Multihomogeneous resultant formulae by means of complexes

We provide conditions and algorithmic tools so as to classify and construct the smallest possible determinantal formulae for multihomogeneous resultants arising from Weyman complexes associated to line bundles in products of projective spaces. We also examine the smallest Sylvester-type matrices, generically of full rank, which yield a multiple of the resultant. We characterize the systems that admit a purely B\'ezout-type matrix and show a bijection of such matrices with the permutations of the variable groups. We conclude with examples showing the hybrid matrices that may be encountered, and illustrations of our Maple implementation. Our approach makes heavy use of the combinatorics of multihomogeneous systems, inspired by and generalizing results by Sturmfels-Zelevinsky, and Weyman-Zelevinsky.Comment: 30 pages. To appear: Journal of Symbolic Computatio

Compact Formulae in Sparse Elimination

International audienceIt has by now become a standard approach to use the theory of sparse (or toric) elimination, based on the Newton polytope of a polynomial, in order to reveal and exploit the structure of algebraic systems. This talk surveys compact formulae, including older and recent results, in sparse elimination. We start with root bounds and juxtapose two recent formulae: a generating function of the m-Bézout bound and a closed-form expression for the mixed volume by means of a matrix permanent. For the sparse resultant, a bevy of results have established determinantal or rational formulae for a large class of systems, starting with Macaulay. The discriminant is closely related to the resultant but admits no compact formula except for very simple cases. We offer a new determinantal formula for the discriminant of a sparse multilinear system arising in computing Nash equilibria. We introduce an alternative notion of compact formula, namely the Newton polytope of the unknown polynomial. It is possible to compute it efficiently for sparse resultants, discriminants, as well as the implicit equation of a parameterized variety. This leads us to consider implicit matrix representations of geometric objects

Bilinear systems with two supports: Koszul resultant matrices, eigenvalues, and eigenvectors

International audienceA fundamental problem in computational algebraic geometry is the computation of the resultant. A central question is when and how to compute it as the determinant of a matrix. whose elements are the coefficients of the input polynomials up-to sign. This problem is well understood for unmixed multihomogeneous systems, that is for systems consisting of multihomogeneous polynomials with the * 1 same support. However, little is known for mixed systems, that is for systems consisting of polynomials with different supports. We consider the computation of the multihomogeneous resultant of bilinear systems involving two different supports. We present a constructive approach that expresses the resultant as the exact determinant of a Koszul resultant matrix, that is a matrix constructed from maps in the Koszul complex. We exploit the resultant matrix to propose an algorithm to solve such systems. In the process we extend the classical eigenvalues and eigenvectors criterion to a more general setting. Our extension of the eigenvalues criterion applies to a general class of matrices, including the Sylvester-type and the Koszul-type ones

Gr\"obner Bases of Bihomogeneous Ideals generated by Polynomials of Bidegree (1,1): Algorithms and Complexity

Solving multihomogeneous systems, as a wide range of structured algebraic systems occurring frequently in practical problems, is of first importance. Experimentally, solving these systems with Gr\"obner bases algorithms seems to be easier than solving homogeneous systems of the same degree. Nevertheless, the reasons of this behaviour are not clear. In this paper, we focus on bilinear systems (i.e. bihomogeneous systems where all equations have bidegree (1,1)). Our goal is to provide a theoretical explanation of the aforementionned experimental behaviour and to propose new techniques to speed up the Gr\"obner basis computations by using the multihomogeneous structure of those systems. The contributions are theoretical and practical. First, we adapt the classical F5 criterion to avoid reductions to zero which occur when the input is a set of bilinear polynomials. We also prove an explicit form of the Hilbert series of bihomogeneous ideals generated by generic bilinear polynomials and give a new upper bound on the degree of regularity of generic affine bilinear systems. This leads to new complexity bounds for solving bilinear systems. We propose also a variant of the F5 Algorithm dedicated to multihomogeneous systems which exploits a structural property of the Macaulay matrix which occurs on such inputs. Experimental results show that this variant requires less time and memory than the classical homogeneous F5 Algorithm.Comment: 31 page

Toric Intersection Theory for Affine Root Counting

Given any polynomial system with fixed monomial term structure, we give explicit formulae for the generic number of roots with specified coordinate vanishing restrictions. For the case of affine space minus an arbitrary union of coordinate hyperplanes, these formulae are also the tightest possible upper bounds on the number of isolated roots. We also characterize, in terms of sparse resultants, precisely when these upper bounds are attained. Finally, we reformulate and extend some of the prior combinatorial results of the author on which subsets of coefficients must be chosen generically for our formulae to be exact. Our underlying framework provides a new toric variety setting for computational intersection theory in affine space minus an arbitrary union of coordinate hyperplanes. We thus show that, at least for root counting, it is better to work in a naturally associated toric compactification instead of always resorting to products of projective spaces

Hybrid resultant matrix algorithm based on the sylvester-bezout formulation

The resultant of a system of polynomial equations is a factor of the determinant of the resultant matrix. The matrix is said to be optimal when its determinant equals exactly the resultant. Other factors besides the determinant are known as extraneous factors and it has been the major interest among researches to seek for a determinantal resultant formula that gives optimal resultant matrix whose determinant exactly equals the resultant. If such determinantal formula does not exist, a formulation that reduces the existence of these extraneous factors is sought. This thesis focuses on the construction and implementations of determinantal formulas that gives exact resultant for certain classes of multihomogeneous multivariate polynomial equations. For the class of multigraded polynomial systems, a Sylvester type formula giving exact resultant can be derived out of certain degree vectors. The first part of this thesis implements the Sylvester type formula for determining the entries and dimension of the Sylvester type matrix of multigraded systems by applying the properties of certain linear maps and permutations of groups of variables. Even though the Sylvester type formula gives exact resultants for multigraded systems, this approach does not take advantage of the sparseness conditions when considering sparse polynomials. Sparse systems can be utilized by considering the underlying properties of its Newton polytopes, the convex hull of the support of the system. Preliminary observations on the properties of mixed volumes of the polytopes in comparison to the degree of the resultant of polynomial systems derived from Sylvester type matrices are used in the determination of whether the resultant matrix is optimal. This research proceeds to construct and implement a new hybrid resultant matrix algorithm based on the Sylvester-B´ezout formulation. The basis of this construction applies some related concepts and tools from algebraic geometry such as divisors, fans and cones, homogeneous coordinate rings and the projective space. The major tasks in the construction are determining the degree vector of the homogeneous variables known as homogeneous coordinates and solving a set of linear inequalities. In this work, the method of solving these equations involves a systematic procedure or combinatorial approach on the set of exponent vectors of the monomials. Two new rules are added as a termination criterion for obtaining the unique solutions for the B´ezout matrix. The implementation of the new algorithm on certain class of unmixed multigraded systems of bivariate polynomial equations with some coefficients being zero suggests conditions that can produce exact resultant. From the results, some theorems on these conditions and properties are proven. An application of the hybrid resultant matrix to solving the multivariate polynomial equations in three variables is discussed. Upon completion of this research two new computer algebra packages have been developed, namely the Sylvester matrix package for multivariate polynomial equations and the hybrid Sylvester- B´ezout matrix package for computing the resultant of bivariate polynomial equations

Multilinear Polynomial Systems: Root Isolation and Bit Complexity

Special Issue of the Journal of Symbolic Computation on Milestones in Computer Algebra (MICA 2016)International audienceWe exploit structure in polynomial system solving by considering polyno-mials that are linear in subsets of the variables. We focus on algorithms and their Boolean complexity for computing isolating hyperboxes for all the isolated complex roots of well-constrained, unmixed systems of multilinear polynomials based on resultant methods. We enumerate all expressions of the multihomogeneous (or multigraded) resultant of such systems as a determinant of Sylvester-like matrices, aka generalized Sylvester matrices. We construct these matrices by means of Weyman homological complexes, which generalize the Cayley-Koszul complex. The computation of the determinant of the resultant matrix is the bottleneck for the overall complexity. We exploit the quasi-Toeplitz structure to reduce the problem to efficient matrix-vector multiplication, which corresponds to multivariate polynomial multiplication, by extending the seminal work on Macaulay matrices of Canny, Kaltofen, and Yagati [9] to the multi-homogeneous case. We compute a rational univariate representation of the roots, based on the primitive element method. In the case of 0-dimensional systems we present a Monte Carlo algorithm with probability of success 1 − 1/2^r, for a given r ≥ 1, and bit complexity O_B (n^2 D^(4+e) (n^(N +1) + τ) + n D^(2+e) r (D +r)) for any e> 0, where n is the number of variables, D equals the multilinear Bézout bound, N is the number of variable subsets, and τ is the maximum coefficient bitsize. We present an algorithmic variant to compute the isolated roots of overdetermined and positive-dimensional systems. Thus our algorithms and complexity analysis apply in general with no assumptions on the input