Length of Polynomial Ascending Chains and Primitive Recursiveness.
Guillermo Moreno Socías · MATHEMATICA SCANDINAVICA · 1992
In a polynomial ring K[X 1 ; : : : ; Xn ] over a field, let I 0 ae I 1 ae \\Delta \\Delta \\Delta ae I s be a strictly ascending chain of ideals, with the condition that every I i can be generated by elements of degree not greater than f(i). A. Seidenberg showed that there is a bound on the length s of such a chain depending only on n and f , which is recursive in f for every n and primitive recursive in f for n = 2. In this paper we give a better bound, expressed in a rather simple way in terms of f , which is attained when f is an increasing function. We prove that it is primitive recursive in f for all n. We also show that, on the contrary, there is no bound which is primitive recursive in n in general. 0. INTRODUCTION Let R = K[X 1 ; : : : ; Xn ] be a polynomial ring over a field K. By definition of noetherianity, the length of any strictly ascending chain I 0 ae I 1 ae \\Delta \\Delta \\Delta ae I i \\Delta \\Delta \\Delta of ideals of R is finite, but to bound it we need some info...