# Solution spaces of algebraic differential equations and derived geometry

We consider potentially non-linear differential equations on the formal one dimensional disc $$\Delta$$. Such equations are given by expressions $$P(z,f,f',f'',...)=0,$$ where $$P$$ is an element of the commutative algebra $$\mathcal{D}:=\mathbb{C}[[z]][x_{0},x_{1},...,x_{n},...]$$ of polynomials in infinitely many variables with coefficients in $$\mathbb{C}[[z]]$$. Note that $$\mathcal{D}$$ is endowed with a derivation $$\delta$$ which acts on $$z$$ as $$\partial_{z}$$ and satisfies $$\delta(x_{i})=x_{i+1}$$. The data of the pair $$(\mathcal{D},\delta)$$ will be called a differential algebra. The element $$P\in \mathcal{D}$$ generates a differential ideal $$\langle P \rangle_{\delta}$$ We call a map of differential algebras $$S: \mathcal{D}/\langle P\rangle_{\delta}\rightarrow\mathbb{C}[[z]]$$ a solution of $$P$$. Note that $$\mathbb{C}[[z]]$$ is considered as a differential algebra with differential $$\partial_{z}$$. We remark that $$S$$ is uniquely determined by the value $$S(x_{0})=f$$ and that indeed we have $$P(z,f,f',f'',...)=0$$.

The space of solutions is defined as the pre-sheaf of spaces, $$\mathbb{S}(P)$$, which takes a derived algebra $$B$$ to the space of maps of differential algebras, $$\mathcal{D}/\langle P \rangle_{\delta}\otimes B\rightarrow\mathbb{C}[[z]].$$

Remark: In down to earth terms $$\mathbb{S}(P)$$ is the space cut out (in the derived sense) by the various equations on coefficients implied by the differential equation.

It is not hard to show that in simple cases $$\mathbb{S}(P)$$ is a classical space, ie there is no derived structure. For example this is the case if $$P$$ is given by a linear ODE. Less obviously there are also equations for which $$\mathbb{S}(P)$$ does have some non-trivial derived structure. A nice example is given by taking $$P$$ to be the minimal algebraic differential equation satisfied by the function $$\exp(\frac{1}{\sqrt{z}})$$.

Question: Do these higher invariants measure something of interest, ideally something that can be formulated in terms of the classical geometry of the differential equation?