Related papers: Positivstellens\"atze for polynomial matrices with…

We prove decomposition theorems for sparse positive (semi)definite polynomial matrices that can be viewed as sparsity-exploiting versions of the Hilbert--Artin, Reznick, Putinar, and Putinar--Vasilescu Positivstellens\"atze. First, we…

We follow a stream of the history of positive matrices and positive functionals, as applied to algebraic sums of squares decompositions, with emphasis on the interaction between classical moment problems, function theory of one or several…

We obtain generalisations of some inequalities for positive unital linear maps on matrix algebra. This also provides several positive semidefinite matrices and we get some old and new inequalities involving the eigenvalues of a Hermitian…

We specialize Schm\"udgen's Positivstellensatz and its Putinar and Jacobi and Prestel refinement, to the case of a polynomial $f\in R[X,Y]+R[Y,Z]$, positive on a compact basic semi algebraic set $K$ described by polynomials in $R[X,Y]$ and…

In this paper, we consider the different eigenvalue condition numbers for matrix polynomials used in the literature and we compare them. One of these condition numbers is a generalization of the Wilkinson condition number for the standard…

This paper reviews some characterizations of positive matrices and discusses which lead to useful parametrizations. It is argued that one of them, which we dub the Schur-Constantinescu parametrization is particularly useful. Two new…

Let M be an archimedean quadratic module of real t-by-t matrix polynomials in n variables, and let S be the set of all real n-tuples where each element of M is positive semidefinite. Our key finding is a natural bijection between the set of…

This paper considers some work done by the author and Catlin [CD1,CD2,CD3] concerning positivity conditions for bihomogeneous polynomials and metrics on bundles over certain complex manifolds. It presents a simpler proof of a special case…

In this work, we generalize the probability simplex constraint to matrices, i.e., $\mathbf{X}_1 + \mathbf{X}_2 + \ldots + \mathbf{X}_K = \mathbf{I}$, where $\mathbf{X}_i \succeq 0$ is a symmetric positive semidefinite matrix of size…

Number theorists have studied extensively the connections between the distribution of zeros of the Riemann $\zeta$-function, and of some generalizations, with the statistics of the eigenvalues of large random matrices. It is interesting to…

We study multiplicative statistics for the eigenvalues of unitarily-invariant Hermitian random matrix models. We consider one-cut regular polynomial potentials and a large class of multiplicative statistics. We show that in the large matrix…

Preordered semialgebras and semirings are two kinds of algebraic structures occurring in real algebraic geometry frequently and usually play important roles therein. They have many interesting and promising applications in the fields of…

We consider the classical problem of invariant generation for programs with polynomial assignments and focus on synthesizing invariants that are a conjunction of strict polynomial inequalities. We present a sound and semi-complete method…

The ensemble covariance matrix of a wide sense stationary signal spatially sampled by a full linear array is positive semi-definite and Toeplitz. However, the direct augmented covariance matrix of an augmentable sparse array is Toeplitz but…

We give a non-commutative Positivstellensatz for CP^n: The (commutative) *-algebra of polynomials on the real algebraic set CP^n with the pointwise product can be realized by phase space reduction as the U(1)-invariant polynomials on…

The prevalence of neural networks in society is expanding at an increasing rate. It is becoming clear that providing robust guarantees on systems that use neural networks is very important, especially in safety-critical applications. A…

We consider nondeterministic probabilistic programs with the most basic liveness property of termination. We present efficient methods for termination analysis of nondeterministic probabilistic programs with polynomial guards and…

Hilbert's Nullstellensatz is one of the most fundamental correspondences between algebra and geometry, and has inspired a plethora of noncommutative analogs. In last two decades, there has been an increased interest in understanding…

We consider a symmetric matrix, the entries of which depend linearly on some parameters. The domains of the parameters are compact real intervals. We investigate the problem of checking whether for each (or some) setting of the parameters,…

We derive inclusion regions for the eigenvalues of matrix polynomials expressed in a general polynomial basis, which can lead to significantly better results than traditional bounds. We present several applications to engineering problems.