new Oscar features

[fieker]

A possibility would be to change the notebook/ implementation to use the recent Oscar features. On Oscar#master this should work

Qt, T = PolynomialRing(QQ, :T=>1:10)
D = free_abelian_group(5)
Q = [...]
w = [D(Q[i, :]) for i=1:10]
R = grade(Qt, w)
a = ideal(R, [T[5]*T[10] - T[6]*T[9] + T[7]*T[8],
T[1]*T[9] - T[2]*T[7] + T[4]*T[5],
T[1]*T[8] - T[2]*T[6] + T[3]*T[5],
T[1]*T[10] - T[3]*T[7] + T[4]*T[6],
T[2]*T[10] - T[3]*T[9] + T[4]*T[8]])

or
a = ideal([R[5] * R[10] ....])

q, mq = quo(R, a)

...
Of course, internally all serious computations in R, a, Qt, are done in Singular, but it looks neater
Also, in the notebook, now automatically, you should see T_1 with proper subscripts

It would be cool if fans, cones, ... would be proper julia structures, then some of the interfacing would be easier to read....

[ThomasBreuer]

This issue has several aspects.

1. Do not specify the subsystem from which some functions or objects are taken (here: QQ instead of Singular.QQ or Ǹemo.QQ or ...) whenever Oscar handles the situation automatically.
2. Put information into the objects whenever possible (here: the graded polynomial ring).
3. Where possible, introduce objects representing mathematical structures (here: fans, cones, ...) instead of just encoding them by some low level data.

All this makes sense, however, the aim of the example notebook is to show the interaction between the subsystems, not to hide them.
Perhaps it is an option to have an alternative variant that focuses on the "Oscar perspective"?

[fieker]

On Wed, Jun 03, 2020 at 01:40:53AM -0700, ThomasBreuer wrote: This issue has several aspects. 1. Do not specify the subsystem from which some functions or objects are taken (here: `QQ` instead of `Singular.QQ` or `Ǹemo.QQ` or ...) whenever Oscar handles the situation automatically. 2. Put information into the objects whenever possible (here: the graded polynomial ring). 3. Where possible, introduce objects representing mathematical structures (here: fans, cones, ...) instead of just encoding them by some low level data. All this makes sense, however, the aim of the example notebook is to show the interaction between the subsystems, not to hide them. Perhaps it is an option to have an alternative variant that focuses on the "Oscar perspective"?

Yep, that would be great for at least two reasons - it would showcase what we are trying to achieve - it would use, extend and debug what we have

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[ThomasBreuer]

Currently I get stuck in the "oscarification" of the code.
The following (taken from the old code) works.

using Singular
R, T = Singular.PolynomialRing(Singular.QQ, ["x1", "x2"]);
z = R(0);
pol = T[1] * T[2]^2;
perm = [2, 1];
Singular.nvars( R )
Singular.substitute_variable(pol, 1, z)
mx = Singular.MaximalIdeal(R, 1)
Singular.satstd(a, mx)
Singular.permute_variables(pol, perm, R)

But when I start with

using Oscar
R, T = PolynomialRing(QQ, ["x1", "x2"]);

then there seem to be no replacements for Singular.substitute_variable, Singular.MaximalIdeal, Singular.satstd, and Singular.permute_variables.

(And when I try the proposed approach using a graded polynomial ring R then also nvars(R) does not work; one can use nvars(R.R) so this is not really a problem.)

[fieker]

You're thinking Singular, not math...
Some of it is genuinely missing (and trivial to add) other I don't even know what it should be, most is available, possibly (for now) not quite as fast as it could be.
substritue_variable => evaluate(f, [1, T[2]])
mx = ideal(R, gens(R)) #in general mxi = ideal(R, monomials(R, i)) (defined in Examples/Reynolds currently, sorry)
saturate(a, mx)
permute_variables(pol, perm) => pol^perm #(thanks to GaloisGroup)

the missing nvars is a error

[ThomasBreuer]

@fieker Thanks.
(Just yesterday I had observed part of what you write.)