On ascending chains of ideals in the polynomial ring

TL;DR

This paper establishes an elementary bound on the length of ascending chains of ideals in polynomial rings over a field, based on degrees of generators, with potential applications in algebraic geometry and computational algebra.

Contribution

It constructs a natural number bound for the length of ascending chains of ideals with degree constraints, providing a new elementary approach to a classical problem.

Findings

Bound on chain length depending on number of variables and degree function

Elementary construction of the bound

Discussion of applications in algebraic geometry and computational algebra

Abstract

Assume that $K$ is a field and $I_{1}\subsetneq ...\subsetneq I_{t}$ is an ascending chain (of length $t$) of ideals in the polynomial ring $K[x_{1},,...,x_{m}]$, for some $m\geq 1$. Suppose that $I_{j}$ is generated by polynomials of degrees less or equal to some natural number $f(j)\geq 1$, for any $j=1,...,t$. In the paper we construct, in an elementary way, a natural number $\mathcal{B}(m,f)$ (depending on $m$ and the function $f$) such that $t\leq\mathcal{B}(m,f)$. We also discuss some possible applications of this result.