Difference equations in Witt vectors
Let \(A\) be a discrete valuation ring of characteristic \(0\) with uniformizing parameter \(t\) and let \(\sigma\) be an automorphism of \(A\). Let \(A[X]_{\sigma}\) denote the ring of polynomials in the indeterminates \(\sigma^i(X)\) for \(i\in{\mathbb Z}\). Assume that \(A\) satisfies an analogue of Hensel's lemma for polynomials in \(A[X]_{\sigma}\). The author sketches a proof of the following analogue of a theorem of \textit{M. J. Greenberg} [Publ. Math., Inst. Hautes Étud. Sci. 31, 563--568 (1966; Zbl 0146.42201)]: given \(f_{1}\), \dots, \(f_{m}\in A[X_{1},\dots,X_{n}]_{\sigma}\), there exists an integer \(N\) such that for every integer \(\alpha>0\) and every \(x_{1}\), \dots, \(x_{n}\in A\) such that \(f_{1}(x_{1},\dots,x_{n})\equiv\dots\equiv f_{m}(x_{1},\dots,x_{n})\equiv0 \mod t^{\alpha N}\), there exist \(y_{1}\), \dots, \(y_{n}\in A\) such that \(f_{1}(y_{1},\dots,y_{n})=\dots=f_{m}(y_{1},\dots,y_{n})=0\) and \(y_{i}\equiv x_{i}\mod t^\alpha\) for \(i=1\), \dots, \(n\). This result is applied to obtain a Nullstellensatz for the field of fractions of the ring of Witt vectors over the algebraic closure of the field of \(p\) elements, with \(\sigma\) the Frobenius automorphism. Detailed proofs will appear elsewhere.
scientific article; zbMATH DE number 2137994
scientific article