FoCM 2014 conference

Workshop B7 - Symbolic Analysis

December 15, 14:30 ~ 14:55 - Room A11

## A decision method for integrability of partial differential algebraic Pfaffian systems.

### Lisi D'Alfonso

### Universidad de Buenos Aires, Argentina - lisi@cbc.uba.ar

Let $m,n\in \mathbb{N}$. Let $x_1,\ldots,x_m$ be independent variables and $\mathbf{y}:=y_1,\ldots,y_n$ be differential unknowns. For each pair $(i,j)$, $1\le i\le n$, $1\le j\le m$, let $f_{ij}$ be a polynomial in $\mathbb{C}[\mathbf{y}]$. A differential algebraic Pfaffian system is a system of differential equations as follows: \[ \Sigma = \left\{ \begin{array}{clll} \ \dfrac{\partial y_i}{\partial x_j} & = & f_{ij}(\mathbf{y}), & \ \ \text{for} \ i=1, \ldots, n \ \text{and}\ j=1\ldots m, \\ \mathbf{g}(\mathbf{y}) & = & 0 &\ \end{array} \right. \] where $\mathbf{g}:=g_1,\ldots,g_s$ are polynomials in $\mathbb{C}[\mathbf{y}]$.

In this work we are interested in the integrability of these systems, that is, in the existence of infinitely differentiable functions over an open set $\mathcal{U}$ of $\mathbb{C}^m$ that are solutions of $\Sigma$. The classical Frobenius Theorem (1877) establishes conditions for a Pfaffian system, without algebraic constraints, to be completely integrable. We focus on the integrability, not necessarily complete, of systems like $\Sigma$.

We associate to each system $\Sigma$ a strictly decreasing chain of algebraic varieties in $\mathbb{C}^{n}$ of length at most $n+1$. We prove that a necessary and sufficient condition for the existence of solutions for $\Sigma$ is that the smallest variety of this chain is nonempty. From this result, we are able to show an effective procedure that allows us to decide whether a Pfaffian system is integrable in triple exponential time in $n$, the number of unknowns.

Joint work with Gabriela Jeronimo (Universidad de Buenos Aires) and Pablo Solernó (Universidad de Buenos Aires).