Related papers: A Complete Characterization of Determinantal Quadr…

The paper develops elementary linear algebra methods to compute the determinants of the tensor symmetrizations of quadratic and hermitian forms over fields of good characteristic. Explicit results are given for the partitions $(n)$,…

A formula expressing the fermionic determinant (a large order polynomial) as an infinite product of smaller determinants is derived and discussed. These smaller determinants are of a fixed size, independent of the size of the lattice and…

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…

The determinantal complexity of a polynomial $P \in \mathbb{F}[x_1, \ldots, x_n]$ over a field $\mathbb{F}$ is the dimension of the smallest matrix $M$ whose entries are affine functions in $\mathbb{F}[x_1, \ldots, x_n]$ such that $P =…

In this paper we give a new and simple algorithm to put any multivariate polynomial into a normal determinant form in which each entry has the form , and in each column the same variable appears. We also apply the algorithm to obtain a…

For bivariate polynomials of degree $n\le 5$ we give fast numerical constructions of determinantal representations with $n\times n$ matrices. Unlike some other available constructions, our approach returns matrices of the smallest possible…

Let $k$ be a field of characteristic two. We prove that a non constant monic polynomial $f\in k[X]$ of degree $n$ is the minimal/characteristic polynomial of a symmetric matrix with entries in $k$ if and only if it is not the product of…

We prove expressions for the inequalities in Hermite's theorem which are conditions for a real polynomial to have real zeros. These expressions generalize the discriminant of a quadratic polynomial and the expression of J. Mar\'ik for a…

Positive and negative quadratic forms are well known and widely used. They are multivariate homogeneous polynomials of degree two taking positive or negative values respectively for any values of their arguments not all zero. In the present…

Consider a convex set S defined by a matrix inequality of polynomials or rational functions over a domain. The set S is called semidefinite programming (SDP) representable or just semidefinite representable if it equals the projection of a…

We study Hermitian non-commutative quadratic polynomials of multiple independent Wigner matrices. We prove that, with the exception of some specific reducible cases, the limiting spectral density of the polynomials always has a square root…

A linear principal minor polynomial or lpm polynomial is a linear combination of principal minors of a symmetric matrix. By restricting to the diagonal, lpm polynomials are in bijection to multiaffine polynomials. We show that this…

The hyperdeterminant of a polynomial (interpreted as a symmetric tensor) factors into several irreducible factors with multiplicities. Using geometric techniques these factors are identified along with their degrees and their…

Here we consider the image of the principal minor map of symmetric matrices over an arbitrary unique factorization domain $R$. By exploiting a connection with symmetric determinantal representations, we characterize the image of the…

We study a graded vector space of polynomials associated to a square matrix, defined by a finite difference condition along the rows. We show this space coincides with one defined by directional derivatives, and prove it is…

This is a continuation of the series of notes on the dynamics of quadratic polynomials. We show the following Rigidity Theorem: Any combinatorial class contains at most one quadratic polynomial satisfying the secondary limbs condition with…

Computing the determinant of a matrix with the univariate and multivariate polynomial entries arises frequently in the scientific computing and engineering fields. In this paper, an effective algorithm is presented for computing the…

Viewing a bivariate polynomial f in R[x,t] as a family of univariate polynomials in t parametrized by real numbers x, we call f real rooted if this family consists of monic polynomials with only real roots. If f is the characteristic…

We introduce a new notion of the determinant, called symmetrized determinant, for a square matrix with the entries in an associative algebra $\AA$. The monomial expansion of the symmetrized determinant is obtained from the standard…

We initiate a study of determinantal representations with symmetry. We show that Grenet's determinantal representation for the permanent is optimal among determinantal representations respecting left multiplication by permutation and…