Noncommutative partially convex rational functions

Positivstellensatzë for noncommutative rational expressions

We derive some Positivstellensatzë for noncommutative rational expressions from the Positivstellensatzë for noncommutative polynomials. Specifically, we show that if a noncommutative rational expression is positive on a polynomially convex set, then there is an algebraic certificate witnessing that fact. As in the case of noncommutative polynomials, our results are nicer when we additionally assume positivity on a convex set-- that is, we obtain a so-called "perfect Positivstellensatz" on convex sets.

Plurisubharmonic Noncommutative Rational Functions

A noncommutative (nc) function in $x_1,\dots,x_g,x_1^*,\dots,x_g$ is called plurisubharmonic (plush) if its nc complex Hessian takes only positive semidefinite values on an nc neighborhood of 0. The main result of this paper shows that an nc rational function is plush if and only if it is a composite of a convex rational function with an analytic (no $x_j^*$) rational function. The proof is entirely constructive. Further, a simple computable necessary and sufficient condition for an nc rational function to be plush is given in terms of its minimal realization.

Hilbert's 17th problem in free skew fields

This paper solves the rational noncommutative analog of Hilbert's 17th problem: if a noncommutative rational function is positive semidefinite on all tuples of hermitian matrices in its domain, then it is a sum of hermitian squares of noncommutative rational functions. This result is a generalization and culmination of earlier positivity certificates for noncommutative polynomials or rational functions without hermitian singularities. More generally, a rational Positivstellensatz for free spectrahedra is given: a noncommutative rational function is positive semidefinite or undefined at every m…

平方和有限和、有限实现与非交换Carathéodory逼近

Finite sum of squares, finite realization and noncommutative Carathéodory approximation

本文在非交换多圆盘上证明了非负遗传有理nc函数的平方和有限和公式，并由此导出收缩nc有理函数的有限维实现公式，进而将Carathéodory经典定理推广到非交换多圆盘上的全纯函数。

Noncommutative rational functions invariant under the action of a finite solvable group

This paper describes the structure of invariant skew fields for linear actions of finite solvable groups on free skew fields in $d$ generators. These invariant skew fields are always finitely generated, which contrasts with the free algebra case. For abelian groups or solvable groups $G$ with a well-behaved representation theory it is shown that the invariant skew fields are free on $|G|(d-1)+1$ generators. Finally, positivity certificates for invariant rational functions in terms of sums of squares of invariants are presented.

Scattering in quantum dots via noncommutative rational functions

In the customary random matrix model for transport in quantum dots with $M$ internal degrees of freedom coupled to a chaotic environment via $N\ll M$ channels, the density $ρ$ of transmission eigenvalues is computed from a specific invariant ensemble for which explicit formula for the joint probability density of all eigenvalues is available. We revisit this problem in the large $N$ regime allowing for (i) arbitrary ratio $ϕ:= N/M \le 1$; and (ii) general distributions for the matrix elements of the Hamiltonian of the quantum dot. In the limit $ϕ\to 0$ we recover the formula for the density $ρ…