Roots of bivariate polynomial systems via determinantal representations

We give two determinantal representations for a bivariate polynomial. They may be used to compute the zeros of a system of two of these polynomials via the eigenvalues of a two-parameter eigenvalue problem. The first determinantal representation is suitable for polynomials with scalar or matrix coefficients, and consists of matrices with asymptotic order $n^2/4$, where $n$ is the degree of the polynomial. The second representation is useful for scalar polynomials and has asymptotic order $n^2/6$. The resulting method to compute the roots of a system of two bivariate polynomials is competitive with some existing methods for polynomials up to degree 10, as well as for polynomials with a small number of terms.Comment: 22 pages, 9 figure

Uniform determinantal representations

The problem of expressing a specific polynomial as the determinant of a square matrix of affine-linear forms arises from algebraic geometry, optimisation, complexity theory, and scientific computing. Motivated by recent developments in this last area, we introduce the notion of a uniform determinantal representation, not of a single polynomial but rather of all polynomials in a given number of variables and of a given maximal degree. We derive a lower bound on the size of the matrix, and present a construction achieving that lower bound up to a constant factor as the number of variables is fixed and the degree grows. This construction marks an improvement upon a recent construction due to Plestenjak-Hochstenbach, and we investigate the performance of new representations in their root-finding technique for bivariate systems. Furthermore, we relate uniform determinantal representations to vector spaces of singular matrices, and we conclude with a number of future research directions.Comment: 23 pages, 3 figures, 4 table

Twenty-Seven Questions about the Cubic Surface

We present a collection of research questions on cubic surfaces in 3-space. These questions inspired a collection of papers to be published in a special issue of the journal Le Matematiche. This article serves as the introduction to that issue. The number of questions is meant to match the number of lines on a cubic surface. We end with a list of problems that are open.Comment: 13 page

Combinatorial and Computational Methods for the Properties of Homogeneous Polynomials

In this manuscript, we provide foundations of properties of homogeneous polynomials such as the half-plane property, determinantal representability, being weakly determinantal, and having a spectrahedral hyperbolicity cone. One of the motivations for studying those properties comes from the ``generalized Lax conjecture'' stating that every hyperbolicity cone is spectrahedral. The conjecture has particular importance in convex optimization and has curious connections to other areas. We take a combinatorial approach, contemplating the properties on matroids with a particular focus on operations that preserve these properties. We show that the spectrahedral representability of hyperbolicity cones and being weakly determinantal are minor-closed properties. In addition, they are preserved under passing to the faces of the Newton polytopes of homogeneous polynomials. We present a proved-to-be computationally feasible algorithm to test the half-plane property of matroids and another one for testing being weakly determinantal. Using the computer algebra system Macaulay2 and Julia, we implement these algorithms and conduct tests. We classify matroids on at most 8 elements with respect to the half-plane property and provide our test results on matroids with 9 elements. We provide 14 matroids on 8 elements of rank 4, including the Vámos matroid, that are potential candidates for the search of a counterexample for the conjecture.:1 Background 1 1.1 Some Properties of Homogeneous Polynomials . . . . . . . . . . 1 Hyperbolic Polynomials . . . . . . . . . . . . . . . . . . . . . . 1 The Half-Plane Property and Stability . . . . . . . . . . . . . . 8 Determinantal Representability . . . . . . . . . . . . . . . . . . 15 Spectrahedral Representability . . . . . . . . . . . . . . . . . . 19 1.2 Matroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Some Operations on Matroids . . . . . . . . . . . . . . . . . . . 29 The Half-Plane Property of Matroids . . . . . . . . . . . . . . . 36 2 Some Operations 43 2.1 Determinantal Representability of Matroids . . . . . . . . . . . 43 A Criterion for Determinantal Representability . . . . . . . . . 46 2.2 Spectrahedral Representability of Matroids . . . . . . . . . . . 50 2.3 Matroid Polytopes . . . . . . . . . . . . . . . . . . . . . . . . . 54 Newton Polytopes of Stable Polynomials . . . . . . . . . . . . . 59 3 Testing the Properties: an Algorithm 61 The Half-Plane Property . . . . . . . . . . . . . . . . . . . . . . 61 Being SOS-Rayleigh and Weak Determinantal Representability 65 4 Test Results on Matroids on 8 and 9 Elements 71 4.1 Matroids on 8 Elements . . . . . . . . . . . . . . . . . . . . . . 71 SOS-Rayleigh and Weakly Determinantal Matroids . . . . . . . 76 4.2 Matroids on 9 Elements . . . . . . . . . . . . . . . . . . . . . . 80 5 Conclusion and Future Perspectives 85 5.1 Spectrahedral Matroids . . . . . . . . . . . . . . . . . . . . . . 86 5.2 Non-negative Non-SOS Polynomials . . . . . . . . . . . . . . . 88 5.3 Completing the Classification of Matroids on 9 Elements and More 89 Bibliography 9