# Generators for the $C^m$-closures of Ideals

## TL;DR

This paper presents an algorithm to compute generators for the $C^m$-closure of ideals in the ring of real polynomials, extending algebraic methods to smooth function contexts.

## Contribution

The paper introduces a novel algorithm for explicitly computing generators of the $C^m$-closure of polynomial ideals.

## Key findings

- Algorithm successfully computes generators for $C^m$-closures
- Extends algebraic ideal theory to smooth function spaces
- Provides a practical tool for analysis in real algebraic geometry

## Abstract

Let $\mathscr{R}$ denote the ring of real polynomials on $\mathbb{R}^{n}$. Fix $m\geq 0$, and let $A_{1},\cdots ,A_{M}\in \mathscr{R}$. The $ C^{m}$-closure of $\left( A_{1},\cdots ,A_{M}\right) $, denoted here by $ \left[ A_{1},\cdots ,A_{M};C^{m}\right] $, is the ideal of all $f\in \mathscr{R}$ expressible in the form $f=F_{1}A_{1}+\cdots +F_{M}A_{M}$ with each $F_{i}\in C^{m}\left( \mathbb{R}^{n}\right) $. In this paper we exhibit an algorithm to compute generators for $\left[ A_{1},\cdots ,A_{M};C^{m}\right] $.