  The bivariate difference field provides an algebraic framework for a sequence
satisfying a recurrence of order two. Based on this, we focus on sequences
satisfying a recurrence of higher order, and consider the multivariate
difference field, in which the summation problem could be transformed into
solving the first order difference equations. We then show a criterion for
deciding whether the difference equation has a rational solution and present an
algorithm for computing one rational solution of such a difference equation, if
it exists. Moreover we get the rational solution set of such an equation.
