It is definitely a bug in the dimension method.
If singular can handle the ring, sage asks singular to compute the
dimension, which does correctly (the -1 is the singular convention for
empty varieties).
The problem is that when the field is not supported by singular (which
happens with QQbar or finite fields of characteristic bigger than 2^31) ,
then sage falls back to its own toy implementation. In that case, it
appears that the empty case is not treated separatedly than the zero
dimensional case.
El viernes, 1 de mayo de 2015, 21:18:55 (UTC+2), gjorgen...@my.fit.edu
escribió:
>
> Hi,
>
> For the following ideal, dimension() returns 0,
> {{{
> R.<s0,s1>=QQbar[]
> I=R.ideal([ s0 + 1, s0*s1 + s0 + s1 + 1, (-2)*s0 + 1, (-10)*s1 + 5, 5*s0^2
> + 10*s0*s1 ])
> I.dimension()
> }}}
> but its variety is empty.
>
> Also for any other ring, dimension() returns -1 for this ideal. Is this a
> bug with dimension()? The documentation for dimension() doesn't seem to
> mention the -1 case.
> It provides the following example,
> {{{
> R.<x,y> = PolynomialRing(GF(2147483659),order='lex')
> I = R.ideal([x*y,x*y+1])
> I.dimension()
> }}}
> which yields dimension 0 for the ideal, yet the corresponding variety is
> empty.
>
> What is the expected behavior for dimension()? When the variety of the
> ideal in question has no points is dimension() always supposed to return -1?
>
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