Solvability of the Cauchy problem for a polynomial difference operator and monomial bases for the quotients of a polynomial ring

We find solvability conditions for a Cauchy problem with a polynomial difference operator and, in particular, give an easy-to-check sufficient condition in terms of the coefficients of the principal symbol of the difference operator. The solvability of the Cauchy problem is shown to be equivalent to the existence of a monomial basis in the quotient ring of the polynomial ring by the ideal generated by the characteristic polynomial.