Connections Between Differential Equation and Algebraic Geometry

Notation

$K$$K$ : an algebraic closed field.

$K[X_1,...,X_n]$$K[X_1,...,X_n]$ : the polynomial ring over $K$$K$.

${\mathbb{A}}_K^n$$\mathbb A_{K}^n$: the Affine Space over $K$$K$.

${\mathbb{P}}_K^n$$\mathbb P^n_K$: the Projective Space over $K$$K$.

$V(f)$$V(f)$: Variety of ideal $(f)$$(f)$

$A(X)$$A(X)$: coordinate ring of $X$$X$

${\partial }_{x_i}$$\part_{x_i}$: $\begin{array}{c}\frac{\partial }{\partial x_i}\end{array}$$\frac{\part}{\part x_i}\\$

$D$$D$: $\begin{array}{c}\frac{d}{dx}\end{array}$$\frac{d}{dx}\\$

$e^{\lambda x}{P}_n(\mathbb{C})$$e^{\lambda x}P_n(\mathbb C)$: $\mathrm{Span}\{e^{\lambda x},e^{\lambda x}x,...,e^{\lambda x}{x}^{n-1}\}$$\mathrm{Span}\{e^{\lambda x},e^{\lambda x}x,...,e^{\lambda x}x^{n-1}\}$

 

Connection Between Differential Equation and Algebraic Geometry

In Algebraic Geometry, we study the variety and the coordinate ring of variety.

In Differential Equations, we could consider sth similar.

For example.

We could consider $V(f)\subset {\mathbb{A}}_{\mathbb{C}}^1$$V(f)\sub\mathbb A_\mathbb C^1$ and the coordinate ring $\mathbb{C}[X]/(f)$$\mathbb C[X]/(f)$. Where $f\in \mathbb{C}[X]$$f\in\mathbb C[X]$ is a polynomial.

The $V(f)$$V(f)$ is the subset of ${\mathbb{A}}_{\mathbb{C}}^1$$\mathbb A_{\mathbb C}^1$ where $(f)$$(f)$ vanish.

What I will do is consider this correspondence.

$$\begin{array}{}\text{(1)}& {\mathbb{A}}_{\mathbb{C}}^n⟷C^{\mathrm{\infty }}({\mathbb{C}}^n)\end{array}$$

and

$$\begin{array}{}\text{(2)}& \mathbb{C}[X_1,...,X_n]⟷\mathbb{C}[{\partial }_{x_1},...,{\partial }_{x_n}]\end{array}$$

In this case,

$$\begin{array}{}\text{(3)}& \mathbb{C}[X_1,...,X_n]\end{array}$$

is polynomial function ring over ${\mathbb{A}}_{\mathbb{C}}^n$$\mathbb{A}_{\mathbb C}^n$.

The

$$\begin{array}{}\text{(4)}& \mathbb{C}[{\partial }_{x_1},...,{\partial }_{x_n}]\end{array}$$

is act on ${\mathbb{C}}^{\mathrm{\infty }}({\mathbb{C}}^n)$$\mathbb C^{\infty}(\mathbb C^n)$.

The trivial ($n=1$$n=1$) work I have already done is ODE:An Algebraic Approach.

That is, the solution of $f(D)=\lambda (D-{\lambda }_1{)}^{j_1}...(D-{\lambda }_n{)}^{j_n}\in \mathbb{C}[D]$$f(D)=\lambda(D-\lambda_1)^{j_1}...(D-\lambda_n)^{j_n}\in\mathbb C[D]$

is

$$\begin{array}{}\text{(5)}& \underset{i=1}{\overset{n}{⨁}}{e}^{{\lambda }_{i}x}{P}_{j_i}\end{array}$$

The key point is the beautiful identity

$$\begin{array}{}\text{(6)}& D-\lambda =e^{\lambda x}\circ D\circ e^{-\lambda x}\end{array}$$

So here we find a corresponding

$$\begin{array}{}\text{(7)}& V(f)⟷\underset{i=1}{\overset{n}{⨁}}{e}^{{\lambda }_{i}x}{P}_{j_i}\end{array}$$

For the coordinate ring

$$\begin{array}{}\text{(8)}& \mathbb{C}[X]/(f(X))⟷\mathbb{C}[D]/(f(D))\end{array}$$

Since I have no idea about $\mathrm{P}\mathrm{D}\mathrm{E}$$\rm PDE$, I do not know could this correspondence works or not in general.

Let us see one application.

We know that the Cauchy-Riemann Condition is

$$\begin{array}{}\text{(9)}& ({\partial }_x+i{\partial }_y)f=0\end{array}$$

Then we could consider the corresponding

$$\begin{array}{}\text{(10)}& X+iY=0⟺(z,iz)\in {\mathbb{C}}^2\end{array}$$

If we write it in a matrix form, i.e.

$$\begin{array}{}\text{(11)}& \left(\begin{array}{cc}a& -b\\ b& a\end{array}\right)\end{array}$$

Which is the matrix representation of complex number!

And $C-R$$C-R$ condition is equivalence to the Jacobi matrix is a complex number!

Readers could compare it with this result.