Free (rational) derivation

Abstract

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.

Keywords

• Hausdorff derivative
• free associative algebra
• free field
• minimal linear representation
• admissible linear system
• free fractions