From: wilker@math.purdue.edu (Clarence Wilkerson)
Subject: Re: Linear factor of a multivariate polynomial
Date: 30 Jan 1999 06:46:51 GMT
Newsgroups: sci.math.symbolic,sci.math.research
Keywords: When does a polynomial split completely into linear factors?
>> Gudbjorn Freyr Jonsson asks:
>>I'd like to know when a polynomial in more than one variable has a
>>linear factor.
Here's a solution to a different problem:
for factoring over the rationals or integers
a homogeneous polynomial in several variables completely
into linear factors:
Let F be the form in Q[x_1,x_2,..x_n].
For all large primes p you must check that in
F_p[x_1,x_2,..x_n]
one has F | P^i F for all i > 0 (actually only have to check
up to i < degree F )
Here P^i F is the i-th mod p Steenrod operation given
by identifying F_p[x_1,x_2,..x_n] as the mod p cohomology
of (CP^\infty})^n.
More explicitly, P^i is determined by
P^1x_j = x_j^p, P^0x_j = x_j, P^ix_j = 0 for i > 1 for all j.
Extend to all of F_p[x_1,x_2,..x_n] by the "Liebnitz" or
Cartan formula that
P^k(uv) = \sum_{i=0}^k P^{i}uP^{k-i}v
This appears in
Wilkerson, Clarence
Classifying spaces, Steenrod operations and algebraic closure.
Topology 16 (1977), no. 3, 227--237.
I don't know if this corresponds to any classical technique
or not.
One can simplify the checking a bit by using the
total Steenrod operation
P_t f= \sum_0^infty P^k(f)t^k
Then P_t(fg) = P_t(f)P_t(g)
Then one just has to check that F | P_t(F) in
F_p[x_1,x_2,..x_n][t].
But P_t(F) is just F( (x_1 + tx_1^p), ... (x_n + tx_n^p)].
This is related to work of Serre:
Serre, Jean-Pierre Sur la dimension cohomologique des
groupes profinis. (French) Topology 3 1965 413--420.
--
Clarence Wilkerson \ HomePage: http://www.math.purdue.edu/~wilker
Prof. of Math. \ Internet: wilker@NOspam.math.purdue.edu
Dept. of Mathematics \ Messages: (765) 494-1903, FAX 494-0548
Purdue University, \
W. Lafayette, IN 47907-1395 \
==============================================================================
From: rusin@math.niu.edu (Dave Rusin)
Subject: Re: Linear factor of a multivariate polynomial
Date: 2 Feb 1999 07:47:09 GMT
Newsgroups: sci.math.symbolic,sci.math.research
Gudbjorn Freyr Jonsson asks:
>I'd like to know when a polynomial in more than one variable has a
>linear factor.
Clarence Wilkerson wrote:
>Here's a solution to a different problem:
>
>for factoring over the rationals or integers
>a homogeneous polynomial in several variables completely
>into linear factors:
I'm not sure if Clarence intended only to look for polynomials with _all_
factors linear, but the same idea tests for the presence of _any_ linear
factors. If L is a linear factor of F, then L must divide P(F) so we
can compute putative linear factors L (modulo any prime with good reduction)
as divisors of gcd(F, P^i(F)).
For example, when p=2 and F = x^2 + y^2 + xy we compute
P(F) = (x+x^2)^2 + (y+y^2)^2 + (x+x^2)(y+y^2)
= (x^2+xy+y^2) + (x^2+xy^2) + (x^2+xy+y^2)^2
but as gcd( x^2+y^2+xy, xy(x+y) ) = 1, we conclude x^2+xy+y^2 has
no linear factors.
But fairly efficient polynomial factorization procedures exist, as can
be seen by asking your favorite symbolic-algebra program to factor a
polynomial. I don't know if it's any easier, in general, to restrict
attention to linear factors, although the Steenrod-algebra approach above
is limited to this case. Check out books on algorithmic aspects of
algebra and number theory, e.g. at
http://www.math.niu.edu/~rusin/known-math/index/12FXX.html
dave