Powers of positive polynomials and codings of Markov chains onto Bernoullishifts

We give necessary and sufficient conditions for a Markov chain to factor on to a Bernoulli shift (i) as an eventual right-closing factor, (ii) by a rig ht-closing factor map, (iii) by a one-to-one a.e. right-closing factor map, and (iv) by a regular isomorphism. We pass to the setting of polynomials i n several variables to represent the Bernoulli shift by a nonnegative polyn omial p in several variables and the Markov chain by a matrix A of such pol ynomials. The necessary and sufficient conditions for each of (i)-(iv) invo lve only an eigenvector r of A and basic invariants obtained from weights o f periodic orbits. The characterizations of (ii)-(iv) are deduced from (i). We formulate (i) as a combinatorial problem, reducing it to certain state- splittings (partitions) of paths of length n. In terms of positive polynomi al masses associated with paths, the aim then becomes the construction of p artitions so that the masses of the paths in each partition element sum to a multiple of p(n), the multiple being prescribed by r. The construction, w hich we sketch, relies on a description of the terms of p(n) and on estimat es of the relative sizes of the coefficients of p(n).