# Opinion 31: Dave Bayer is Right When He Said That
We Should Sometimes Forget the Sacrosanct Principle of
Minimality

## By Doron Zeilberger

Written: Oct. 26, 1998

One of the best (on many-levels!) talks that I have
ever heard was Dave Bayer's invited talk, delivered
at the MSRI workshop on Symbolic Computation in Geometry
and Analysis (Oct. 12-16, 1998, Oct. 13, 9:30-10:30,
soon to be posted at the MSRI site).
In addition to the mathematics itself, that while beautiful,
will probably not influence my own research very much,
the meta-mathematics, the rich metaphors, and the
general style of DOING and LOVING mathematics, certainly will.

One very wise suggestion was to take a break from hiking, and
pause and enjoy the view. Most of us, myself included,
are too busy trying to prove theorems, and leave too little
time for retrospection.

But, perhaps the most useful lesson that I got out of
Dave Bayer's outstanding talk was the advice that
it sometimes could be useful to abandon the sacrosanct
obsession, that we mathematicians have, with minimality.
In other words, we are slaves of Occam's razor.
We always want the shortest possible proof, the
sharpest estimates with as few assumptions as possible,
a canonical base that is minimal, the most succinct
formula, the most efficient algorithm etc.

This was even quantified by Gregory Chaitin who defined
a program to be `most elegant' if it is as short
as possible.

In Dave Bayer's talk he mentioned, that he himself,
as the pioneering co-developer (with Mike Stillman)
of the Macaulay system, was always using Groebner
bases because these are canonical and, in a certain sense,
minimal (if they are reduced). While COMPUTATIONALLY,
of course, Groebner bases (the amazing brainchild of
Bruno Buchberger), are the bases of choice,
for theoretical development, there may be non-minimal, and
not-necessarily-canonical bases that may be better.

It so happened that Dominique Foata and I have spent
the last summer trying, so far in vain, to prove
Mark Haiman's notorious (n+1)^(n-1) conjecture.
Our approach was to construct explicitly (recursively)
a Groebner basis for the relevant ideal. But the
Groebner basis seems to be a mess. Dave Bayer's
advice gave us renewed hope, and now we are searching
for other, not necessarily canonical, and not necessarily
reduced, bases, that would, who knows?, prove the conjecture.

So Sometimes (mathematical) FAT is BEAUTIFUL,
and LONG can be SWEET.

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