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.
Categories: math.AC