Subject: Re: [HM] Roots of polynomials
From: William Tait (wwtx@midway.uchicago.edu)
Date: Fri Apr 28 2000 - 10:30:08 EDT
In my posting on roots of polynomials last evening I said that the
question of the exostence of roots of an arbitrary poynomial of
degree 1 is precisely the question of whether or not an arbitrary
real is = 0 -- I of course meant degree 0.
But, perhaps more convincing in the general case is this: Sturm's
Theorem asserts that the number of distinct roots of the polynomial f
in the interval (a,b) is w(b)-w(a), where w(c) is the number of
changes of sign in the sequence X_1(c), ..., X_n(c), where the X_j
are certain polynomials depending on f, and *where all zeros are
dropped from the sequence*. It is this last part which is not
constructive when arbitrary real coefficients are admitted. (Of
course, when a, b and the coefficients of f are rational, there is no
problem.)
Bill Tait
William W. Tait
Professor Emeritus of Philosophy
University of Chicago
wwtx@midway.uchicago.edu
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