Constructive proof of Hilbert’s theorem on ascending chains

Abstract:

In a previous note it was shown that if a bound $f(i)$ is placed on the degrees of the elements in some basis of an ideal ${A_i}$ in the polynomial ring $k[{X_1}, \cdots ,{X_n}]$ over an explicitly given field $k,i = 0,1,2, \cdots$, then a bound can be (and was) constructed for the length of a strictly ascending chain ${A_0} < {A_1} < \cdots$. This result is now obtained using a strictly finitist argument. A corollary is a finitist version of Hilbert’s theorem on ascending chains.