Suppose you are given a family of differential equations with rational coefficients: $\partial_x^ny+a_1(x,x_1,\dots,x_m)\partial_x^{n-1}y+\dots+a_n(x,x_1,\dots, x_m)y=0$. At each point $X=(x_1,\dots, x_m)\in\mathbb C^m$, we can attach a vector space of solutions to the differential equation above. This would be a vector bundle if for each $X$, we had a local trivialization, i.e. a neighborhood $U$ of $X$ such that we had $n$ continuous maps $\Phi_i:U\to\mathcal O(\mathbb C)$, from $U$ to the meromorphic functions on $\mathbb C$, such that at each $Y\in U$, $\{\Phi_i(Y)\}_{i=1}^n$ is a basis of solutions of the ODE over $\mathbb C$. But this is not the case.

For example, with constant coefficients, i.e. the family of differential equations $\partial_x^ny+a_1\partial_x^{n-1}y+\dots+a_ny=0$, with $a_i\in\mathbb C$, it is standard from an introductory course on ODEs that the points $a=(a_1,\dots, a_n)$ where there does not exist a local trivialization are precisely where the discriminant $\Delta$ of the polynomial $p(\lambda)=\lambda^n+a_1\lambda^{n-1}+\dots+a_n$ is zero: $\Delta=0$. Away from this algebraic set, the solutions are the $n$ linearly independent functions $e^{rx}$, where $r$ is one of the $n$ roots of $p$. On the set $\{\Delta=0\}$, you get solutions of the form $xe^{rx}$. Thus, we obtain a vector bundle over $\mathbb C^n\setminus\{\Delta=0\}$. A similar result holds for equations $\partial_x^ny+\frac{a_1}{x}\partial_x^{n-1}y+\dots+\frac{a_n}{x^n}y=0$ (parameterized by $a=(a_1,\dots, a_n)\in\mathbb C$); the singular points $a$ are exactly those which satisfy a certain algebraic equation.

My question: has the singularities described above and the vector bundles constructed in this way (using other families of ODEs) been studied? Do you have any resources?


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