. We design a
randomized sparse homotopy algorithm for computing the isolated points in
which takes advantage of the determinantal
structure of the system defining . Its complexity
is polynomial in the maximum number of isolated solutions to such systems
sharing the same sparsity pattern and in some combinatorial quantities attached
to the structure of such systems. It is the first algorithm which takes
advantage both on the determinantal structure and sparsity of input
polynomials. We also derive complexity bounds for the particular but important
case where and the columns of satisfy weighted degree
constraints. Such systems arise naturally in the computation of critical points
of maps restricted to algebraic sets when both are invariant by the action of
the symmetric group
Computing critical points for invariant algebraic systems
Let be a field and , in
be multivariate polynomials (with )
invariant under the action of , the group of permutations of
. We consider the problem of computing the points at which
vanish and the Jacobian matrix associated to is
rank deficient provided that this set is finite. We exploit the invariance
properties of the input to split the solution space according to the orbits of
. This allows us to design an algorithm which gives a triangular
description of the solution space and which runs in time polynomial in ,
and where is the maximum degree of the
input polynomials. When are fixed, this is polynomial in while when
is fixed and this yields an exponential speed-up with respect
to the usual polynomial system solving algorithms
Homotopy algorithms for solving structured determinantal systems
Multivariate polynomial systems arising in numerous applications have special structures. In particular, determinantal structures and invariant systems appear in a wide range of applications such as in polynomial optimization and related questions in real algebraic geometry. The goal of this thesis is to provide efficient algorithms to solve such structured systems.
In order to solve the first kind of systems, we design efficient algorithms by using the symbolic homotopy continuation techniques. While the homotopy methods, in both numeric and symbolic, are well-understood and widely used in polynomial system solving for square systems, the use of these methods to solve over-detemined systems is not so clear. Meanwhile, determinantal systems are over-determined with more equations than unknowns. We provide probabilistic homotopy algorithms which take advantage of the determinantal structure to compute isolated points in the zero-sets of determinantal systems. The runtimes of our algorithms are polynomial in the sum of the multiplicities of isolated points and the degree of the homotopy curve. We also give the bounds on the
number of isolated points that we have to compute in three contexts: all entries of the input are in classical polynomial rings, all these polynomials are sparse, and they are weighted polynomials.
In the second half of the thesis, we deal with the problem of finding critical points of a symmetric polynomial map on an invariant algebraic set. We exploit the invariance properties of the input to split the solution space according to the orbits of the symmetric group. This allows us to design an algorithm which gives a triangular description of the solution space and which runs in time polynomial in the number of points that we have to compute. Our results are illustrated by applications in studying real algebraic sets defined by invariant polynomial systems by the means of the critical point method