Re: JSH: Where in the world is Carmen San Diego ?


J.S.Harris Fan wrote: [slightly edited: ce]

James' most admirable attribute is his relentless tenacity. Math and
science, all knowledge, is the cumulative result of a half million years
of arguing, squabbling, and general disagreement over what constitutes
fact. Without such a process all knowledge would quickly evaporate
due to lack of interest.

When tenacity is offset by willful refusal to learn from ones mistakes,
it makes no positive contribution to the process of acquiring
knowledge.

Rather James' best attribute is asking the odd question, challenging
us to try and construct a context in which the question makes sense.

How many times have we (in vain) asked James, What ring are you
working in? Yet it is interesting to try and answer "for him".

Here's an example that came up about 2 months ago. James wants
to define something stronger than irreducible about polynomials, in a
thread he started on Sept. 11, 2006 called "JSH: Problem with ring of
algebraic integers".

He wants f(x) to remain irreducible "no matter what" is substituted
for x. Challenge: For a sensible notion of "no matter what", find out
whether such polynomials exist.

I'm paraphrasing. James' exact words were:

"That is, P(x) can't be factorable into polynomial factors nor factors
that can be turned into polynomial with a variable substitution. What
does that mean exactly?"

Indeed, what does that mean exactly? As is often the case, James
resorts at that point to giving an example, x -1 becomes reducible
when y^2 is substituted for x. The rest is left to the reader's
imagination.

Some interesting work was done on this by sttscitrans, who showed
that if rational polynomial substitutions are allowed, then we can get
any irreducible quadratic (over the integers) to become a reducible
quartic (over the rationals), but his(her?) reply was only to alt.math
and did not propagate to sci.math and alt.math.recreational as JSH
had originally cross-posted. I suspect that if we limit ourselves to
integer polynomial substitutions, though, the answer will be that
there are irreducible polynomials like x+2 that remain irreducible
under substitutions.

IMHO a nice solution to this question would be publishable, at
least as an expository article in a journal like the Math Monthly.

While the majority of JSH's posts do not inspire any meaningful
mathematics, even in refracted form as above, enough of them
do to provide intellectual stimulation lasting years. Truly, JSH
is a unique individual who seems to enjoy the attention his rants
get on various newsgroups.


regards, chip

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