The main result of this note is a tracial Nullstellensatz for free noncommutative polynomials evaluated at tuples of matrices of all sizes: Suppose $f_1, \dots, f_r, f$ are free polynomials, and ${\mathrm tr(f)}$ vanishes whenever all ${\mathrm tr(f_j)}$ vanish. Then either 1 or $f$ is a linear combination of the $f_j$ modulo sums of commutators.
COBISS.SI-ID: 17146969
In real algebraic geometry there are several notions of the radical of an ideal $I$. There is the vanishing radical $\sqrt{I}$ defined as the set of all real polynomials vanishing on the real zero set of $I$, and the real radical $\sqrt[re]{I}$ defined as the smallest real ideal containing $I$. (Neither of them is to be confused with the usual radical from commutative algebra.) By the real Nullstellensatz, $\sqrt{I} = \sqrt[re]{I}$. This paper focuses on extensions of these to the free algebra $\mathbb{R} \langle x, x^\ast \rangle$ of noncommutative real polynomials in $x=(x_1, \dots ,x_g)$ and $x=(x_1^\ast, \dots ,x_g^\ast)$. We work with a natural notion of the (noncommutative real) zero set $V(I)$ of a left ideal $I$ in $\mathbb{R} \langle x, x^\ast \rangle$. The vanishing radical $\sqrt{I}$ of $I$ is the set of all $p \in \mathbb{R} \langle x, x^\ast \rangle$ which vanish on $V(I)$. The earlier paper [J. Cimprič, J.W. Helton, S. McCullough, C. Nelson, A Non-commutative Real Nullstellensatz Corresponds to a Non-commutative Real Ideal; Algorithms, Proc. Lond. Math. Soc., 106 (2013), pp. 1060-1086] gives an appropriate notion of $\sqrt[re]{I}$ and proves $\sqrt{I} = \sqrt[re]{I}$ when $I$ is a finitely generated left ideal, a free $\ast$-Nullstellensatz. However, this does not tell us for a particular ideal I whether or not $I = \sqrt[re]{I}$, and that is the topic of this paper. We give a complete solution for monomial ideals and homogeneous principal ideals. We also present the case of principal univariate ideals with a degree two generator and find that it is very messy. We discuss an algorithm to determine if $I = \sqrt[re]{I}$ implemented under NCAlgebra) with finite run times and provable effectiveness.
COBISS.SI-ID: 16793945
Ever since the historical Sklar's theorem in 1959 copulas have been one of the main tools in modelling dependence of random variables. With the range of applications in applied mathematics expanding and varying from mathematics of finance through system theory to fuzzy set theory, there is a growing need for new types of copulas that could serve as appropriate models in these applications. It is our aim to turn a new page in constructing copulas by setting a counterpart to the famous Marshall copulas (an extension of Marshall-Olkin copulas) that are typically applied to model lifetime of a two-component system. Even a small but essential change of assumptions that the model is applied to such a system with one of the components having a recovery option turns into a substantially different problem on the level of copulas. We give a full study of the augmented case by introducing a new type of copulas, called maxmin, together with concrete applications in server-side web framework, in an economic model and in fuzzy set theory.
COBISS.SI-ID: 17160793