#19595: Implement a check that a hyperplane arrangement is free
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Reporter: tscrim | Owner: tscrim
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-7.0
Component: geometry | Resolution:
Keywords: free, hyperplane | Merged in:
arrangement | Reviewers:
Authors: Travis Scrimshaw | Work issues:
Report Upstream: N/A | Commit:
Branch: | b10a0c819f1b50869ba620be849bbd12d199c146
public/hyperplanes/check_free-19595| Stopgaps:
Dependencies: |
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Comment (by mmarco):
I have just given a very quick look at the code, but it seems to me that
you are actually computing a free resolution of the logarithmic derivation
module. Or at least the first steps of it. Is that correct?
If this is the case, maybe it would be worth doing so by using Singular,
which already has this implemented.
Something like
{{{
sage: R.<x,y,z,t> = QQ[]
sage: f = x^3 + y^3 +z^3+ t^3
sage: I = f.jacobian_ideal()
sage: IS = I._singular_()
sage: res = IS.mres(0)
sage: res
[1]:
_[1]=t^2
_[2]=z^2
_[3]=y^2
_[4]=x^2
[2]:
_[1]=z^2*gen(1)-t^2*gen(2)
_[2]=y^2*gen(1)-t^2*gen(3)
_[3]=y^2*gen(2)-z^2*gen(3)
_[4]=x^2*gen(1)-t^2*gen(4)
_[5]=x^2*gen(2)-z^2*gen(4)
_[6]=x^2*gen(3)-y^2*gen(4)
[3]:
_[1]=y^2*gen(1)-z^2*gen(2)+t^2*gen(3)
_[2]=x^2*gen(1)-z^2*gen(4)+t^2*gen(5)
_[3]=x^2*gen(2)-y^2*gen(4)+t^2*gen(6)
_[4]=x^2*gen(3)-y^2*gen(5)+z^2*gen(6)
[4]:
_[1]=x^2*gen(1)-y^2*gen(2)+z^2*gen(3)-t^2*gen(4)
sage: len(res)
4
}}}
should give you a minimal free resolution of the logarithmic derivation
module of every homogenous polynomial f, so it would be free iff
len(res)<=2 .
For the non homogenous case, one might need to work a little bit with the
homogenization of the polynomial, but it can also be done.
Could you please compare the timings of your code with the singular
method?
--
Ticket URL: <http://trac.sagemath.org/ticket/19595#comment:10>
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