Ansatz for cinf-structures

Motivation

The general determining equations for the functions ${\lambda }_1,\dots ,{\lambda }_{m-1}$ are a system of coupled, non-linear partial differential equations. Solving them in full generality is often computationally intractable. To overcome this, one imposes constraints—an ansatz—to reduce the complexity of the system.

Common Strategies

1. variable restriction: The most common ansatz involves assuming that the functions ${\lambda }_i$ depend on a subset of the jet bundle coordinates (e.g., assuming ${\lambda }_i={\lambda }_i(x,u)$ instead of the full dependency ${\lambda }_i(x,u,\dots ,u_{m-1})$).
2. Polynomial forms: Another approach is to assume ${\lambda }_i$ is a polynomial in the derivative coordinates $u_1,\dots ,u_{m-1}$.
3. Assuming, for example, ${\lambda }_2=-{\lambda }_1+\eta (x,u,\dots )$, where $\eta$ has a restricted dependency.

The Trade-off

Using an ansatz simplifies the PDEs significantly, often transforming them into an overdetermined linear system that can be solved algorithmically. However, this comes with a drawback: by enforcing strong constraints, one reduces the probability that a valid ${\mathcal{C}}^{\mathrm{\infty }}$-structure exists within the search space

See also: determining equations for cinf-structures, canonical cinf-structure, recursive prolongation formula