Representations of fundamental groups of 3-manifolds into PGL(3,C): Exact computations in low complexity

In this paper we are interested in computing representations of the fundamental group of a 3-manifold into PSL(3;C) (in particular in PSL(2;C); PSL(3;R) and PU(2; 1)). The representations are obtained by gluing decorated tetrahedra of flags. We list complete computations (giving 0-dimensional or 1-dimensional solution sets) for the first complete hyperbolic non-compact manifolds with finite volume which are obtained gluing less than three tetrahedra with a description of the computer methods used to find them

Solving rank-constrained semidefinite programs in exact arithmetic

We consider the problem of minimizing a linear function over an affine section of the cone of positive semidefinite matrices, with the additional constraint that the feasible matrix has prescribed rank. When the rank constraint is active, this is a non-convex optimization problem, otherwise it is a semidefinite program. Both find numerous applications especially in systems control theory and combinatorial optimization, but even in more general contexts such as polynomial optimization or real algebra. While numerical algorithms exist for solving this problem, such as interior-point or Newton-like algorithms, in this paper we propose an approach based on symbolic computation. We design an exact algorithm for solving rank-constrained semidefinite programs, whose complexity is essentially quadratic on natural degree bounds associated to the given optimization problem: for subfamilies of the problem where the size of the feasible matrix is fixed, the complexity is polynomial in the number of variables. The algorithm works under assumptions on the input data: we prove that these assumptions are generically satisfied. We also implement it in Maple and discuss practical experiments.Comment: Published at ISSAC 2016. Extended version submitted to the Journal of Symbolic Computatio

On the Complexity of Solving Zero-Dimensional Polynomial Systems via Projection

Given a zero-dimensional polynomial system consisting of n integer polynomials in n variables, we propose a certified and complete method to compute all complex solutions of the system as well as a corresponding separating linear form l with coefficients of small bit size. For computing l, we need to project the solutions into one dimension along O(n) distinct directions but no further algebraic manipulations. The solutions are then directly reconstructed from the considered projections. The first step is deterministic, whereas the second step uses randomization, thus being Las-Vegas. The theoretical analysis of our approach shows that the overall cost for the two problems considered above is dominated by the cost of carrying out the projections. We also give bounds on the bit complexity of our algorithms that are exclusively stated in terms of the number of variables, the total degree and the bitsize of the input polynomials

Parts of Quantum States

It is shown that generic N-party pure quantum states (with equidimensional subsystems) are uniquely determined by their reduced states of just over half the parties; in other words, all the information in almost all N-party pure states is in the set of reduced states of just over half the parties. For N even, the reduced states in fewer than N/2 parties are shown to be an insufficient description of almost all states (similar results hold when N is odd). It is noted that Real Algebraic Geometry is a natural framework for any analysis of parts of quantum states: two simple polynomials, a quadratic and a cubic, contain all of their structure. Algorithmic techniques are described which can provide conditions for sets of reduced states to belong to pure or mixed states.Comment: 10 pages, 1 figur

Algorithms for zero-dimensional ideals using linear recurrent sequences

Inspired by Faug\`ere and Mou's sparse FGLM algorithm, we show how using linear recurrent multi-dimensional sequences can allow one to perform operations such as the primary decomposition of an ideal, by computing the annihilator of one or several such sequences.Comment: LNCS, Computer Algebra in Scientific Computing CASC 201

Symbolic Methods for Solving Algebraic Systems of Equations and Applications for Testing the Structural Stability

International audienceIn this work, we provide an overview of the classical symbolic techniques for solving algebraic systems of equations and show the interest of such techniques in the study of some problems in dynamical system theory, namely testing the structural stability of multidimensional systems

A Special Homotopy Continuation Method For A Class of Polynomial Systems

A special homotopy continuation method, as a combination of the polyhedral homotopy and the linear product homotopy, is proposed for computing all the isolated solutions to a special class of polynomial systems. The root number bound of this method is between the total degree bound and the mixed volume bound and can be easily computed. The new algorithm has been implemented as a program called LPH using C++. Our experiments show its efficiency compared to the polyhedral or other homotopies on such systems. As an application, the algorithm can be used to find witness points on each connected component of a real variety

Sparse Rational Univariate Representation

International audienceWe present explicit worst case degree and height bounds for the rational univariate representation of the isolated roots of polynomial systems based on mixed volume. We base our estimations on height bounds of resultants and we consider the case of 0-dimensional, positive dimensional, and parametric polynomial systems