We prove that Schur polynomials in Chern forms of Nakano and dual positive vector bundles are as differential forms. Moreover, modulo a statement about the positivity “double mixed discriminant” linear operators on matrices, which preserve cone definite we establish Griffiths weakly This provides differential-geometric versions Fulton–Lazarsfeld inequalities for ample bundles. An interpretation...

In these notes we review the main concepts about Determinantal Point Processes. Determinantal point processes are of considerable current interest in Probability theory and Mathematical Physics. They were first introduced by Macchi ([8]) and they arise naturally in Random Matrix theory, non-intersecting paths, certain combinatorial and stochastic growth models and representation theory of large...

D.V. Chudnovsky and G.V. Chudnovsky [CH] introduced a generalization of the FrobeniusStickelberger determinantal identity involving elliptic functions that generalize the Cauchy determinant. The purpose of this note is to provide a simple essentially non-analytic proof of this evaluation. This method of proof is inspired by D. Zeilberger’s creative application in [Z1]. AMS Subject Classificatio...

Kabanets and Impagliazzo [KI04] show how to decide the circuit polynomial identity testing problem (CPIT) in deterministic subexponential time, assuming hardness of some explicit multilinear polynomial family {fm}m≥1 for arithmetic circuits. In this paper, a special case of CPIT is considered, namely non-singular matrix completion (NSMC) under a low-individual-degree promise. For this subclass ...

Abstract: We give a probabilistic introduction to determinantal and permanental point processes. Determinantal processes arise in physics (fermions, eigenvalues of random matrices) and in combinatorics (nonintersecting paths, random spanning trees). They have the striking property that the number of points in a region D is a sum of independent Bernoulli random variables, with parameters which a...

We show that any loop-free Markov chain on a discrete space can be viewed as a determinantal point process. As an application we prove central limit theorems for the number of particles in a window for renewal processes and Markov renewal processes with Bernoulli noise. Introduction Let X be a discrete space. A (simple) random point process P on X is a probability measure on the set 2 of all su...

We prove that the determinantal complexity of a hypersurface of degree d > 2 is bounded below by one more than the codimension of the singular locus, provided that this codimension is at least 5. As a result, we obtain that the determinantal complexity of the 3×3 permanent is 7. We also prove that for n > 3, there is no nonsingular hypersurface in Pn of degree d that has an expression as a dete...

The problem of writing real zero polynomials as determinants of linear matrix polynomials has recently attracted a lot of attention. Helton and Vinnikov [9] have proved that any real zero polynomial in two variables has a determinantal representation. Brändén [2] has shown that the result does not extend to arbitrary numbers of variables, disproving the generalized Lax conjecture. We provide a ...

DMITRY KERNER AND VICTOR VINNIKOV Abstract. Let M be a d×d matrix whose entries are linear forms in the homogeneous coordinates of P2. Then M is called a determinantal representation of the curve {det(M) = 0}. Such representations are well studied for smooth curves. We study determinantal representations of curves with arbitrary singularities (mostly reduced). The kernel of M defines a torsion ...

Adding a column of numbers produces “carries” along the way. We show that random digits produce a pattern of carries with a neat probabilistic description: the carries form a one-dependent determinantal point process. This makes it easy to answer natural questions: How many carries are typical? Where are they located? We show that many further examples, from combinatorics, algebra and group the...