ALL libraries (COBIB.SI union bibliographic/catalogue database)
13.
On real one-sided ideals in a free algebra
Cimprič, Jaka ...
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.
It was not necessary to add the material to the shelf.
Bibliographic material was successfully added on shelf.
Adding bibliographical material on shelf was not successful.
Material is already on the shelf.
Permalink
URL:
Impact factor
Access to the JCR database is permitted only to users from Slovenia. Your current IP address is not on the list of IP addresses with access permission, and authentication with the relevant AAI accout is required.
Year
Impact factor
Edition
Category
Classification
JCR
SNIP
JCR
SNIP
JCR
SNIP
JCR
SNIP
Select the library membership card:
DRS,
in which the journal is indexed
Database name
Field
Year
Links to authors' personal bibliographies
Links to information on researchers in the SICRIS system
Subject headings in COBISS General List of Subject Headings
Select pickup location
The material from the parent unit is free. If the material is delivered to the pickup location from another unit, the library may charge you for this service.
