On the image conjecture (Q661890)

The Image Conjecture is as follows: Let \(k\) be a field and \(A\) be a \(k\) -algebra, and let \(B=A[z_{1},\ldots ,z_{n}].\) Then for \(a_{1},\ldots ,a_{n}\in A\) a regular sequence, the image of the \(A\)-linear map \( B^{n}\rightarrow B\) defined by \(\mathcal{D=(\partial }_{z_{1}}-a_{1},\ldots , \mathcal{\partial }_{z_{n}}-a_{n}\mathcal{)}\) is a Mathieu subspace in \(B\) (by definition a sub \(k\)-vector space \(\mathcal{M}\) of \(B\) is called a Mathieu subspace if for all \(f\in B,\) if \(f^{m}\in \mathcal{M}\) for all \( m\geq 1\) then for any \(g\in B\) we have \(f^{m}g\in \mathcal{M}\) for all \(m\gg 0).\) The Image Conjecture is important because it implies the Jacobian Conjecture [\textit{W. Zhao}, J. Algebra 324, No. 2, 231--247 (2010; Zbl 1197.14064)]. The authors prove various cases of the Image Conjecture (for instance for \(k\) of positive characteristic) and connect it to the following Factorial Conjecture: if \(f\in \mathbb{C}[z_{1},\ldots ,z_{n}]\) satisfies \( \mathcal{L}(f^{m})=0\) for all \(m\geq 1\) then \(f=0,\) where \(\mathcal{L}: \mathbb{C}[z_{1},\ldots ,z_{n}]\rightarrow \mathbb{C}\) is the \(\mathbb{C}\) -linear map defined on monomials by \(\mathcal{L}(z_{1}^{l_{1}}\cdots z_{n}^{l_{n}}):=l_{1}!\cdots l_{n}!\) They prove various cases of the Factorial Conjecture.