On logarithmic differential operators and equations in the plane
Let $k$ be a field of characteristic zero, $f\in\mathbf{A}:=k[x_{0}, y_{0}]$ an irreducible polynomial and $D = \sum_{i, j\geq 0}a_{i,j}(x_{0}, y_{0})\partial_{x_{0}}^{i}\partial_{y_{0}}^{j}$ a differential operator with coefficients in $\mathbf{A}$. Then a differential equation of the form $D(f) = a(x_{0}, y_{0})f$ is said to be a \textit{logarithmic} polynomial PDE along $\{f=0\}$. The polynomial $\mathrm{Sb}(D) = \sum_{i, j\geq 0}a_{i,j}x_{1}^{i}y_{1}^{j}\in\mathbf{A_{1}}:=k[x_{0}, y_{0}, x_{1}, y_{1}]$ is called a \textit{symbol} of $D$ (or a symbol of the PDE). The operator $D$ (and the PDE) is said to be \textit{homogeneous} if $\mathrm{Sb}(D)$ is a homogeneous polynomial in $x_{1}$ and $y_{1}$ (with coefficients in $\mathbf{A}$). The set of logarithmic polynomial PDEs can be identified, via symbols, with an ideal of $\mathbf{A_{1}}$ (the inverse image of the ideal $\langle f\rangle\mathbf{A}$ under the $k$-algebra morphism $\mathbf{A_{1}}\rightarrow\mathbf{A}$ defined by $x_{1}\mapsto\partial_{x_{0}}(f)$, $y_{1}\mapsto\partial_{y_{0}}(f)$) denoted by $\mathcal{E}_{k}(f)$. \par The paper under review studies the associate homogeneous ideal $\hat{\mathcal{E}}_{k}(f) = \bigoplus_{i\geq 0}\mathcal{E}_{k}^{(i)}(f)$ in the ring $\mathbf{A}_{1}$, where $\mathcal{E}_{k}^{(i)}(f)$ ($i\in\mathbb{N}$) is the $\mathbf{A}$-module generated by the elements of $\mathcal{E}_{k}(f)$ homogeneous of degree $i$. The investigation of this object allows the author to obtain new information about the space of polynomial PDEs of order one in the plane, which admit $f$ as a solution. In particular, the author obtains the following result that gives a characterization of $\hat{\mathcal{E}}_{k}(f)$. (In what follows, $\mathcal{C}$ denotes the integral affine curve $\mathrm{Spec}(\mathbf{A}/\langle f\rangle)$ of the affine plane $\mathbf{A}_{k}^{2}$ and $V_{s}^{\mathcal{C}}$ denotes the component $\{D\in\mathcal{D}\mid \forall l\in\mathbb{Z}, D(\langle f\rangle^{l})\subset \langle f\rangle^{l-s}\}$ of the \textit{$V$-filtration along $\mathcal{C}$} of the ring of differential operators $\mathcal{D}$ with coefficients in $\mathbf{A}$.) Let $P = \sum_{j=0}^{d}a_{j}(x_{0}, y_{0})x_{1}^{j}y_{1}^{d-j}$ ($d\in\mathbb{N}$) be a homogeneous polynomial in $\mathbf{A}_{1}$ with total degree (in $x_{1}, y_{1}$) equal to $d$. Let $D_{P} = \sum_{j=0}^{d}a_{j}(x_{0}, y_{0})\partial_{x_{0}}^{j}\partial_{y_{0}}^{d-j}$ be the associated differential operator. Then the following statements are equivalent: (i) $P$ is the symbol of a homogeneous element of ${\mathcal{E}}_{k}(f)$ of degree $d$, that is, $P(\partial_{x_{0}}(f), \partial_{y_{0}}(f))\in\langle f\rangle\mathbf{A}$. (ii) $D_{P}\in V_{d-1}^{\mathcal{C}}$. (iii) $D_{P}(f^{d})\in\langle f\rangle$. \par Another central result of the paper states that if $\tau:\mathbf{A}_{1}\rightarrow \mathbf{A}_{1}$is the $\mathbf{A}$-automorphism defined by $\tau(x_{1}) = y_{1}$, $\tau(y_{1}) = -x_{1}$ (we use the above notation), $\Delta(f) = \partial_{x_{0}}(f)x_{1} + \partial_{y_{0}}(f)y_{1}$, and $\partial(f)$ a nonzero partial derivative of $f$, then $\tau(\hat{\mathcal{E}}_{k}(f)) = \langle f, \Delta(f)\rangle:\partial(f)^{\infty}$. The paper also contains some consequences of the above-mentioned results, including a characterization of the general component of the tangent space of ${\mathcal{C}}$.
scientific article; zbMATH DE number 7036784