Singular

Avoiding trivialities can be done by using saturation, so relative to ideals such as ideal j=u-v. This can be found in Singular.

Finding varieties that are not 0-dimensional from a lex GB is more problematic.
These are usually not implemented.

Look up elimination and extension on the web.
I wrote Macaulay2 code for this in a 2019 paper (on arXiv) using multi-homogeneous coordinates rather than projective ones,
as I thought this did what was in Cox, Little, O'Shea without having to deal with exceptions.
But I'm sure there are other papers on the subject as well.

I'm completely new to Singular, and wish to obtain "non-trivial" solutions to a system of equations. More precisely, I create a ring
[code]ring r = 0,(u(1..5),v(1..5),x(1..5),y(1..5)),lp;[/code]
and an ideal
[code]ideal i = ...;[/code]
and its Gröbner basis. I want to find u(...),v(...),x(...),y(...) that satisfy the equations in i. Two problems:
(1) i has positive dimension (actually projective of dimension 11, degree 25), so [code]LIB "solve.lib"; solve(i);[/code] does not work
(2) there are some trivial solutions, for example u=x,v=y which I want to avoid

I tried to specialize some variables at random (by defining ideal j = i,u(1)-1,...), but didn't manage to find a non-trivial 0-dimensional ideal in this manner, so something more systematic is welcome!