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arXiv · 2509.13118

Elementary differentials from multi-indices to rooted trees

Yvain Bruned·Paul Laubie
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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.

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Yvain Bruned, Paul Laubie. 2025-09-16. Elementary differentials from multi-indices to rooted trees. https://arxiv.org/abs/2509.13118

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