Root Isolation of Zero-dimensional Polynomial Systems with Linear Univariate Representation

In this paper, a linear univariate representation for the roots of a zero-dimensional polynomial equation system is presented, where the roots of the equation system are represented as linear combinations of roots of several univariate polynomial equations. The main advantage of this representation is that the precision of the roots can be easily controlled. In fact, based on the linear univariate representation, we can give the exact precisions needed for roots of the univariate equations in order to obtain the roots of the equation system to a given precision. As a consequence, a root isolation algorithm for a zero-dimensional polynomial equation system can be easily derived from its linear univariate representation.Comment: 19 pages,2 figures; MM-Preprint of KLMM, Vol. 29, 92-111, Aug. 201

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

Evaluation properties of invariant polynomials

AbstractA polynomial invariant under the action of a finite group can be rewritten using generators of the invariant ring. We investigate the complexity aspects of this rewriting process; we show that evaluation techniques enable one to reach a polynomial cost

Homotopy techniques for solving sparse column support determinantal polynomial systems

Let $\mathbf{K}$ be a field of characteristic zero with $\overline{\mathbf{K}}$ its algebraic closure. Given a sequence of polynomials $\mathbf{g} = (g_1, \ldots, g_s) \in \mathbf{K}[x_1, \ldots , x_n]^s$ and a polynomial matrix $\mathbf{F} = [f_{i,j}] \in \mathbf{K}[x_1, \ldots, x_n]^{p \times q}$, with $p \leq q$, we are interested in determining the isolated points of $V_p(\mathbf{F},\mathbf{g})$, the algebraic set of points in $\overline{\mathbf{K}}$ at which all polynomials in $\mathbf{g}$ and all $p$-minors of $\mathbf{F}$ vanish, under the assumption $n = q - p + s + 1$. 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 $V_p(\mathbf{F},\mathbf{g})$ which takes advantage of the determinantal structure of the system defining $V_p(\mathbf{F}, \mathbf{g})$. 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 $\mathbf{g}$ and the columns of $\mathbf{F}$ 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

Root Isolation of Zero-dimensional Polynomial Systems with Linear Univariate Representation

Abstract In this paper, a linear univariate representation for the roots of a zero-dimensional polynomial equation system is presented, where the roots of the equation system are represented as linear combinations of roots of several univariate polynomial equations. The main advantage of this representation is that the precision of the roots can be easily controlled. In fact, based on the linear univariate representation, we can give the exact precisions needed for isolating the roots of the univariate equations in order to obtain the roots of the equation system to a given precision. As a consequence, a root isolation algorithm for a zero-dimensional polynomial equation system can be easily derived from its linear univariate representation

Bit complexity for multi-homogeneous polynomial system solving Application to polynomial minimization

International audienceMulti-homogeneous polynomial systems arise in many applications. We provide bit complexity estimates for solving them which, up to a few extra other factors, are quadratic in the number of solutions and linear in the height of the input system under some genericity assumptions. The assumptions essentially imply that the Jacobian matrix of the system under study has maximal rank at the solution set and that this solution set if finite. The algorithm is probabilistic and a probability analysis is provided. Next, we apply these results to the problem of optimizing a linear map on the real trace of an algebraic set. Under some genericity assumptions, we provide bit complexity estimates for solving this polynomial minimization problem

Analysis of Randomized Algorithms in Real Algebraic Geometry

Consider the problem of computing at least one point in each connected component of a smooth real algebraic set. This is a basic and important operation in real and semi-algebraic geometry: it gives an upper bound on the number of connected components of the algebraic set, it can be used to decide if the algebraic set has real solutions, and it is also used as a subroutine in many higher-level algorithms. We consider an algorithm for this problem by Safey El Din and Schost: "Polar varieties and computation of one point in each connected component of a smooth real algebraic set," (ISSAC'03). This algorithm uses random changes of variables that are proven to generically ensure certain desirable geometric properties. The cost of the algorithm was given in an algebraic complexity model, and the analysis of the bit complexity and the error probability were left for future work. We also consider another algorithm that solves a special case of the problem. Namely, when the algebraic set is a compact hypersurface. We determine the bit complexity and error probability of these algorithms. Our main contribution is a quantitative analysis of several genericity statements, such as Thom's weak transversality theorem and Noether normalization properties for polar varieties. Furthermore, in doing this work, we have developed techniques that can be used in the analysis of further randomized algorithms in real algebraic geometry, which rely on related genericity properties.