Factorization of Multivariate Positive Laurent Polynomials

Recently Dritschel proves that any positive multivariate Laurent polynomial can be factorized into a sum of square magnitudes of polynomials. We first give another proof of the Dritschel theorem. Our proof is based on the univariate matrix Fejer-Riesz theorem. Then we discuss a computational method to find approximates of polynomial matrix factorization. Some numerical examples will be shown. Finally we discuss how to compute nonnegative Laurent polynomial factorizations in the multivariate setting

Finite automata and algebraic extensions of function fields

We give an automata-theoretic description of the algebraic closure of the rational function field F_q(t) over a finite field, generalizing a result of Christol. The description takes place within the Hahn-Mal'cev-Neumann field of "generalized power series" over F_q. Our approach includes a characterization of well-ordered sets of rational numbers whose base p expansions are generated by a finite automaton, as well as some techniques for computing in the algebraic closure; these include an adaptation to positive characteristic of Newton's algorithm for finding local expansions of plane curves. We also conjecture a generalization of our results to several variables.Comment: 40 pages; expanded version of math.AC/0110089; v2: refereed version, includes minor edit