arXiv subjects
Publications and source records attributed to Konrad Schrempf.
By representing elements in free fields (over a commutative field and a finite alphabet) using Cohn and Reutenauer's linear representations, we provide an algorithmic construction for the (partial) non-commutative (or Hausdorff-) derivative and show how it can be applied to the non-commutative version of the Newton iteration to find roots of matrix-valued rational equations.
We describe a standard form for the elements in the universal field of fractions of free associative algebras (over a commutative field). It is a special version of the normal form provided by Cohn and Reutenauer and enables the use of linear algebra techniques for the construction of minimal linear representations (in standard form) for the sum and the product of two elements (given in standard form). This completes "minimal" arithmetics in free fields since "minimal" constructions for the inverse are already known. The applications are wide: linear algebra (over the free field), rational identities, computing the left gcd of two non-commutative polynomials, etc.
Long before we learn to construct the field of rational numbers (out of the ring of integers) at university, we learn how to calculate with fractions at school. When it comes to "numbers", we are used to a commutative multiplication, for example 2*3=6=3*2. On the other hand --even before we can write-- we learn to talk (in a language) using words, consisting of purely non-commuting "letters" (or symbols), for example "xy" is not equal to "yx" (with the concatenation as multiplication). Now, if we combine numbers (from a field) with words (from the free monoid of an alphabet) we get non-commutative polynomials which form a ring (with "natural" addition and multiplication), namely the free associative algebra. Adding or multiplying polynomials is easy, for example (2/3*xy+z)+1/3*xy=xy+z or 2*x(yx+3*z)=2*xyx+6*xz. Although the integers and the non-commutative polynomials look rather different, they share many properties, for example the unique number of irreducible factors: x(1-yx)=x-xyx=(1-xy)x. However, the construction of the universal field of fractions (aka "free field") of the free associative algebra is highly non-trivial (but really beautiful). Therefore we provide techniques (building on the work of Cohn and Reutenauer) to calculate with free fractions (representing elements in the free field or "skew field of non-commutative rational functions") to be able to explore a fascinating non-commutative world.
Although in general there is no meaningful concept of factorization in fields, that in free associative algebras (over a commutative field) can be extended to their respective free field (universal field of fractions) on the level of minimal linear representations. We establish a factorization theory by providing an alternative definition of left (and right) divisibility based on the rank of an element and show that it coincides with the classical left (and right) divisibility for non-commutative polynomials. Additionally we present an approach to factorize elements, in particular rational formal power series, into their (generalized) atoms. The problem is reduced to solving a system of polynomial equations with commuting unknowns.
Since more than three decades, interior-point methods proved very useful for optimization, from linear over semidefinite to conic (and partly beyond non-convex) programming; despite the fact that already in the semidefinite case (even when strong duality holds) "hard" problems are known. We shade a light on a rather surprising restriction in the non-commutative world (of semidefinite programming), namely "commutative" paths and propose a new family of solvers that is able to use the full richness of "non-commutative" search directions: (primal) feasible-interior-point methods. Beside a detailed basic discussion, we illustrate some variants of "non-commutative" paths and provide a simple implementation for further (problem specific) investigations.
By viewing non-commutative polynomials, that is, elements in free associative algebras, in terms of linear representations, we generalize Horner's rule to the non-commutative (multivariate) setting. We introduce the concept of Horner systems (which has parallels to that of companion matrices), discuss their construction and show how they enable the efficient evaluation of non-commutative polynomials by matrices.
We describe a simple approach to factorize non-commutative (nc) polynomials, that is, elements in free associative algebras (over a commutative field), into atoms (irreducible elements) based on (a special form of) their minimal linear representations. To be more specific, a correspondence between factorizations of an element and upper right blocks of zeros in the system matrix (of its representation) is established. The problem is then reduced to solving a system of polynomial equations (with at most quadratic terms) with commuting unknowns to compute appropriate transformation matrices (if possible).
We describe a solution of the word problem in free fields (coming from non-commutative polynomials over a commutative field) using elementary linear algebra, provided that the elements are given by minimal linear representations. It relies on the normal form of Cohn and Reutenauer and can be used more generally to (positively) test rational identities. Moreover we provide a construction of minimal linear representations for the inverse of non-zero elements.