arXiv subjects
Publications and source records attributed to Anatolii Grinshpan.
Hilbert's ternary quartic theorem states that every nonnegative degree 4 homogeneous polynomial in three variables can be written as a sum of three squares of homogeneous quadratic polynomials. We give a linear-algebraic approach to Hilbert's theorem by showing that a structured cone of positive semidefinite matrices is generated by rank 1 elements.
We prove that every matrix-valued rational function $F$, which is regular on the closure of a bounded domain $\mathcal{D}_\mathbf{P}$ in $\mathbb{C}^d$ and which has the associated Agler norm strictly less than 1, admits a finite-dimensional contractive realization $$F(z)= D + C\mathbf{P}(z)_n(I-A\mathbf{P}(z)_n)^{-1} B. $$ Here $\mathcal{D}_\mathbf{P}$ is defined by the inequality $\|\mathbf{P}(z)\|<1$, where $\mathbf{P}(z)$ is a direct sum of matrix polynomials $\mathbf{P}_i(z)$ (so that appropriate Archimedean and approximation conditions are satisfied), and $\mathbf{P}(z)_n=\bigoplus_{i=1}^k\mathbf{P}_i(z)\otimes I_{n_i}$, with some $k$-tuple $n$ of multiplicities $n_i$; special cases include the open unit polydisk and the classical Cartan domains. The proof uses a matrix-valued version of a Hermitian Positivstellensatz by Putinar, and a lurking contraction argument. As a consequence, we show that every polynomial with no zeros on the closure of $\mathcal{D}_\mathbf{P}$ is a factor of $\det (I - K\mathbf{P}(z)_n)$, with a contractive matrix $K$.
We establish the existence of a finite-dimensional unitary realization for every matrix-valued rational inner function from the Schur--Agler class on a unit square-matrix polyball. In the scalar-valued case, we characterize the denominators of these functions. We also show that every polynomial with no zeros in the closed domain is such a denominator. One of our tools is the Korányi--Vagi theorem generalizing Rudin's description of rational inner functions to the case of bounded symmetric domains; we provide a short elementary proof of this theorem suitable in our setting.
We show that an irreducible polynomial $p$ with no zeros on the closure of a matrix unit polyball, a.k.a. a cartesian product of Cartan domains of type I, and such that $p(0)=1$, admits a strictly contractive determinantal representation, i.e., $p=\det(I-KZ_n)$, where $n=(n_1,...,n_k)$ is a $k$-tuple of nonnegative integers, $Z_n=\bigoplus_{r=1}^k(Z^{(r)}\otimes I_{n_r})$, $Z^{(r)}=[z^{(r)}_{ij}]$ are complex matrices, $p$ is a polynomial in the matrix entries $z^{(r)}_{ij}$, and $K$ is a strictly contractive matrix. This result is obtained via a noncommutative lifting and a theorem on the singularities of minimal noncommutative structured system realizations.
For every bivariate polynomial $p(z_1, z_2)$ of bidegree $(n_1, n_2)$, with $p(0,0)=1$, which has no zeros in the open unit bidisk, we construct a determinantal representation of the form $$p(z_1,z_2)=\det (I - K Z),$$ where $Z$ is an $(n_1+n_2)\times(n_1+n_2)$ diagonal matrix with coordinate variables $z_1$, $z_2$ on the diagonal and $K$ is a contraction. We show that $K$ may be chosen to be unitary if and only if $p$ is a (unimodular) constant multiple of its reverse. Furthermore, for every bivariate real-zero polynomial $p(x_1, x_2),$ with $p(0,0)=1$, we provide a construction to build a representation of the form $$p(x_1,x_2)=\det (I+x_1A_1+x_2A_2),$$ where $A_1$ and $A_2$ are Hermitian matrices of size equal to the degree of $p$. A key component of both constructions is a stable factorization of a positive semidefinite matrix-valued polynomial in one variable, either on the circle (trigonometric polynomial) or on the real line (algebraic polynomial).
For every multivariable polynomial $p$, with $p(0)=1$, we construct a determinantal representation $$p=\det (I - K Z),$$ where $Z$ is a diagonal matrix with coordinate variables on the diagonal and $K$ is a complex square matrix. Such a representation is equivalent to the existence of $K$ whose principal minors satisfy certain linear relations. When norm constraints on $K$ are imposed, we give connections to the multivariable von Neumann inequality, Agler denominators, and stability. We show that if a multivariable polynomial $q$, $q(0)=0,$ satisfies the von Neumann inequality, then $1-q$ admits a determinantal representation with $K$ a contraction. On the other hand, every determinantal representation with a contractive $K$ gives rise to a rational inner function in the Schur--Agler class.