has received many studies [@GMS92; @Seidenberg1971].

The computing is based on Moreno-Socías’s approach [@GMS92], which consists of the following three steps:

-   Reduce computing the bound on $f$-bounded polynomial ideal chains to computing the bound on $f$-generating sequences of monomials, which is obtained by Moreno-Socías’s result [@GMS92].

-   Compute the longest homogeneous $f$-generating sequence, which is achieved directly by using Hilbert’s function and Macaulay’s theorem. This step is different from Moreno-Socias’s, as his result on this step (i.e. Proposition 4.3 in [@GMS92]) is wrong.

-   Prove that the bound of $f$-generating sequences of monomials is exactly same as the length of the longest homogeneous $f$-generating sequence obtained in (ii), which is trivially achieved by introducing a fresh variable.

For any increasing function $f:{\mathbb{N}}\rightarrow{\mathbb{N}}$ (that is, $f(x)\leq f(y)$ for all $x\leq y$), an ascending chain $I_1\subseteq I_2\subseteq\cdots\subseteq I_n$ of polynomial ideals of ${\mathbb{K}}[{\pmb{x}}]$ is called $f$-bounded, if $\operatorname{gdeg}(I_i)\leq f(i)$ for all $i\geq 1$. Denote by $L(d,f)$ the greatest length of all strictly ascending chains of ${\mathbb{K}}[{\pmb{x}}]$ which are $f$-bounded.

1.  The condition of $f$-boundedness is necessary to define the greatest length, as the length of chains with unbounded degrees could be arbitrarily large (for instance, the length of $\langle x^n\rangle\subset\langle x^{n-1}\rangle\subset\cdots\subset\langle 