number theory

On Conservative Matrix Fields: Continuous Asymptotics and Arithmetic

arXiv:2507.08138

summary

The paper introduces Conservative Matrix Fields as a framework for analyzing D‑finite functions and uses them to derive asymptotic properties of linear forms in periods, such as multivariate Mellin integrals, and proposes conjectures on their continuous asymptotic and arithmetic behavior.

Abstract

We present the Conservative Matrix Field (CMF) as a tool for the analysis and computation of D-finite functions. We use conservative matrix fields to establish asymptotic properties of families of linear forms in periods, including (but not limited to) multivariate Mellin integrals, via a discrete Levinson-type framework due to Benzaid and Lutz. Finally, we present an experimental analysis of the families of linear forms generated by these objects and formalize the resulting observations as conjectures on their continuous asymptotic and arithmetic properties.

Revised structure and exposition. Added a proof of a statement previously formulated as a conjecture. This version has been submitted for publication

Topics & keywords

#conservative matrix fields#d-finite functions#asymptotic analysis#linear forms in periods#mellin integrals#symbolic computationconservative matrix fieldD-finitelinear forms in periodsmultivariate Mellin integraldiscrete Levinson methodarithmetic conjectures