Subject: Re: [HM] Roots of polynomials
From: Michael Detlefsen (detlefsen.1@nd.edu)
Date: Fri Apr 28 2000 - 15:23:24 EDT
Thanks to William Waterhouse, John Conway and Bill Tait for their replies
to my inquiry re. laws determining the number of real roots of real
polynomials. They were each helpful in various ways. Actually, I had
thought of Sturm's Theorem ... though I was thinking of it mainly as a law
for determining the number of real roots of a real polynomial within a
given interval. What I did not know, but wondered about, was that, as John
Conway put it, Sturm's Theorem "burst like a bombshell on the 19th century
algebraists, none of whom expected such a result to exist".
This actually highlights the historical puzzles that inspired my earlier
question (and which form part of a project I've been working on for the
past few months). Indeed, they add something more to it. I'll try to state
these puzzles briefly here (though to state them adequately and with full
force requires a fair amount of stage-setting ... and a large number of
quotations and references).
Puzzle I: Sturm proved his theorem in 1829. Yet well into the twentieth
century one finds mathematicians making statements such as this:
"The ordinary complex magnitudes of algebra ... are an instance of the use
of ideal elements; they serve to simplify the theorems on the existence and
number of roots of an equation." (Hilbert 1925 (On the Infinite)).
Huntington also made such remarks in his 1911 treatise on the fundamental
propositions of algebra ... and there are quite a few other such statements
that I know of as well.
I find such statements puzzling. In particular, they raise the following
questions:
(a) (i) Are we to understand Hilbert (and the others who make similar
present-tense claims) as holding the view that Sturm's Theorem (and related
theorems by Cauchy and others) is a significantly less simple law than the
simple form of the FTA that Descartes stated for polynomials with real
coefficients? If so, what is the notion of complexity (simplicity) that is
involved?
(ii) Was Sturm's Theorem not well known or appreciated for a long time? Was
there perhaps some quite complicated other law that was commonly taught or
used to find the number of roots of a real polynomial? If so, what was this
law, and by whom (and when) was it first (or most influentially) stated?
(b)And (a new question, related to (a)(ii) above, and prompted by John
Conway's remark) was the view enunciated by Hilbert still the dominant view
in his time? If so, in what sense did Sturm's Theorem "hit the mathematical
community like a bombshell"?
Most importantly ...
(c) Why, supposing Sturm's Theorem to be rightly viewed as providing a
SIMPLE law for determining the number of real roots of a real polynomial,
didn't the mathematical community go back to thinking of real polynomials
as having ONLY real roots? Why didn't they simply eliminate the
controversial imaginary and complex roots as not affording any particular
advantages of simplicity after all?
[Note: It may be, of course, that Hilbert's use of the present tense ("The
ordinary complex magnitudes of algebra ... ARE an instance of the use of
ideal elements; they SERVE to simplify the theorems on the existence and
number of roots of an equation.") are not to be taken literally ... that
they were just intended to describe the situation in mathematics up until
the time of Sturm's theorem. I don't think this is a plausible reading, but
it is, I suppose, a possible one.]
Puzzle II (related to Puzzle I): In 1829, when Sturm's Theorem was
published, the search for a geometrical interpretation of the complex
numbers was not yet accomplished ... or, better, the geometrical
interpretation that had been found (by Caspar Wessels and a couple others)
was not yet widely known. Gauss' influential geometrical interpretation was
not produced until 1831. But, this being so, ...
(a) Supposing the impetus behind the search for a geometrical
interpretation was to find a way to make use of imaginary and complex
numbers respectable so that THE ADVANTAGES OF THEIR SIMPLICITY ... the kind
of simplicity referred to by Hilbert in the remark quoted above ... could
be retained, why didn't the search for a geometrical interpretation of the
complexes end when Sturm's theorem was proved?
(b) Did Gauss and the others not know of Sturm's work?
(c) Did they not trust it (since it wasn't proved by Sturm when it was
first published)?
(d) Did they know of and trust Sturm's Theorem, but take it to be
significantly less simple than the FTA?
Or
(e) Did they want a geometrical interpretation of the complexes and
imaginaries for reasons other than the foundational purposes of making them
intellectually respectable? (Note: Up to that time, the central issue
concerning the imaginaries and complexes seems clearly to have been their
respectability or legitimacy.)
Once again, thanks to Bill, John and Bill for their responses ...
From a still sunny and fragrant South Bend,
Mic Detlefsen
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