The existence of solutions of abstract partial difference polynomials.

Abstract:

A partial difference (p.d.) ring is a commutative ring $R$ together with a (finite) set of isomorphisms (called transforming operators) of $R$ into $R$ which commute under composition. It is shown here that (contrary to the ordinary theory [R. M. Cohn, Difference algebra]) there exist nontrivial algebraically irreducible abstract p.d. polynomials having no solution and p.d. fields having no algebraically closed p.d. overfield. If $F$ is a p.d. field with two transforming operators, then the existence of a p.d. overfield of $F$ whose underlying field is an algebraic closure of that of $F$ is a necessary and sufficient condition for every nontrivial algebraically irreducible abstract p.d. polynomial $P$ in the p.d. polynomial ring $F\{ {y^{(1)}},{y^{(2)}}, \ldots ,{y^{(n)}}\}$ to have a solution $\eta$ (in some p.d. overfield of $F$) such that: $\eta$ has $n - 1$ transformal parameters, $\eta$ is not a proper specialization over $F$ of any other solution of $P$, and, if $Q$ is a p.d. polynomial whose indeterminates appear effectively in $P$ and $Q$ is annulled by $\eta$, then $Q$ is a multiple of $P.P$ has at most finitely many isomorphically distinct such solutions. Necessity holds if $F$ has finitely many transforming operators.