Re: multivariable polynomials

[Bill Allombert <allomber@math.u-bordeaux.fr>]

On Tue, Aug 20, 2002 at 06:30:52PM -0400, Siman Wong wrote:
> Hi,
> 
> I'm using pari-2.1.4 to work with two-variable polynomials, and I got a 
> strange result:
> 
> 	? f = x^5 - y^234*x^2 - 3*y^7
> 	%1 = x^5 - y^234*x^2 - 3*y^7
> 	? f % (x^3-1)
> 	%2 = (-y^234 + 1)*x^2 - 3*y^7
> 	? f % (y^3-1)
> 	%3 = 0
> 
> In order words, I cannot "mod y" !  Is that because pari can only hand
> polynomials with x as the variable?

GP knows only Euclidean division for polynomial over a field. Since y has
lower priority than x, GP make the computation in the field Q(y)[x], in which
(y^3-1) is a unit, so  f % (y^3-1) is 0.

If you want to work in the field  Q(x)[y] instead, you can replace x
with a variable having higher priority than y, say z.

? subst(f,x,z)%(y^3-1) 
%4 = -3*y + (z^5 - z^2) 

To explicitly state that you want to apply '%(y^3-1)' to all coefficients
of f as a polynomial in x, do:
? lift(f*Mod(1,y^3-1))
%5 = x^5 - x^2 - 3*y
which work independently of the priorities of x and y.

Note that it is very different from
? lift(Mod(f,y^3-1))
%6 = 0

Please see
??"Multivariate objects"@4

The documentation has been extended in the development version GP 2.2.4.

Cheers,

Bill.