Partial Denominator Bounds for Partial Linear Difference Equations

Several algorithms in symbolic computation depend on a subroutine for finding the rational solutions of an ordinary linear difference (or differential) equation (OLDE), and several algorithms are known for implementing such a subroutine [1,2,4,11,13,14,6,8,9]. On a conceptual level, the typical approach for finding rational solutions can be divided into three steps. In the first step, one constructs a polynomial Q such that the denominator q of any potential solution p/q must divide Q. This polynomial Q is called universal denominator or denominator bound. In the second step, the universal denominator is used to transform the given equation into a new equation such that P is a polynomial solution of the new equation if and only if P/Q is a rational solution of the original one. In the third and final step, the polynomial solutions P of the transformed equation are determined.

The first algorithm for computing universal denominators in the case of OLDEs with polynomial coefficients was proposed in 1971 by Abramov [1] (see Section 2 below for a summary). It has been generalized to q-difference equations [3], to matrix equations [5], and also to equations whose coefficients belong to domains other than polynomials. For example, Bronstein [7] and Schneider [12] have observed that a universal denominator can be constructed also when the coefficient domains are difference fields which can be used for representing nested sums and products (ΠΣ-fields). For such domains, the situation is more involved. There is a need to distinguish between “normal” factors of the universal denominator which can be found very much like in the usual polynomial case, and “special” factors which have to be constructed by some other means.

In the present article, we consider partial (i.e., multivariate) linear difference equations with polynomial coefficients (PLDEs). Our ultimate goal is the construction of a universal denominator for potential rational solutions of a given PLDE. Like in the univariate case with sophisticated coefficient domains, there are two kinds of factors to be distinguished. As a matter of fact, some parts of the denominator cannot be bound at all. For example, the equation

has (n+k) -α as a rational solution, for any α ∈ AE, and there is obviously no finite polynomial Q that would be a multiple of (n + k) α for all α ∈ AE. We will call factors that may exhibit such “special” behaviour periodic. Our main result is that we can construct for any given PLDE a polynomial d such that every aperiodic factor of any potential solution p/q must divide d.

Such a bound on the aperiodic factors of the denominators does not directly give rise to a full algorithm for finding rational solutions of PLDEs, but it can be considered as a step in this direction. For a full algorithm, besides of the bounding of the periodic parts of the denominator, also the entire question of how to find (all) polynomial solutions of a PLDE in the third step is wide open and far from being settled. But even if these parts have to remain open for now, our aperiodic denominator bound is useful in practice. When it comes to solving an actual equation, possible periodic factors in a solution can often be guessed by inspection, their multiplicities can be determined by trial and error, and degree bounds for polynomial solutions can be established heuristically. A reasonably tight universal denominator, on the other hand, cannot be as easily obtained on heuristic grounds.

Before entering the multivariate setting, let us summarize Abramov’s classical denominator bound for univariate equations. We will introduce on the fly some notions and notations needed later.

Let à be a field of characteristic zero and let Ã[n] and Ã(n) denote the ring of univariate polynomials and the field of rational functions in n with coefficients in Ã, respectively. Write N for the shift operator acting on Ã[n] and Ã(n) via N q(n) := q(n + 1).

The objects of interest are difference equations of the form

where a0, . . . , am, f ∈ Ã[n] (a0, am = 0) are given and y ∈ Ã(n) is unknown.

The denominator bounding problem is as follows: given

Abramov’s denominator bounding algorithm [1] is an efficient way of computing gcd(

where

is the dispersion of a0 and N -m am. It is efficient in the sense that the gcd is constructed without explicitly calculating the products.

To see that this bound is correct, write (1) in the form

Shifting this equation by s gives

By repeatedly using the recurrence, the terms N i+s y appearing on the right hand side can be reduced to smaller shifts of y so that for certain polynomials b, b0, . . . , bm-1 we have

At this point we rely on the following result.

Theorem 1. [1] For any solution y = p q ∈ Ã(n) of (1), max{i ≥ 0 : gcd(q, N i q) = 1} ≤ s.

This theorem ensures that the denominator of a solution y cannot contain two factors u, v with u = N s+1 v, and this in turn implies that no denominator of any of the N i y on the right can have a common

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