Denominator bounds for partial linear difference equations.

We report on joint work with Carsten Schneider about generalizing Abramov's classical denominator bounding technique to the multivariate case. The question thus is, given a linear difference operator L in K(n₁,...,n_r)[S_n1,..., S_nr], to determine a polynomial Q such that for any rational function y=p/q with L(y)=0 we have q | Q. In contrast to the univariate case, such a polynomial Q does not exist in general. We introduce the notion of aperiodic polynomials and show that it is always possible to find a polynomial Q' which predicts all the aperiodic factors in the denominator q of a solution y=p/q. (ISSAC 2010.) Next, we show that with a refined version of the same technique we are also able to deduce at least some partial information about the periodic factors of q. (ISSAC 2011.)