Re: PolynomialReduce
- To: mathgroup at smc.vnet.net
- Subject: [mg38685] Re: [mg38670] PolynomialReduce
- From: Andrzej Kozlowski <andrzej at platon.c.u-tokyo.ac.jp>
- Date: Sun, 5 Jan 2003 06:33:43 -0500 (EST)
- Sender: owner-wri-mathgroup at wolfram.com
There is no bug; you just have to understand what PolynomialReduce and
Groebner basis do. Roughly, PolynomialReduce generalizes usual
polynomial division, giving you a remainder of the lowers degree with
respect to a particualr monomial ordering. So when you divide 1 by a
polynomial of degree > 0 the quotient is 0 and the remainder is 1 and
in the same way PolynomialReduce[1,{poly1,poly2,poly3,...},{x,y}] will
return {{0,0,0,...},1}.
On the other hand, the fact that GroebnerBasis[{poly1,poly2...},{x,y}]
returns {{1}} means that the ideal generated by the polynomials
{poly1,poly2...} is indeed the entire ring of polynomials(or
alternatively, that the polynomials have no common complex roots). That
in turn does imply that you can write 1 as a linear sum of the
{poly1,poly2...} with polynomial coefficients, which is exactly what
you want to do. However, to find such an expression you need a
function ExtendedPolynomialGCD , like the one in the package
Algebra`ExtendedPolynomialGCD` but without the bug it contains and for
multivariate polynomials. But this seems to me that this must be rather
hard thing to do efficiently since none of the symbolic algebra
programs I know has one.
Andrzej Kozlowski
Yokohama, Japan
http://www.mimuw.edu.pl/~akoz/
http://platon.c.u-tokyo.ac.jp/andrzej/
On Saturday, January 4, 2003, at 09:25 PM, Michael R wrote:
> There seems to be a bug in PolynomialReduce. I want to write 1 as a
> linear combination (with polynomial coefficients) of polynomials, say
> a list polys={poly1,poly2,poly3,poly4...}.
>
> I use the command PolynomialReduce[1,{poly1,poly2,poly3,...},{x,y}],
> but it returns {{0,0,0,...},1}, which is not what I want.
> Groebner[{poly1,poly2...},{x,y}] returns {{1}}. What is wrong with
> what I am doing?
>
>
>