Factorization of Z-homogeneous polynomials in the First (q)-Weyl Algebra

We present algorithms to factorize weighted homogeneous elements in the first polynomial Weyl algebra and $q$-Weyl algebra, which are both viewed as a $\mathbb{Z}$-graded rings. We show, that factorization of homogeneous polynomials can be almost completely reduced to commutative univariate factorization over the same base field with some additional uncomplicated combinatorial steps. This allows to deduce the complexity of our algorithms in detail. Furthermore, we will show for homogeneous polynomials that irreducibility in the polynomial first Weyl algebra also implies irreducibility in the rational one, which is of interest for practical reasons. We report on our implementation in the computer algebra system \textsc{Singular}. It outperforms for homogeneous polynomials currently available implementations dealing with factorization in the first Weyl algebra both in speed and elegancy of the results.Comment: 26 pages, Singular implementation, 2 algorithms, 1 figure, 2 table

Geometry of free loci and factorization of noncommutative polynomials

The free singularity locus of a noncommutative polynomial f is defined to be the sequence $Z_n(f)=\{X\in M_n^g : \det f(X)=0\}$ of hypersurfaces. The main theorem of this article shows that f is irreducible if and only if $Z_n(f)$ is eventually irreducible. A key step in the proof is an irreducibility result for linear pencils. Apart from its consequences to factorization in a free algebra, the paper also discusses its applications to invariant subspaces in perturbation theory and linear matrix inequalities in real algebraic geometry.Comment: v2: 32 pages, includes a table of content

On real one-sided ideals in a free algebra

In classical and real algebraic geometry there are several notions of the radical of an ideal I. There is the vanishing radical defined as the set of all real polynomials vanishing on the real zero set of I, and the real radical defined as the smallest real ideal containing I. By the real Nullstellensatz they coincide. This paper focuses on extensions of these to the free algebra R of noncommutative real polynomials in x=(x_1,...,x_g) and x^*=(x_1^*,...,x_g^*). We work with a natural notion of the (noncommutative real) zero set V(I) of a left ideal I in the free algebra. The vanishing radical of I is the set of all noncommutative polynomials p which vanish on V(I). In this paper our quest is to find classes of left ideals I which coincide with their vanishing radical. We completely succeed 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. Also we give an algorithm (running under NCAlgebra) which checks if a left ideal is radical or is not, and illustrate how one uses our implementation of it.Comment: v1: 31 pages; v2: 32 page

Feynman integrals and motives

This article gives an overview of recent results on the relation between quantum field theory and motives, with an emphasis on two different approaches: a "bottom-up" approach based on the algebraic geometry of varieties associated to Feynman graphs, and a "top-down" approach based on the comparison of the properties of associated categorical structures. This survey is mostly based on joint work of the author with Paolo Aluffi, along the lines of the first approach, and on previous work of the author with Alain Connes on the second approach.Comment: 32 pages LaTeX, 3 figures, to appear in the Proceedings of the 5th European Congress of Mathematic