Iterative differential embedding problems in positive characteristic (Q2253053)

The paper under review studies an inverse problem of differential Galois theory with iterated derivation. Let \(R\) be a commutative ring. A family \(\partial^{\ast} = (\partial^{(k)})_{k\in \mathbb{N}}\) of additive maps \(\partial^{(k)}:R\rightarrow R\) is called an iterative derivation of \(R\) if \(\partial^{(0)} = \mathrm{id}_{R}\), \(\partial^{(k)}(ab) = \sum_{i+j=k}\partial^{(i)}(a)\partial^{(j)}(b)\), and \(\partial^{(j)}\circ\partial^{(i)} = {{i+j}\choose i}\partial^{(i+j)}\). The pair \((R, \partial^{\ast})\) is called an iterated differential ring or an ID-ring. If \(R\) is a field, it is called an ID-field. The concepts of an ID-homomorphism, ID-module, ID-field extension \(E/F\) and the iterative differential Galois group \(\mathrm{Gal}_{\text{ID}}(E/F)\), etc. are defined as analogues of the corresponding concepts of the classical differential algebra. An iterative differential equation over an ID-field \(F\) is defined as a certain matrix equation of the form \(\partial^{(p^{l})}(\mathbf{y}) = A_{l}\mathbf{y}\), and the iterative Picard-Vessiot ring (IPV-ring) for this equation, is introduced as an analogue of the classical Picard-Vessiot extension in the differential Galois theory. Let \(K\) be a field of constants of an ID-field \(F\) and \(E/F\) an IPV-extension with Galois group \(\mathcal{G}(K)\cong \mathrm{Gal}_{\text{ID}}(E/F)\), where \(\mathcal{G}(K)\) is a reduced linear algebraic group over \(K\); the category of such groups is denoted by \(\mathbf{AffGr}^{\text{red}}_{K}\). (It is shown that \(\mathrm{Gal}_{\text{ID}}(E/F)\) is isomorphic to the \(K\)-rational points of an affine group scheme defined over \(K\).) Let \(\beta:\tilde{\mathcal{G}}\rightarrow \mathcal{G}\) be an epimorphism in \(\mathbf{AffGr}^{\text{red}}_{K}\). The corresponding \textit{ID-embedding problem} asks for the existence of an IPV-extension \(\tilde{E}/F\) and a monomorphism \(\tilde{\alpha}\) of \(\mathrm{Gal}_{\text{ID}}(\tilde{E}/F)\) onto a closed subgroup of \(\tilde{\mathcal{G}}\) such that \(\beta\circ\tilde{\alpha} = \alpha\circ({\text{res}}:\mathrm{Gal}_{\text{ID}}(\tilde{E}/F)\rightarrow \mathrm{Gal}_{\text{ID}}(E/F))\). \(\tilde{\alpha}\) is called a \textit{solution} of the D-embedding problem. If \(\tilde{\alpha}\) is an isomorphism, it is called a \textit{proper solution} of the problem. Note that if \(\mathcal{A}(K)\) denotes the kernel of \(\beta\), then \(\tilde{\mathcal{G}}\) and \(\mathcal{G}\) are reduced, whereas \(\mathcal{A}\) need not be reduced. The main result of the paper under review is the following theorem: Let \(F\) be an algebraic function field in one variable over an algebraically closed field \(K\) of positive characteristic. Then every ID-embedding problem in \(\mathbf{AffGr}^{\text{red}}_{K}\) over \(F\) with reduced kernel \(\mathcal{A}(K)\) has a proper solution. It should be noted that the question whether a similar theorem holds for fields of zero characteristic is still open.