Let $R$ be a noetherian ring. By the Hilbert Basis Theorem the polinomial ring $R[x_1, ... , x_n]$ is also a noetherian ring. What can we say about the number of generators of an ideal $I$ of $R[x_1, ... , x_n]$? (We can suppose that every ideal in $R$ is principal)
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Nothing. Assume $R=k$, a field, for specificity. Then $k[x_1]$ is a principal ideal domain, as you know, but $k[x_1,x_2]$ has ideals with unbounded number of generators. Specifically, $(x_1,x_2)^n$ is minimally generated by $n+1$ elements for all $n$. One can get higher rates of growth by adding more variables. |
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Perhaps I should mention that within certain classes of ideals (for example ideals with the same integral closure or radical or some other closure) people do certainly study the minimal number of generators. For integral closure of ideals (working locally and assuming that the residue field is infinite), then that minimal number of generators of all ideals which have the same integral closure, is called the analytic spread of an ideal. For radical, certainly people are interested in showing various things are set theoretic complete intersections (in other words, that there is an ideal with the same radical as the given one defined by the minimal number of generators one expects by dimension). I'd see the work of Neil Epstein who if I recall correctly developed a theory of when closure operations on ideals have reasonable "analytic spread" behaviors. |
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