Make refine_embedding into a method of number fields instead of stand-alone

[cremona]

The stand-alone 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.

[cremona]

New commits:

​87cb9be

#18836: mvoe refine_embedding function into NumberField class

[vdelecroix]

Hello John,

• Could you be more uniform in notations: you wrote RR and CC 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 like

     diff_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

[cremona]

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!

[git]

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

​450209d	#18836: mvoe refine_embedding function into NumberField class
​accaa9f	#18836: simplified code after review
​87e8c14	Merge branch 'u/cremona/18836-refine_embedding' of trac.sagemath.org:sage into refine_embedding

[cremona]

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!

[git]

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

​ee94ab4

Merge branch 'master' (6.10) into refine_embedding

[cremona]

I have merged this with 6.10 to make it possible to continue with it.

[cremona]

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 low-precision 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 = 3-2*i
print("Original ei: %s with parent %s" % ([e1,e2,e3],parent(e1)))
x = polygen(K)
pol = (x-e1)*(x-e2)*(x-e3)

# 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(e1-e2) from these:                                                                    

d = e1-e2
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)

[nbruin]

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.

[cremona]

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.

[vdelecroix]

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

[cremona]

Thanks, I did not know of the meta-ticket #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).

[git]

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

​94a1d56

Merge branch 'develop (7.1.beta3)' into 18836

[cremona]

I am about to see if the fix at #20064 deals with the remaining issues.

[cremona]

Replying to cremona:

I am about to see if the fix at #20064 deals with the remaining issues.

It does, and now has a positive review, so I will set that as a dependency of this and ask for another review of this one.

[lftabera]

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

[cremona]

Are those failures the ones which are fixed by #20064? Did you apply #20064 first -- it is merged but not yet in the latest beta, I think.

[lftabera]

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 p-adic 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