Minimal determinantal representations of bivariate polynomials

Abstract

For a square-free bivariate polynomial p of degree n we introduce a simple and fast numerical algorithm for the construction of n×n matrices A, B, and C such that det(A+x B+ y C) = p (x, y). This is the minimal size needed to represent a bivariate polynomial of degree n. Combined with a square-free factorization one can now compute n × n matrices for any bivariate polynomial of degree n. The existence of such symmetric matrices was established by Dixon in 1902, but, up to now, no simple numerical construction has been found, even if the matrices can be nonsymmetric. Such representations may be used to efficiently numerically solve a system of two bivariate polynomials of small degree via the eigenvalues of a two-parameter eigenvalue problem. The new representation speeds up the computation considerably.

Explicit determinantal representations of up to quintic bivariate polynomials

“…We know from [3] that a representation of the minimum size always exists, but a different construction needs to be applied. For instance, the construction from [11] gives an n × n representation for a square-free bivariate polynomial of degree n, i.e., a polynomial that is not a multiple of a square of a non-constant polynomial. 9.…”

“…In this example we generated random bivariate polynomials whose coefficients are random real numbers uniformly distributed on [0, 1] or random complex numbers, such that real and imaginary parts are both uniformly distributed on [0, 1]. We compared Lin345 to Lin2 from [13], which returns matrices of size 3, 5 and 8 for a generic bivariate polynomial or degree 3, 4, and 5, respectively, and to MinRep from [11], which returns matrices of the same size as the degree of a square-free polynomial. These are the only two methods that we compared Lin345 to, since other methods for solving systems of bivariate polynomials (for example [11] and [13]) return representations of bigger sizes and moreover turn out to be slower.…”

“…The above requirements are most closely met by a recent algorithm in [11] that, using only simple numerical computations, returns a determinantal representation with n × n matrices of a square-free bivariate polynomial of degree n and a representation with (2n − 2) × (2n − 2) matrices of a non square-free polynomial. It is important that it does not require any symbolic computation, which usually is the bottleneck for this kind of algorithms.…”

“…all (s, t) ∈ P 1 if and only if the left matrix in (2.6) is one of the following form p 2 is then respectively a 20 x 2 + a 11 x y + a 02 y 2 , x(a 20 x + a 11 y + a 10 z) or y(a11 x + a 02 y + a 01 z).Secondly, when q 2 (x, y, z) = x 2 , all (s, t) ∈ P 1 if and only if the left matrix in (2.7) is one of the following…”

“…+ a11 x y + a 10 xz + a 00 z 2 with the tangent {x = 0} at (0, 1, 0), and in the second case p 2 has equation a 20 x 2 + a 11 x y + a 02 y 2 + a 10 xz + a 01 yz +…”