# How to solve a formal PDE

Introduction

A multivariate formal series $f\in\mathbb{Q}[[x_1,\ldots ,x_n]]$ can be written as $$f(x_1,\ldots ,x_n) = \sum_{i_1,\ldots ,i_n\geq 0}c(i_1,\ldots ,i_n)x_1^{i_1}\cdots x_n^{i_n} \quad .$$ When $1\leq k\leq n$ denote $f_k$ the partial derivative of $f$ with respect to the variable $x_k$, and when $1\leq k_1,\ldots k_t\leq n$ denote $f_{k_1\cdots k_t}$ the order $t$ partial derivative with respect to the variables $x_{k_1},\ldots ,x_{k_t}$. Since two formal power series are equal if all the coefficients of their monomials are equal, a partial differential equation for $f$ (formal PDE) is equivalent to a system of polynomial equations for the coefficients $c(i_1,\ldots ,i_n)$.

Example of mixed exponential type

Consider $n=2$ variables, and the formal power series $$f(x_1,x_2)=\sum_{d\geq 1}N(d)e^{dx_1}\frac{x_2^{3d-1}}{(3d-1)!} \quad .$$ This is an example of formal series of "mixed exponential type", because its presentation involves exponentials of some of the variables. By expanding the exponential and renaming indices, it can be written as above with $c(i_1,i_2)=(\frac{i_2+1}{3})^{i_1}N(\frac{i_2+1}{3})/(i_1!i_2!)$ if $i_2 \equiv 2 \; (\textrm{mod 3})$ and $c(i_1,i_2)=0$ otherwise. The PDE $$f_{222} = f_{112}^2 -f_{111}f_{122}$$ is equivalent to the polynomial equations for $d\geq 2$ $$N(d)=\sum_{d_1+d_2=d}N(d_1)N(d_2)\left( d_1^2d_2^2 \binom{3d-4}{3d_1-2} -d_1^3d_2\binom{3d-4}{3d_1-1}\right)$$ and this sequence is uniquely determined by the value of $N(1)$. As discovered by Kontsevich, if $N(1)=1$ then $N(d)$ is the number of rational curves of degree $d$ through $3d-1$ generic points in the complex projective plane.

Question

Can one use SageMath to solve formal PDEs? By solving I mean the following: given the formal PDE and a degree bound $b \in \mathbb{N}$, compute all numbers $c(i_1,\ldots ,i_n)\in\mathbb{Q}$ with $i_1+\ldots +i_n\leq b$ for the solution (assuming that it exists and is unique). In practice, it would be nice if the proposed method allowed to work with formal series of "mixed exponential type" too. A good test would be to see if in the example above the method computes, given $b\in\mathbb{N}$, all numbers $N(d)$ with $d \leq b$ as listed in OEIS A013587 (cannot post link due to low karma).

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Do you mean a function like pdsolve/series in Maple?

( 2022-12-14 01:53:07 +0200 )edit

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This functionality can be implemented via solving a system of polynomial equations:

L = 6        # number of terms to compute
R = PolynomialRing(QQ,L,'N')
N = R.gens()
K.<x1,x2> = PowerSeriesRing(R,default_prec=3*L)
f = sum(N[d]*((d+1)*x1).exp()*x2^(3*(d+1)-1)/factorial(3*(d+1)-1) for d in range(L))
p = f.polynomial()    # differentiation seems to be broken for power series
x = p.parent().gens()
eq = p.derivative( x[1], 3 ) - p.derivative( x[0], x[0], x[1] )^2 + p.derivative( x[0], x[0], x[0] )*p.derivative( x[0], x[1], x[1] )
J = R.ideal( list(eq.dict().values())+[N[0]-1] )
print( J.variety() )


Intermediate conversion to polynomial is needed here due to the bug https://trac.sagemath.org/ticket/34846

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