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DeterminantalRepresentations :: DeterminantalRepresentations

DeterminantalRepresentations -- computing determinantal representations of polynomials

Description

The goal of this package is to compute determinantal representations of polynomials. To be precise, a polynomial f in ℝ[x1, ..., xn] of total degree d (not necessarily homogeneous) is called determinantal if f is the determinant of a matrix of linear forms - in other words, there exist matrices A0, ..., An ∈ℝd×d such that f(x1, ..., xn) = det(A0 + x1A1 + ...+ xnAn). The matrix pencil A0 + x1A1 + ...+ xnAn is said to give a determinantal representation of f of size d. If the matrices Ai can be chosen to be all symmetric, then the determinantal representation is called symmetric. The determinantal representation is called definite if A0 is positive definite, and monic if A0 = Id is the identity matrix.

Deciding whether or not a degree d polynomial has a determinantal representation of size d is in general difficult, and computing such a representation even more so. Computing (monic) symmetric determinantal representations (even in 2 variables) is of interest owing to a connection with real-zero and hyperbolic polynomials, due to a celebrated theorem of Helton-Vinnikov. In general, determinantal polynomials also have connections to convex algebraic geometry and semidefinite programming.

Currently, the functions in this package are geared towards computing monic symmetric determinantal representations of quadrics, as well as plane curves of low degree (i.e. cubics and quartics). The algorithms implemented in this package can be found in [1], [2].

Additionally, a number of helper functions are included for creating/working with various classes of matrices, which may be of general interest (and are not limited to the scope of determinantal representations). These include: creating/testing orthogonal, symmetric, doubly stochastic, unipotent, and positive semidefinite matrices, Hadamard products, Cholesky decomposition, and lifting/rounding matrices from CC to RR/QQ.

References:

  • [1] Dey, P., Definite Determinantal Representations via Orthostochastic Matrices
  • [2] Dey, P., Definite Determinantal Representations of Multivariate Polynomials

Authors

Version

This documentation describes version 1.0.0 of DeterminantalRepresentations.

Source code

The source code from which this documentation is derived is in the file DeterminantalRepresentations.m2.

Exports