factoring homogeneous polynomials

[Robert I McLachlan]

An embarrassingly simple question:

Homogeneous polynomials of degree m in 2 variables can be uniquely factored
(over the complex numbers) into m linear factors.

Polynomials in n>2 variables cannot always be so factored, for example,
x^2 + y^2 + z^2 has no linear factor.

What is the general situation here? What are the prime homogeneous
polynomials? (Actually, I particularly need to do cubics, but
the general case sounds interesting too.)

Thanks,

Robert McLachlan
R.McL...@massey.ac.nz

[Clarence Wilkerson]

Since you can move from inhomogeneous polynomials in n-1
variables to homogeneous polynomials in n variables by
just sticking in a dummy variable to make the terms have the
same total degree, I don't see how there could be anything
special about the homogeneous case.

Clarence Wilkerson

[David L. Johnson]

Robert I McLachlan wrote:
>
> An embarrassingly simple question:
>
With what may seem to be an incredibly difficult answer.

> Homogeneous polynomials of degree m in 2 variables can be uniquely factored
> (over the complex numbers) into m linear factors.
>
> Polynomials in n>2 variables cannot always be so factored, for example,
> x^2 + y^2 + z^2 has no linear factor.
>
> What is the general situation here? What are the prime homogeneous
> polynomials? (Actually, I particularly need to do cubics, but
> the general case sounds interesting too.)

The zeroes of a homogeneous polynomial describes an algebraic variety in
complex projective space. The fact that, in two variables, the solution is
easy is equivalent to saying that the co-dimension 1 subvarieties of P^1 are
easy to understand (they are all finite sets of points, the points being the
solutions of each of the linear factors). In higher dimensions, for more than
two variables (so the variety is in P^n for n+1=#variables), the question of
finding all irreducible polynomials is the question of finding irreducible
co-dimension one subvarieties. This is a very subtle question.

--

David L. Johnson dl...@lehigh.edu, dl...@netaxs.com
Department of Mathematics http://www.lehigh.edu/~dlj0/dlj0.html
Lehigh University
14 E. Packer Avenue (610) 758-3759
Bethlehem, PA 18015-3174

[Alan D. Burns]

Hi,
As other replies indicate, your 'embarrasingly simple question' is in
fact one of the main problems in the huge field of algebraic geometry.
Homogeneous polynomials in n variables are related in a simple way
to general polynomials in n-1 variables. Hence, the n=2 case is easy.
However, the n>2 case is related to the classification of hypersurfaces
in (n-1)-dimensional complex affine space and n-dimensional complex
projective space.
A very accessible introduction to this field is given in the book:

Ideals, Varieties and Algorithms, by Cox, Little and O'Shea,
Undergraduate Texts in Mathematics, Springer-Verlag,
ISBN 0-387-97847-X

--
Alan D. Burns