Blaschke–singular–outer factorization of free non-commutative functions

Factorization of Non-Commutative Polynomials

We describe an algorithm for the factorization of non-commutative polynomials over a field. The first sketch of this algorithm appeared in an unpublished manuscript (literally hand written notes) by James H. Davenport more than 20 years ago. This version of the algorithm contains some improvements with respect to the original sketch. An improved version of the algorithm has been fully implement...

The Hopf Algebra of Non-commutative Symmetric Functions and Quasi-symmetric Functions Are Free and Cofree

We uncover the structure of the space of symmetric functions in non-commutative variables by showing that the underlined Hopf algebra is both free and co-free. We also introduce the Hopf algebra of quasi-symmetric functions in non-commutative variables and define the product and coproduct on the monomial basis of this space and show that this Hopf algebra is free and cofree. In the process of l...

Dual bases for non commutative symmetric and quasi-symmetric functions via monoidal factorization

where O(P ) is the (multi-)set of roots of P (a polynomial) invites to consider Λj(?) as a “multiset (endo)functor” rather than a function K → K (K is a field where P splits). But, here, Λk(X) = 0 whenever |X | < k and one would like to get the universal formulas i.e. which hold true whatever the cardinality of |X |. This set of formulas is obtained as soon as the alphabet is infinite and, ther...

Factorization in Commutative Rings

Non-commutative holomorphic functions on operator domains.

We characterize functions of d-tuples of bounded operators on a Hilbert space that are uniformly approximable by free polynomials on balanced open sets.