py3: richcmp for ideals of multivariate polynomials

[chapoton]

split off from #23787

refactoring of the comparison of ideals in multivariate polynomials

• to get rid of cmp
• consistent comparison by containment
• when a total order is needed, compare instead by gens

[chapoton]

New commits:

​f077b00

trac 23920 richcmp for ideals of multi-polynomials

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​b5f4dd4

trac 23920 fixing some doctests (order changed)

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​4b74c31

trac 23920 chasing triggered bugs all around the place

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​7f8200b

trac 23920 fix last doctest failing + py3 warning

[chapoton]

This should be ready to go. This refactors the comparison of ideals in multi-variate polynomial rings. Comparison is now meaning "for containment" in all cases.

Besides, it tries to avoid computing Grobner bases, by first comparing the generators.

Please review.

[chapoton]

green bot!

[chapoton]

now this is going to conflict with #23814

[chapoton]

Could someone please review this? there should be no real conflict with #23814.

[tscrim]

Rather than comparing gens as a list, I would compare them as sets. That way, reordering of gens doesn't require a GB computation, e.g.:

sage: R.<x,y> = ZZ[]
sage: I = R.ideal([x^2, x*y - y^2 + 2])
sage: Ip = R.ideal([x*y - y^2 + 2, x^2])
sage: I.gens()
[x^2, x*y - y^2 + 2]
sage: Ip.gens()
[x*y - y^2 + 2, x^2]
sage: I == Ip   # Avoid GB here
True

Also, what is the rationale for this change:

@@ -532,8 +529,13 @@ class SchemeMorphism_polynomial_affine_space(SchemeMorphism_polynomial):
 
         #homogenize the domain and codomain
         A = self.domain().projective_embedding(ind[0]).codomain()
-        B = self.codomain().projective_embedding(ind[1]).codomain()
-        H = Hom(A, B)
+        if self.is_endomorphism():
+            B = A
+            H = End(A)
+        else:
+            B = self.codomain().projective_embedding(ind[1]).codomain()
+            H = Hom(A, B)
+
         newvar = A.ambient_space().coordinate_ring().gen(ind[0])
 
         N = A.ambient_space().dimension_relative()

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​bb784e1

trac 23920 using set(gens()) == set(gens())

[chapoton]

• I have modifed the comparison to use sets of gens.

• The changes in scheme_morphisms and dynamical systems could probably be made independently now, even if at some point they were intertwined with the main issue here.

• the precise change you talk about was made so that one could replace comparison of domains (in dynamical systems) with a call to is_endomorphism.

[tscrim]

Okay. LGTM. Thanks.