2025/09/16 by Yvain Bruned, Bruned, Yvain, Paul Laubie +1
Economics, Econometrics and Finance · Physics and Astronomy · #Combinatorics (math.CO) #FOS: Mathematics #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Probability (math.PR) #Representation Theory (math.RT) #Statistical Mechanics and Entropy #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2509.13118
openalex publication_date 2025/09/16 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28
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≠ 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.