Polynomial Solutions to the First Order Difference Equations in the Bivariate Difference Field

The bivariate difference filed $(\mathbb{F}(\alpha ,\beta ),\sigma )$ provides an algebraic framework for a sequence satisfying a recurrence of order two and it could transform the summation involving a sequence satisfying a recurrence of order two into the first order difference equations in the bivariate difference field. Based on it, we present an algorithm for finding all the polynomial solutions of such equations in the bivariate difference field, and show an upper bound on the degree for polynomial solutions which is sufficient to compute polynomial solution by using the undetermined method.