Opened 2 years ago

Closed 8 months ago

Characteristic polynomial of central Hyperplane arrangement returns wrong result?

Reported by: Owned by: Jean-Philippe Labbé major sage-9.6 geometry hyperplane arrangements, regions gh-kliem, Laith Rastanawi, Julian Ritter Jonathan Kliem Travis Scrimshaw N/A d74929d d74929dfb74820f79d574189a6bc5803dc00dd8d

A central hyperplane arrangement must have an even number of regions by central symmetry... yet the one below gets 31 regions(!).

```R.<y> = QQ[]
v1 = AA.polynomial_root(AA.common_polynomial(y^2 - 3), RIF(RR(1.7320508075688772), RR(1.7320508075688774)))
v2 = QQbar.polynomial_root(AA.common_polynomial(y^4 - y^2 + 1), CIF(RIF(RR(0.8660254037844386), RR(0.86602540378443871)), RIF(-RR(0.50000000000000011), -RR(0.49999999999999994))))
my_vectors = (vector(AA, [-v1, -1, 1]), vector(AA, [0, 2, 1]), vector(AA,[v1, -1, 1]), vector(AA, [1, 0, 0]), vector(AA, [1/2, AA(-1/2*v2^3 + v2),0]), vector(AA, [-1/2, AA(-1/2*v2^3 + v2), 0]))

sage: H = HyperplaneArrangements(AA,names='xyz')
sage: x,y,z = H.gens()
sage: A = H(backend="normaliz")
sage: for v in my_vectors:
....:     a,b,c = v
....:     A = A.add_hyperplane(a*x + b*y + c*z)
sage: A
Arrangement of 6 hyperplanes of dimension 3 and rank 3
sage: A.n_regions()
31
sage: A.is_central()
True
```

Here is another failure in characteristic polynomial:

```sage: tau = AA((1+sqrt(5))/2)
sage: ncn = [[2*tau+1,2*tau,tau],[2*tau+2,2*tau+1,tau+1],[1,1,1],[tau+1,tau+1,tau],[2*tau,2*tau,tau],[tau+1,tau+1,1],[1,1,0],[0,1,0],[1,0,0],[tau+1,tau,tau]]
sage: H = HyperplaneArrangements(AA,names='xyz')
sage: x,y,z = H.gens()
sage: A = H()
sage: for v in ncn:
....:     a,b,c = v
....:     A = A.add_hyperplane(a*x + b*y + c*z)
....:
sage: A.n_regions()
Traceback (most recent call last):
...
ValueError: arrangement cannot simultaneously have h and -h as hyperplane
```

#30078 fixes this and we add another doctest here.

comment:1 Changed 2 years ago by Frédéric Chapoton

Description: modified (diff)

comment:2 follow-up:  3 Changed 2 years ago by Frédéric Chapoton

The char poly gives the same answer

```sage: p = A.characteristic_polynomial() : p
x^3 - 6*x^2 + 13*x - 11
sage: p(-1)
-31
```

comment:3 in reply to:  2 Changed 2 years ago by Jean-Philippe Labbé

The char poly gives the same answer

```sage: p = A.characteristic_polynomial() : p
x^3 - 6*x^2 + 13*x - 11
sage: p(-1)
-31
```

Yes, this is how the number of regions is computed up to sign. The hyperplane arrangement seems to be sane: computing `.regions()` computes the expected (number of) regions.

I asked the number of regions to double-check something and then got this scary answer...

Last edited 2 years ago by Jean-Philippe Labbé (previous) (diff)

comment:4 Changed 2 years ago by Jean-Philippe Labbé

Summary: Characteristic polynomial of central Hyperplane arrangement results wrong result? → Characteristic polynomial of central Hyperplane arrangement returns wrong result?

comment:5 follow-up:  10 Changed 2 years ago by Julian Ritter

I encountered this problem in some example a long while ago, before I was contributing to Sage myself. A simple workaround is to have `n_regions` output `len(self.regions())`, which was good enough for me. Of course, this is much slower if one doesn’t want to compute the regions anyway. I don’t understand the characteristic polynomial well enough to figure out the error in there.

comment:6 Changed 2 years ago by Jean-Philippe Labbé

Description: modified (diff)

comment:7 Changed 22 months ago by Matthias Köppe

Milestone: sage-9.3 → sage-9.4

Setting new milestone based on a cursory review of ticket status, priority, and last modification date.

comment:8 Changed 16 months ago by Matthias Köppe

Milestone: sage-9.4 → sage-9.5

comment:9 Changed 12 months ago by Matthias Köppe

Milestone: sage-9.5 → sage-9.6

comment:10 in reply to:  5 Changed 9 months ago by Julian Ritter

I don’t understand the characteristic polynomial well enough to figure out the error in there.

The characteristic polynomial is obtained through recursive deletions and contractions of hyperplane arrangements. During this process, it may occur that some hyperplane `h` is included in an arrangement multiple times with different scalings of the defining linear expression which are not properly detected as duplicates. This is due to a defect in the method `hyperplane.primitive()` addressed in #30078.

comment:11 Changed 9 months ago by Matthias Köppe

Dependencies: → #30078

comment:12 Changed 8 months ago by Matthias Köppe

Is this defect solved by #30078?

comment:13 Changed 8 months ago by Travis Scrimshaw

It seems like it is. Using the first example, I get

```sage: A.n_regions()
24
sage: len(A.regions())
24
sage: A.characteristic_polynomial()(-1)
-24
```

So it is now consistent. We probably just want to add a doctest for this ticket.

comment:14 Changed 8 months ago by gh-kliem

Authors: → Jonathan Kliem → public/30749 → 1492ed150e76f36d09317428d9e642986554ec6b #30078 modified (diff) new → needs_review

#30078 already adds one doctest. So we can either close it as duplicate or add another doctest. Either way is fine with me.

New commits:

 ​1492ed1 `add another doctest for 30749`

• sage/geometry/hyperplane_arrangement/arrangement.py

 a sage: H = HyperplaneArrangements(AA, names='xyz') sage: x,y,z = H.gens() sage: A = H(backend="normaliz")  # optional - pynormaliz sage: for v in my_vector:        # optional - pyrormaliz sage: for v in my_vectors:       # optional - pynormaliz ....:     a, b, c = v ....:     A = A.add_hyperplane(a*x + b*y + c*z) sage: A.n_regions()              # optional - pyrormaliz sage: A.n_regions()              # optional - pynormaliz 32 """ if self.base_ring().characteristic() != 0:

comment:16 Changed 8 months ago by git

Commit: 1492ed150e76f36d09317428d9e642986554ec6b → d74929dfb74820f79d574189a6bc5803dc00dd8d

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

 ​d74929d `typos`

comment:17 Changed 8 months ago by Travis Scrimshaw

Reviewers: → Travis Scrimshaw needs_review → positive_review

Thank you. LGTM.