Ideals and differential operators in the ring of polynomials of infinitely many variables
Given an ideal \(J\) of the polynomial ring \(\mathbb{C}[x_1,x_2,\cdots,x_n]\), a theorem due to Krull states that for evey \(f\in \mathbb{C}[x_1,x_2,\cdots,x_n]\setminus J\), there exists a differential operator \(D\) and a complex vector \(c=(c_1,c_2,\cdots,c_n)\) such that \(Dp(c)=0\) for every \(p\in J\) and \(Df(c)\neq 0\). M. Lefranc took profit from this result to prove that the spectral synthesis holds on \(\mathbb{Z}^n\). It is worth noting that the above Krull's theorem does not hold in the polynomial ring \(D:=\mathbb{C}[x_1,x_2,\cdots,x_n,\cdots]\) with countably infinite variables. The paper under review main's goal is to prove that Krull's theorem remains valid in \(D\) via using appropriate generalization of the notion of differential operator. Moreover, the author extends Lefranc's theorem to countable abelian groups by using local polynomials.
polynomials of infinitely many variables
scientific article; zbMATH DE number 6384180