FOM: Some reflections on Rota

Stephen G Simpson simpson at
Fri Apr 23 17:19:46 EDT 1999

Quoting Gian-Carlo Rota, Walter Whiteley writes:

 > "Hlibert's theorem on finite generation of the ring of
 > invariants can be recast in the language of umbrae
 > and can be given a simple combinatorial proof that 
 > dispenses with the Hilbert Basis Theorem"

Years ago I published a paper on reverse mathematics aspects of the
Hilbert Basis Theorem.  (It turns out that HBT is logically equivalent
to well-orderedness of the ordinal number omega^omega, in a certain
sense.  The equivalence is via a combinatorial lemma known as
Dickson's Lemma: for all n, the monomials on n commuting variables are
well-partially-ordered under divisibility.)  But I never looked at
Hilbert's theorem on finite generation of the ring of invariants in
this light.  

Walter, can you supply a reference for the proof that Rota refers to?

It was a shock to hear of Rota's death.  He was one of my revered
teachers at MIT.  We kept in touch off and on over the years.  Last
fall we sat next to each other at a banquet at Penn State.  He seemed
chipper as always.  We had a lively conversation about all kinds of
things, including his book ``Indiscrete Thoughts'', which I had just

-- Steve

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