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Transactions of the American Mathematical Society

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The existence of solutions of abstract partial difference polynomials.

Author: Irving Bentsen
Journal: Trans. Amer. Math. Soc. 158 (1971), 373-397
MSC: Primary 12.80
MathSciNet review: 0279078
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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.

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Keywords: Partial difference ring, ring isomorphisms commuting under composition, inversive closure, abstract partial difference polynomial, existence of solutions of abstract partial difference polynomials, algebraic closure, partial difference kernel, prolongation of partial difference kernel, realizations of partial difference kernels, specialization over partial difference field, compatibility property $ {\mathcal{P}^ \ast }$, stepwise compatibility condition, transformally independent set, primary difference field extension, regular difference field extension, quasi-linearly disjoint field extensions
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