Asymptotic ideals (ideals in the ring of Colombeau generalized constants with continuous parametrization)

TL;DR

This paper investigates the structure of asymptotic ideals in the ring of continuous functions on (0, 1], characterizing maximal and prime ideals via filters, with implications for understanding asymptotic behavior at zero.

Contribution

It introduces a novel framework for analyzing asymptotics at zero using ideals and filters in the ring of continuous functions, extending previous algebraic approaches.

Findings

Characterization of maximal ideals via maximal filters

Identification of prime ideals through prime filters

Establishment of a correspondence between ideals and filters

Abstract

We study the asymptotics at zero of continuous functions on (0, 1] by means of their asymptotic ideals, i.e., ideals in the ring of continuous functions on (0, 1] satisfying a polynomial growth condition at 0 modulo rapidly decreasing functions at 0. As our main result, we characterize maximal and prime ideals in terms of maximal and prime filters.