Definite determinantal representations via orthostochastic matrices

Determinantal inequalities for positive definite matrices

Let Ai , i = 1, . . . ,m , be positive definite matrices with diagonal blocks A ( j) i , 16 j 6 k , where A ( j) 1 , . . . ,A ( j) m are of the same size for each j . We prove the inequality det( m ∑ i=1 A−1 i ) > det( m ∑ i=1 (A (1) i ) −1) · · ·det( m ∑ i=1 (A (k) i ) −1) and more determinantal inequalities related to positive definite matrices.

Determinantal representations of hyperbolic forms via weighted shift matrices

Representations of Braid Groups via Determinantal Rings

We construct representations for braid groups Bn via actions of Bn on a determinantal ring, thus mirroring the setting of the classical representation theory for GLn. The representations that we construct fix a certain unitary form.

Determinantal representations of elliptic curves via Weierstrass elliptic functions

Helton and Vinnikov proved that every hyperbolic ternary form admits a symmetric derminantal representation via Riemann theta functions. In the case the algebraic curve of the hyperbolic ternary form is elliptic, the determinantal representation of the ternary form is formulated by using Weierstrass ℘-functions in place of Riemann theta functions. An example of this approach is given.

Sturm and Sylvester algorithms revisited via tridiagonal determinantal representations

There are several methods to count the number of real roots of an univariate polynomial p(x) ∈ R[x] of degree n (for details we refer to [BPR]). Among them, the Sturm algorithm says that the number of real roots of p(x) is equal to the number of Permanence minus the number of variations of signs which appears in the leading coefficients of the signed remainders sequence of p(x) and p(x). Anothe...