Re: elimination using GroebnerBasis

• To: mathgroup at smc.vnet.net
• Subject: [mg49374] Re: [mg49352] elimination using GroebnerBasis
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Sat, 17 Jul 2004 06:38:38 -0400 (EDT)
• References: <200407161006.GAA24818@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```On 16 Jul 2004, at 19:06, Gareth Owen wrote:
>
> Am I right in thinking that
>
> GroebnerBasis[{poly1, poly2, poly3},{x,y,z},{z}]
>
> will give a polynomial that is the result of eliminating x and y from
> the 3
> polynomials together?
> --
> Gareth Owen
> Usenet is like Tetris for people who still remember how to read
>
>
You have got the last two arguments wrong way round. You will get two
polynomials from which z has been eliminated, e.g.

poly1 = x^2 - y^2 - x*y + z^2; poly2 = x + y + z;
poly3 = x - y + z;

GroebnerBasis[{poly1, poly2, poly3}, {x, y, z}, {z}]

{y, x^2}

If you want to elimnate x and y you should use:

GroebnerBasis[{poly1, poly2, poly3}, {x, y, z}, {x, y}]

{z^2}

or (sometimes more effciently I think)

GroebnerBasis[{poly1, poly2, poly3}, {x, y, z}, {x,
y},MonomialOrder->EliminationOrder]

{z^2}

another equivalent approach is

First[Eliminate[{poly1, poly2, poly3} == 0, {x, y}]]

z^2

Andrzej Kozlowski
Chiba, Japan
http://www.mimuw.edu.pl/~akoz/

```

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