We study bihomogeneous systems defining, non-zero dimensional, biprojective varieties for which the projection onto the first group of variables results in a finite set of points. To compute (with) the 0-dimensional projection and the corresponding quotient ring, we introduce linear maps that greatly extend the classical multiplication maps for zero-dimensional systems, but are not those associated to the elimination ideal; we also call them multiplication maps. We construct them using linear algebra on the restriction of the ideal to a carefully chosen bidegree or, if available, from an arbitrary Gröbner basis. The multiplication maps allow us to compute the elimination ideal of the projection, by generalizing FGLM algorithm to bihomogenous, non-zero dimensional, varieties. We also study their properties, like their minimal polynomials and the multiplicities of their eigenvalues, and show that we can use the eigenvalues to compute numerical approximations of the zero-dimensional projection. Finally, we establish a single exponential complexity bound for computing multiplication maps and Gröbner bases, that we express in terms of the bidegrees of the generators of the corresponding bihomogeneous ideal.

Solving systems of polynomial equations is a central problem in nonlinear and computational algebra. Since Buchberger's algorithm for computing Gröbner bases in the 60s, there has been a lot of progress in this domain. Moreover, these equations have been employed to model and solve problems from diverse disciplines such as biology, cryptography, and robotics. Currently, we have a good understanding of how to solve generic systems from a theoretical and algorithmic point of view. However, polynomial equations encountered in practice are usually structured, and so many properties and results about generic systems do not apply to them. For this reason, a common trend in the last decades has been to develop mathematical and algorithmic frameworks to exploit specific structures of systems of polynomials.

Grö bner bases is one the most powerful tools in algorithmic nonlinear 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. We consider sparse systems where the input polynomials have a few non-zero terms. Our approach to exploit sparsity is to embed the systems in a semigroup algebra and to compute Grö bner 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 Grö bner basis in the standard algebra and solve sparse polynomials systems over the torus (\mathbbC ^*)^n. The complexity of the algorithm depends on the Newton polytopes.

A 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 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.

One of the biggest open problems in computational algebra is the design of efficient algorithms for Gröbner basis computations that take into account the sparsity of the input polynomials. We can perform such computations in the case of unmixed polynomial systems, that is systems with polynomials having the same support, using the approach of Faugère, Spaenlehauer, and Svartz [ISSAC'14]. We present two algorithms for sparse Gröbner bases computations for mixed systems. The first one computes with mixed sparse systems and exploits the supports of the polynomials. Under regularity assumptions, it performs no reductions to zero. For mixed, square, and 0-dimensional multihomogeneous polynomial systems, we present a dedicated, and potentially more efficient, algorithm that exploits different algebraic properties that performs no reduction to zero. We give an explicit bound for the maximal degree appearing in the computations.

Symmetric Tensor Decomposition is a major problem that arises in areas such as signal processing, statistics, data analysis and computational neuroscience. It is equivalent to write a homogeneous polynomial in $n$ variables of degree $D$ as a sum of $D$-th powers of linear forms, using the minimal number of summands. This minimal number is called the rank of the polynomial/tensor. We consider the decomposition of binary forms, that corresponds to the decomposition of symmetric tensors of dimension $2$ and order $D$. This problem has its roots in Invariant Theory, where the decompositions are known as canonical forms. As part of that theory, different algorithms were proposed for the binary forms. In recent years, those algorithms were extended for the general symmetric tensor decomposition problem. We present a new randomized algorithm that enhances the previous approaches with results from structured linear algebra and techniques from linear recurrent sequences. It achieves a softly linear arithmetic complexity bound. To the best of our knowledge, the previously known algorithms have quadratic complexity bounds. We compute a symbolic minimal decomposition in O(M(D) log(D)) arithmetic operations, where M(D) is the complexity of multiplying two polynomials of degree D. We approximate the terms of the decomposition with an error of 2-ε, in O(D log2(D) (log2(D) + log(ε))) arithmetic operations. To bound the size of the representation of the coefficients involved in the decomposition, we bound the algebraic degree of the problem by min(rank, D-rank+1). When the input polynomial has integer coefficients, our algorithm performs, up to poly-logarithmic factors, OB(D l + D4 + D3 τ) bit operations, where τ is the maximum bitsize of the coefficients and 2-l is the relative error of the terms in the decomposition.