Opened 7 years ago
Last modified 6 years ago
#18836 needs_work enhancement
Make refine_embedding into a method of number fields instead of standalone
Reported by:  cremona  Owned by:  

Priority:  minor  Milestone:  sage6.8 
Component:  number fields  Keywords:  
Cc:  Merged in:  
Authors:  John Cremona  Reviewers:  Vincent Delecroix 
Report Upstream:  N/A  Work issues:  
Branch:  u/cremona/18836refine_embedding (Commits, GitHub, GitLab)  Commit:  94a1d567b17399c9e39517ab6943004210ee23ba 
Dependencies:  #20064  Stopgaps: 
Description
The standalone function {{{refine_embedding}} has been around for a while, right at the bottom of sage/rings/number_field/number_field.py, not imported into the global namespace and hence having to be imported whenever needed. More seriously, it is very easy for users and developers to miss its existence entirely! (Proof: today I learned that one of my students needed such a function and wrote his own since he did not know that I had written the existing function a few years ago!)
I am moving the function into the class NumberField?, so that one can now say
embx = K.refine_embedding(emb, prec)
instead of
from sage.rings.number_field.number_field import refine_embedding embx = refine_embedding(emb,prec)
There are only about 3 places in the Sage library where this is used, which need small changes.
Change History (18)
comment:1 Changed 7 years ago by
 Branch set to u/cremona/18836refine_embedding
 Commit set to 87cb9bec69f77d1a3b4f33dacef1b7d6043c5062
 Status changed from new to needs_review
comment:2 Changed 7 years ago by
 Reviewers set to Vincent Delecroix
 Status changed from needs_review to needs_work
Hello John,
 Could you be more uniform in notations: you wrote
RR
andCC
but``AA``
and``QQbar``
.
if not self is e.domain()
>if self is not e.domain()
 it is crazy that this function needs to compute all embeddings to refine one!! (might be for another ticket)
 This is very obfuscated
diffs = [(RC(ee(self.gen()))old_root).abs() for ee in elist] return elist[min(izip(diffs,count()))[1]]
Could you make it more readable likediff_min = Infinity ans = None for ee in elist: diff = (RC(ee(self.gen()))  old_root).abs() if diff < diff_min: ans = ee diff_min = diff return ans
Vincent
comment:3 Changed 7 years ago by
OK, I am working on it. On your second point I am not sure that it is so wasteful to find all roots of a polynomial in a real / complex field than to refine one approximate root to higher precision, but I will experiment to see if that is possible. And of course, this code (mostly by me) has been in Sage for a few years already, all I did here was to move it!
comment:4 Changed 7 years ago by
 Commit changed from 87cb9bec69f77d1a3b4f33dacef1b7d6043c5062 to 87e8c14747eb0f5bf9b877b971b2ab2d5edbbffd
comment:5 Changed 7 years ago by
I did what was asked for (almost) and rebased onto 6.8.beta7. There does not seem to be a way to ask for one root in a field close to a given approximation, so it still finds all roots in the higher precision field  but only the roots, it does not contruct all teh associated embeddings, just the closest one. I think the code is less obfuscated now too.
There is still a problem,though: in testing I get two errors (from essentially the same test) in sage/schemes/elliptic_curves/ell_point.py and period_lattice.py. These occur when a complex embedding has been refined to infinite precision (i.e. to an embedding into QQbar), but an error is caused when an element of QQbar has its square root taken: its _value is an element of sage.rings.real_mpfi.RealIntervalFieldElement? which is surely wrong  it should be the corresponding complex type  and does not like its argument being asked for.
I have not been able to track this down, so I have not set the ticket to "needs review" as I know that these tests fail, but I am guessing that I have somehow stumbled across a bug which has nothing to do with this ticket. Help appreciated!
comment:6 Changed 6 years ago by
 Commit changed from 87e8c14747eb0f5bf9b877b971b2ab2d5edbbffd to ee94ab4b6da7a2bdf20b6fc50eefb2dba9190494
Branch pushed to git repo; I updated commit sha1. New commits:
ee94ab4  Merge branch 'master' (6.10) into refine_embedding

comment:7 Changed 6 years ago by
I have merged this with 6.10 to make it possible to continue with it.
comment:8 Changed 6 years ago by
This code was written while debugging the above. It runs fine unless you comment out the two lines near the end. I would say that this is a bug in QQbar.sqrt!
K.<i> = QuadraticField(1) # define a lowprecision embedding from K to CC: emb = K.embeddings(CC)[1] # extend this to the closest embedding into QQbar: old_gen = emb(K.gen()) rr = K.defining_polynomial().roots(QQbar, multiplicities=False) diffs = [(CC(r)old_gen).abs() for r in rr] new_gen = rr[diffs.index(min(diffs))] emb0 = K.hom([new_gen], check=False) # Take a polynomial with 3 roots in K: e1 = 4+i e2 = 1+i e3 = 32*i print("Original ei: %s with parent %s" % ([e1,e2,e3],parent(e1))) x = polygen(K) pol = (xe1)*(xe2)*(xe3) # Find the roots again in QQbar: pol0 = PolynomialRing(QQbar,'x')([emb0(c) for c in list(pol)]) e1, e2, e3 = pol0.roots(QQbar,multiplicities=False) print("Roots ei: %s with parent %s" % ([e1,e2,e3],parent(e1))) # Attempt to compute sqrt(e1e2) from these: d = e1e2 print("d=%s with parent %s" % (d,d.parent())) # If the next 2 lines are commented out, an error is raised in the sqrt! s = d.imag().is_zero() print("d.imag().is_zero()=%s" % s) print("d=%s with parent %s" % (d,d.parent())) d = d.sqrt() print("d=%s" % d)
comment:9 Changed 6 years ago by
The fact that _value
becomes a real interval element is probably due to this implementation (line 7431 of sage/rings/qqbar.py
)
def _interval_fast(self, prec): gen_val = self._generator._interval_fast(prec) v = self._value.polynomial()(gen_val) if self._exactly_real and is_ComplexIntervalFieldElement(v): return v.real() return v
It could well be that this is really the intention (and there could be other places where the interval is set to be a real thing!), in which case the bug is indeed in sqrt
, which should avoid relying on "argument" being available.
comment:10 Changed 6 years ago by
That is good, I thought that you (Nils) would be able to help with this. I think we should fix this on #20068, which I made a dependency for this ticket, so I will copy your comments there.
comment:11 Changed 6 years ago by
Hello,
I followed the comment from #18333. It would actually make sense for elements in QQbar
to always have their intervals being in some complex interval field. Since the zero is exact in interval arithmetic we can check if the number is real with
sage: a = CIF(5) sage: a.imag().is_zero() True
Though there is a lot of code shared between AA
and QQbar
and the change might be less trivial than it seems.
Vincent
comment:12 Changed 6 years ago by
Thanks, I did not know of the metaticket #18333. If this issue (now at #20064) can be resolved as part of that, then good, but I hope that a quick resolution will be possible since #20064 is a real bug while it looks as if many of the issues at #18333 are less urgent (though worth doing, certainly).
comment:13 Changed 6 years ago by
 Commit changed from ee94ab4b6da7a2bdf20b6fc50eefb2dba9190494 to 94a1d567b17399c9e39517ab6943004210ee23ba
Branch pushed to git repo; I updated commit sha1. New commits:
94a1d56  Merge branch 'develop (7.1.beta3)' into 18836

comment:14 followup: ↓ 15 Changed 6 years ago by
I am about to see if the fix at #20064 deals with the remaining issues.
comment:15 in reply to: ↑ 14 Changed 6 years ago by
 Dependencies set to #20064
 Status changed from needs_work to needs_review
comment:16 Changed 6 years ago by
 Status changed from needs_review to needs_work
I got doctest failing for this one.
sage t long src/sage/schemes/elliptic_curves/ell_point.py # 1 doctest failed sage t long src/sage/schemes/elliptic_curves/period_lattice.py # 2 doctests failed
comment:17 Changed 6 years ago by
comment:18 Changed 6 years ago by
You are right, I thought #20064 was already merged, while it is not. Everything is fine now.
 The code is ok, I miss extending embeddings from number to padic fields, but this issue is out of scope for this ticket.
 Please document that precision cannot decrease.
sage: N=QQ[sqrt(2)] sage: e=N.embeddings(RR)[0] sage: N.refine_embedding(e,16) is e True sage: e Ring morphism: From: Number Field in sqrt2 with defining polynomial x^2  2 To: Real Field with 53 bits of precision Defn: sqrt2 > 1.41421356237310
Also, it would be nice to note that, when the refinement is not unique, an "arbitrary" one is returned:
sage: N=NumberField(x^4  199999999/50000000*x^2 + 40000000400000001/10000000000000000,'a') sage: e = N.embeddings(ComplexField(16))[1] sage: e Ring morphism: From: Number Field in a with defining polynomial x^4  199999999/50000000*x^2 + 40000000400000001/10000000000000000 To: Complex Field with 16 bits of precision Defn: a > 1.414 sage: N.refine_embedding(e,32) Ring morphism: From: Number Field in a with defining polynomial x^4  199999999/50000000*x^2 + 40000000400000001/10000000000000000 To: Complex Field with 32 bits of precision Defn: a > 1.41421356  0.0000988882552*I sage: len(N.embeddings(ComplexField(16))) 2 sage: len(N.embeddings(ComplexField(32))) 4
New commits:
#18836: mvoe refine_embedding function into NumberField class