Abstract: Rational (formal) power series (in one variable over an arbitrary ground field) are the smallest subring of the ring of power series that contains the polynomials and is closed under the inversion (of invertible power series). Equivalently, it is the ring formed by Taylor expansions of rational functions that are regular at $0$. It is completely characterized by a classical theorem of Kronecker. The characterization is by means of linear recurrence relations (with constant coefficients). Alternatively, the infinite Hankel matrix $(r_{i+j})_{i,j=0}^\infty$ associated to the power series $r=\sum_{i=0}^\infty r_i x^i$ is to have finite rank. A consequence or reformulation of this result is that a power series is rational iff there exists a square matrix $A$ over the ground field s.t. for all $i$, $r_i$ is the $(1,1)$ elements of the matrix $A^i$ --- these are called realizations in systems and control and linearizations in other areas and are very useful since they allow us to reduce many questions about $r$ to linear algebra.

I will discuss some of the generalizations of Kronecker's theorem to the setting of power series and polynomials in (free) noncommuting variables and of the free skew field (the universal field of fractions of the free associative algebra), starting with the results of Schuetzenberger from the 1960s and of Fliess from the 1970s and continuing with more recent work (joint with I. Klep, M. Porat, and Ju. Volcic) in the area of free noncommutative analysis and geometry.