Re: An observation of Weil's
- From: Gerry Myerson <gerry@xxxxxxxxxxxxxxxxxxxxxxxxx>
- Date: Mon, 27 Mar 2006 01:20:30 GMT
In article
<28965307.1143331209578.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxx>,
Maury Barbato <mauriziobarbato@xxxxxxxx> wrote:
Dave Rusin wrote:
In article
<3622522.1143102807039.JavaMail.jakarta@xxxxxxxxxxxxxx
orum.org>,
Do you know how to use the Euclidean algorithm to compute the
greatest common divisor d of two integers p,q ? You need the
division algorithm, that is, you need to know how to find s and
r so that p = q s + r and so that r is smaller than q. You
then iterate the algorithm until you come to the gcd, d. If you
keep track of these equations ("p = q s + r"), you can reassemble
the data at the end to produce an equation d = a p + b q for
some integers a and b. (There are ways to find a and
b which are more efficient than I am suggesting here.)
That same algorithm can be used to compute the gcd of two
polynomials in one variable, because we have a division algorithm
there, too. This time the remainder is "smaller" in the sense of
having a smaller degree. You may know that the division algorithm
requires, in general, that you be willing to write down
coefficients which are fractions (you have to divide by the leading
coefficient of the divisor polynomial), but if you like you can
just sprinkle some powers of that leading coefficient here and
there and keep all the equations and coefficients integral. At the
end of the day you then end up with an equation relating several
integer-coefficient polynomials:
d(X) = a(X) p(X) + b(X) q(X) .
Of course if p(X) and q(X) have no common factors in the first
place, then d will just be a constant polynomial. Usually we say
we can just take d = 1 but if you are trying to avoid division by
the coefficients of p and q
then you will probably get larger integer for d . For example,
X+1 and X^2+1 are coprime polynomials, but if you carry out this
process the smallest integer d you can get is
2 = (1).(X^2+1) + (-X+1).(X+1)
You can treat the coefficients of a(X) and b(X)
as unknowns;
then the equation we're trying to end up with is a linear system in
all those unknowns. So there is a way to take all the coefficients
of p(X) and q(X) and write down a big matrix with these things
as entries, in such a way that the determinant is this constant d
when p and q are coprime (and the determinant is zero when p
and q have a common factor).
Hmm, are you saying that the determinant of the Sylvester
matrix is exactly d (the smallest integer d)?
I think you are asking the following question:
Let p and q be monic polynomials, relatively prime, with integer
coefficients. Let R be the resultant (that is, the determinant
of the Sylvester matrix, described above), and let d be the
smallest positive integer in the ideal p and q generate in Z[x].
Is it necessarily true that |R| = d?
The answer, in general, is no. In my paper, On resultants,
Proc. Amer. Math. Soc. 89 (1983), no. 3, 419--420, MR 84j:13004,
I prove that R divides d^n, where n is the smaller of deg p,
deg q.
--
Gerry Myerson (gerry@xxxxxxxxxxxxxxx) (i -> u for email)
.
- References:
- Re: An observation of Weil's
- From: Dave Rusin
- Re: An observation of Weil's
- From: Maury Barbato
- Re: An observation of Weil's
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