Elementary differentials from multi-indices to rooted trees
Yvain Bruned, Paul Laubie
TL;DR
The paper investigates whether there exist intermediate combinatorial systems between rooted trees and multi-indices that encode elementary differentials in fixed dimensions. It proves, for $d\neq 1$, that no faithful or naturally compatible description exists beyond rooted trees when constrained to align with both RT and MI, and strengthens this with a Witt algebra-based non-existence in dimension 2. It introduces $\ell$-labellings to broaden the framework to $\ell$-weak descriptions and shows nuanced results: $2$-weak descriptions face strong obstructions, while a faithful $1$-weak description relative to symmetry-quotiented MI is conceptually possible but not explicitly constructed for $d\neq 1$. Overall, the work argues that meaningful fixed-dimension alternatives to rooted trees are unlikely under natural compatibility requirements, though certain weaker or symmetry-adapted frameworks may still hold potential. The findings have implications for algebraic approaches to B-series, SPDE expansions, and the design of combinatorial tools in numerical analysis.
Abstract
Rooted trees are essential for describing numerical schemes via the so-called B-series. They have also been used extensively in rough analysis for expanding solutions of singular Stochastic Partial Differential Equations (SPDEs). When one considers scalar-valued equations, the most efficient combinatorial set is multi-indices. In this paper, we investigate the existence of intermediate combinatorial sets that will lie between multi-indices and rooted trees. We provide a negative result stating that there is no combinatorial set encoding elementary differentials in dimension $d\neq 1$, and compatible with the rooted trees and the multi-indices aside from the rooted trees. This does not close the debate of the existence of such combinatorial sets, but it shows that it cannot be obtained via a naive and natural approach.