Subject: Re: [HM] Roots of polynomials
From: William C Waterhouse (wcw@math.psu.edu)
Date: Thu Apr 27 2000 - 15:43:03 EDT
On April 27, Michael Detlefsen <detlefsen.1@nd.edu> asked about
"the laws governing numbers of roots of polynomials." The
first question is:
"I. Is there a (possibly quite complicated) law that someone has stated to
determine the number of REAL roots of polynomials with real coefficients?
If so, where could I find a statement of it? What is its earliest
appearance?"
The standard rule for this was discovered by Sturm in "Me'moire sur la
re'solution des e'quations nume'riques" (Me'm. de l'acade'mie de Paris.
Sav. e'trag. VI, 1835). Just to mention books I have at hand, it
can be found in N. Jacobson, Basic Algebra I, and in L. Childs, A
Concrete Introduction to Higher Algebra. Jacobson says that his
presentation closely follows Weber's Lehrbuch der Algebra (1898).
Basically, it proceeds by applying the Euclidean algorithm
to the polynomial f(x) and its derivative f'(x). Then (usually
after some sign adjustments for simplicity) it looks at the signs
of f, f', first remainder, second remainder, ... at a and at b
and reads off from that the number of distinct real roots
between a and b. If you want only the total number of real roots,
then of course the answer depends only on the signs of the
leading coefficients in these polynomials. The proof just examines
how the sign sequence changes as you pass a root of some one of
the polynomials when you move from a toward b.
Tarski extended this to a sort of "decision method" for systems of
polynomials over real closed fields, though I've never worked
with that. Jacobson has some material on it.
The second question is,
"II. How do the laws governing numbers of roots of a polynomial change as a
function of allowing different types of coefficients? Are there
interestingly DISTINCT laws pertaining to (i) polynomials with only
positive coefficients (to take the first case of historical interest ...
concerning the controversy over whether negative numbers should be taken as
existing), (ii) polynomials with integral coefficients, (iii) polynomials
with rational coefficients, and (iv) polynomials with real coefficients?"
I think there really isn't anything gained by the restrictions
(except, of course, that polynomials with only positive coefficients
cannot have positive roots). Suppose we start with a real
polynomial with degree n and k real roots r_1,...,r_k. If we change
the variable from x to y = x+M, we will translate the roots from
r_i to r_i - M, so translation lets us assume that all the real roots
are negative. Now the complex factors are y^2 + b_jy + c_j, with
c_j necessarily positive; changing the sign of b_j will keep the
quadratic irreducible, so we can make all the b_j positive without
changing the real roots. Multiplying, we see that now all the
coefficients are positive, and we have the same roots as before up to
translation.
Similarly, we can approximate the real roots and the coefficients
in the quadratic factors by rationals and multiply out to get a
polynomial with rational coefficients that has the same number of
real roots (and has them as close as we like to the original roots).
Finally, we can clear denominators and then change the variable by a
scaling factor to get a monic polynomial with integer coefficients
and the same number of roots (and just scaled by a constant factor).
William C. Waterhouse
Penn State
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