Uniform Determinantal Representations

Abstract

Abstract. The problem of expressing a specific polynomial as the determinant of a square matrix of affine-linear forms arises from algebraic geometry, optimization, 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 and 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.

Minimal determinantal representations of bivariate polynomials

“…Here is a list of some available determinantal representations for generic bivariate polynomials. The first group of determinantal representations has the property that the elements of matrices A, B, and C depend affine-linearly on the coefficients of the polynomial p. Such determinantal representations are named uniform in [3]. The first such representation of order n 2 is given by Khazanov in [7] as a special case of a linearization of a multiparameter polynomial matrix.…”

“…A smaller nonsymmetric uniform representation of the same asymptotic order is described in [16]. Recently, a uniform representation of order 2n−1 was presented in [3], which is the first uniform representation such that the order of matrices grows linearly and not quadratically with n. All uniform representations do not require any computation, the construction is very simple and fast as one just puts the coefficients of p on prescribed places in the matrices A, B, and C.…”

“…We compute γ 2 = −6/(ξ 1 ξ 2 ) = −3 from (25) to annihilate the term y 2 in s (3) . The residual after step k = 3 is…”

Solving Singular Generalized Eigenvalue Problems by a Rank-Completing Perturbation

“…We take example C3 from [12] that comes from control theory and belongs to a set of examples C1, C2, and C3, where each has successively more ill-conditioned eigenvalues. The pencil has the form Its KCF contains blocks L 2 , J 1 (1), and J 1 (2). As the pencil is rectangular, we add a zero line to make it square.…”

“…Using a uniform determinantal representation from [2], we write the above system as a 2EP of the form where p i (λ, µ) = det(A i + λB i + µC i ) for i = 1, 2. The obtained 2EP is singular and has 9 regular eigenvalues (λ j , µ j ) which are exactly the 9 solutions of the initial polynomial system.…”