another naive question

[Fernando Gouvea]

I'm working with ideals in the polynomial ring in three variables.

sage> R.<u,v,w>=QQ[]
sage> f=u*v-w
sage> g=u^2-v
sage> I=Ideal(f,g)
sage> I.is_prime()
    True
sage> I.associated_primes()
    [Ideal (v^2 - u*w, u*v - w, u^2 - v) of Multivariate Polynomial Ring in u, v, w over Rational Field]

That last ideal is the same as I, since u(uv - w) - v(u^2-v) = v^2-uw. So why do I get an extra generator?

Thanks,

Fernando

--

=============================================================
Fernando Q. Gouvea         http://www.colby.edu/~fqgouvea
Carter Professor of Mathematics
Dept. of Mathematics and Statistics
Colby College              
5836 Mayflower Hill        
Waterville, ME 04901       

So if a man's wit be wandering,let him study the mathematics; for in
demonstrations, if his wit be called away never so little, he must
begin again.
  -- Francis Bacon, "Of Studies"

[Dima Pasechnik]

On Tue, Apr 7, 2020 at 2:29 AM Fernando Gouvea <fqgo...@colby.edu> wrote:
>
> I'm working with ideals in the polynomial ring in three variables.
>
> sage> R.<u,v,w>=QQ[]
> sage> f=u*v-w
> sage> g=u^2-v
> sage> I=Ideal(f,g)
> sage> I.is_prime()
> True
> sage> I.associated_primes()
> [Ideal (v^2 - u*w, u*v - w, u^2 - v) of Multivariate Polynomial Ring in u, v, w over Rational Field]
>
> That last ideal is the same as I, since u(uv - w) - v(u^2-v) = v^2-uw. So why do I get an extra generator?

I think that it's a Groebner basis (w.r.t. the default monomial
ordering) of the ideal in question.

>
> Thanks,
>
> Fernando
>
>
> --
>
> =============================================================
> Fernando Q. Gouvea http://www.colby.edu/~fqgouvea
> Carter Professor of Mathematics
> Dept. of Mathematics and Statistics
> Colby College
> 5836 Mayflower Hill
> Waterville, ME 04901
>
> So if a man's wit be wandering,let him study the mathematics; for in
> demonstrations, if his wit be called away never so little, he must
> begin again.
> -- Francis Bacon, "Of Studies"
>

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