p-adic square root

[742ccff4-512f-43e4-bdd0-5df91c799a52]

This ticket implements square root over p-adic fields (i.e. finite extension of Qp)

CC: @roed314 @xcaruso

Component: padics

Author: Xavier Caruso

Branch/Commit: 3085538

Reviewer: David Lubicz

Issue created by migration from https://trac.sagemath.org/ticket/23344

[kedlaya]

comment:1

Doesn't this already work? This field L is an example from the docstring for Qp.extension.

sage: F = list(Qp(19)['x'](cyclotomic_polynomial(5)).factor())[0][0]
sage: L = Qp(19).extension(F, names='a')
sage: L
Unramified Extension of 19-adic Field with capped relative precision 20 in a defined by (1 + O(19^20))*x^2 + (5 + 2*19 + 10*19^2 + 14*19^3 + 7*19^4 + 13*19^5 + 5*19^6 + 12*19^7 + 8*19^8 + 4*19^9 + 14*19^10 + 6*19^11 + 5*19^12 + 13*19^13 + 16*19^14 + 4*19^15 + 17*19^16 + 8*19^18 + 4*19^19 + O(19^20))*x + (1 + O(19^20))
sage: u = L(4+19).sqrt(); u
2 + 5*19 + 3*19^2 + 11*19^3 + 12*19^4 + 13*19^5 + 11*19^6 + 15*19^7 + 12*19^8 + 18*19^9 + 15*19^10 + 2*19^11 + 9*19^12 + 4*19^13 + 19^15 + 19^16 + 13*19^18 + 3*19^19 + O(19^20)
sage: u^2
4 + 19 + O(19^20)

The syntax sqrt(L(4+19)) also works.

[742ccff4-512f-43e4-bdd0-5df91c799a52]

comment:2

If I am not mistaken, with the version of sage I am working with (8.0.rc0), the square root is working fine for elements of Qp (even if Qp is embeded in an extension) but not for elements of an extension of Qp (not in Q_p). Continuing your example :

sage: a=L.gen()
sage: sqrt(4+19*a)
---------------------------------------------------------------------------
NotImplementedError                       Traceback (most recent call last)
<ipython-input-5-842761a28969> in <module>()
----> 1 sqrt(Integer(4)+Integer(19)*a)

/home/lubicz/sagemath/sage/local/lib/python2.7/site-packages/sage/functions/other.pyc in sqrt(x, *args, **kwds)
   2002         except (AttributeError, TypeError):
   2003             pass
-> 2004         return _do_sqrt(x, *args, **kwds)
   2005 
   2006 # register sqrt in pynac symbol_table for conversion back from other systems

/home/lubicz/sagemath/sage/local/lib/python2.7/site-packages/sage/functions/other.pyc in _do_sqrt(x, prec, extend, all)
   1903             z = I
   1904         else:
-> 1905             z = SR(x) ** one_half
   1906 
   1907         if all:

/home/lubicz/sagemath/sage/src/sage/symbolic/expression.pyx in sage.symbolic.expression.Expression.__pow__ (/home/lubicz/sagemath/sage/src/build/cythonized/sage/symbolic/expression.cpp:25800)()
   3882                            relational_operator(base._gobj))
   3883         else:
-> 3884             x = g_pow(base._gobj, nexp._gobj)
   3885         return new_Expression_from_GEx(base._parent, x)
   3886 

/home/lubicz/sagemath/sage/local/lib/python2.7/site-packages/sage/rings/padics/CR_template.pxi in sage.rings.padics.qadic_flint_CR.CRElement.__pow__ (/home/lubicz/sagemath/sage/src/build/cythonized/sage/rings/padics/qadic_flint_CR.c:20012)()
    672         elif exact_exp:
    673             # exact_pow_helper is defined in padic_template_element.pxi
--> 674             right = exact_pow_helper(&ans.relprec, self.relprec, _right, self.prime_pow)
    675             if ans.relprec > self.prime_pow.ram_prec_cap:
    676                 ans.relprec = self.prime_pow.ram_prec_cap

/home/lubicz/sagemath/sage/local/lib/python2.7/site-packages/sage/rings/padics/padic_template_element.pxi in sage.rings.padics.qadic_flint_CR.exact_pow_helper (/home/lubicz/sagemath/sage/src/build/cythonized/sage/rings/padics/qadic_flint_CR.c:15216)()
    545         exp_val = mpz_get_si((<Integer>right.valuation(p)).value)
    546     elif isinstance(_right, Rational):
--> 547         raise NotImplementedError
    548     ansrelprec[0] = relprec + exp_val
    549     if exp_val > 0 and mpz_cmp_ui(p.value, 2) == 0 and relprec == 1:

NotImplementedError: 

Another example which motivates my tickets since I am implementing in sagemath an improved version of the AGM algorithm :

sage: R.<x>=Zq(2^10, 20)
sage: sqrt(1+8*x)
---------------------------------------------------------------------------
NotImplementedError                       Traceback (most recent call last)
<ipython-input-10-25d75e2ffffe> in <module>()
----> 1 sqrt(Integer(1)+Integer(8)*x)

/home/lubicz/sagemath/sage/local/lib/python2.7/site-packages/sage/functions/other.pyc in sqrt(x, *args, **kwds)
   2002         except (AttributeError, TypeError):
   2003             pass
-> 2004         return _do_sqrt(x, *args, **kwds)
   2005 
   2006 # register sqrt in pynac symbol_table for conversion back from other systems

/home/lubicz/sagemath/sage/local/lib/python2.7/site-packages/sage/functions/other.pyc in _do_sqrt(x, prec, extend, all)
   1903             z = I
   1904         else:
-> 1905             z = SR(x) ** one_half
   1906 
   1907         if all:

/home/lubicz/sagemath/sage/src/sage/symbolic/expression.pyx in sage.symbolic.expression.Expression.__pow__ (/home/lubicz/sagemath/sage/src/build/cythonized/sage/symbolic/expression.cpp:25800)()
   3882                            relational_operator(base._gobj))
   3883         else:
-> 3884             x = g_pow(base._gobj, nexp._gobj)
   3885         return new_Expression_from_GEx(base._parent, x)
   3886 

/home/lubicz/sagemath/sage/local/lib/python2.7/site-packages/sage/rings/padics/CR_template.pxi in sage.rings.padics.qadic_flint_CR.CRElement.__pow__ (/home/lubicz/sagemath/sage/src/build/cythonized/sage/rings/padics/qadic_flint_CR.c:20012)()
    672         elif exact_exp:
    673             # exact_pow_helper is defined in padic_template_element.pxi
--> 674             right = exact_pow_helper(&ans.relprec, self.relprec, _right, self.prime_pow)
    675             if ans.relprec > self.prime_pow.ram_prec_cap:
    676                 ans.relprec = self.prime_pow.ram_prec_cap

/home/lubicz/sagemath/sage/local/lib/python2.7/site-packages/sage/rings/padics/padic_template_element.pxi in sage.rings.padics.qadic_flint_CR.exact_pow_helper (/home/lubicz/sagemath/sage/src/build/cythonized/sage/rings/padics/qadic_flint_CR.c:15216)()
    545         exp_val = mpz_get_si((<Integer>right.valuation(p)).value)
    546     elif isinstance(_right, Rational):
--> 547         raise NotImplementedError
    548     ansrelprec[0] = relprec + exp_val
    549     if exp_val > 0 and mpz_cmp_ui(p.value, 2) == 0 and relprec == 1:

NotImplementedError: 

[roed314]

comment:3

Replying to @dlubicz:

If I am not mistaken, with the version of sage I am working with (8.0.rc0), the square root is working fine for elements of Qp (even if Qp is embeded in an extension) but not for elements of an extension of Qp (not in Q_p).

I agree. The current implementation just calls off to Pari (see line 2774 in padic_generic_element.pyx). It would be good if we could have a native implementation that worked for general extensions (not just unramified ones).

[742ccff4-512f-43e4-bdd0-5df91c799a52]

Branch: u/lubicz/ramified_extensions_of_general_p_adic_rings_and_fields

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

e64d891

added square for unramified 2-adic fields

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Commit: e64d891

[roed314]

Changed branch from u/lubicz/ramified_extensions_of_general_p_adic_rings_and_fields to u/roed/ramified_extensions_of_general_p_adic_rings_and_fields

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from e64d891 to e3a1c86

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. This was a forced push. Last 10 new commits:

ce61f64	Improve base ring injections for relative extensions of p-adic fields
16477d3	Make conversion from residue field work for two-step extensions of p-adics
1c7cc71	Fix typo in section method for base ring injection in p-adic two-step extensions
8be1504	Fix bugs in creduce and ccoefficients for two-step p-adic extensions
d53df12	Add coerce_list back in to the `_populate_coercion_lists_` call in pAdicExtensionGeneric.__init__
e68beee	Fix some p-adic doctests
635021a	Change _poly_rep to always return the polynomial representing the element, not the unit
67c1f14	Change the internal base ring for two-step extensions to not show precision when printing, fix bug in base ring coercion
8939981	Fix problems in expansion code
e3a1c86	Merge commit '8939981e4b02ac12b86f943d93a8baef25e9af51' of git://trac.sagemath.org/sage into t/23218/general_extensions

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

e64d891	added square for unramified 2-adic fields
65f2853	Merge commit 'e64d891c19b4b5bb1b2b8d666f16f02ad405c67e' of git://trac.sagemath.org/sage into t/23344/sqrt_2

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from e3a1c86 to 65f2853

[742ccff4-512f-43e4-bdd0-5df91c799a52]

Changed branch from u/roed/ramified_extensions_of_general_p_adic_rings_and_fields to u/lubicz/ramified_extensions_of_general_p_adic_rings_and_fields

[roed314]

Changed branch from u/lubicz/ramified_extensions_of_general_p_adic_rings_and_fields to u/roed/ramified_extensions_of_general_p_adic_rings_and_fields

[roed314]

Changed branch from u/roed/ramified_extensions_of_general_p_adic_rings_and_fields to u/roed/23344/2adic_sqrt

[xcaruso]

comment:12

Several comments:

• you don't need your function _artin_schreier since Sage has already methods (e.g. roots) for solving algebraic equations in finite fields

• in the square_root method, prec shouldn't be the precision cap of the parent but the actual precision of the element, no?

• in the square_root method, use the sqrt method for computing the square root modulo p (by the way, I don't understand why this part is needed)

• ceil((prec+1)/2.0) is also 1 + prec // 2, isn't it?

• I would say that our recursive implementation is slower that an iterative one, but I'm not sure

• maybe, you could at the same time complete the documentation of the method sqrt

[xcaruso]

comment:13

By the way, this ticket only implements sqrt for unramified extensions of Q2, is that right? (It's not what is written in the description of the ticket.)

    sage: R.<a> = Zq(5^2)
    sage: sqrt(1+5*a)
    Traceback (most recent call last):
    ...
    NotImplementedError

[roed314]

comment:14

Replying to @xcaruso:

By the way, this ticket only implements sqrt for unramified extensions of Q2, is that right? (It's not what is written in the description of the ticket.)

Yeah, I think that's correct. It would be nice to have it also include an implementation for other primes, but that's not currently in the code.

[742ccff4-512f-43e4-bdd0-5df91c799a52]

comment:15

My answer to the first comments of Xavier:

1) artin schreier is very efficient to solve equations of the form x2=ax+b by using a square and multiply type algorithm it shoud perform in ln(n)2 (where n is the degree of Q_q / Q_p) operations in Q_2. I don't know what is the algorithm implemented in roots but it should be less efficient, no ?

2) yes I will make the change

3) I have search through the square_root method and I could find only to lines with the keyword sqrt :

• ans = self.parent()(self.pari().sqrt())
• z = self._inv_sqrt(prec) the first one is for calling the pari implementation and the second one is a call to the inverse square root method.

4) yes I will make the change

5) ok

6) Ok

7) I will extend the implementation for odd primes p. Thanks for your comments and you help !

[xcaruso]

Changed branch from u/roed/23344/2adic_sqrt to u/caruso/23344/2adic_sqrt

[xcaruso]

Changed commit from 65f2853 to 61ba701

[xcaruso]

comment:17

I've pushed a new implementation which works over any extension of Qp for p > 2 and any unramified extension of Q2 (the case of ramified extensions of Q2 is much more subtle).

PS: I've deleted your code; feel free to revert my changes if you think that it's appropriate.

---

New commits:

830dbba	Merge branch 'develop' into t/23344/23344/2adic_sqrt
61ba701	square root over almost all p-adic fields (except ramified extension of Q2)

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

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

88e08b2

square root for all p-adic fields

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 61ba701 to 88e08b2

[xcaruso]

comment:19

I've extended my implementation to all p-adic fields (including ramified extensions of Q_2).

Needs review.

[742ccff4-512f-43e4-bdd0-5df91c799a52]

Changed branch from u/caruso/23344/2adic_sqrt to u/lubicz/23344/2adic_sqrt

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

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

332fe2c

positive review

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 88e08b2 to 332fe2c

[742ccff4-512f-43e4-bdd0-5df91c799a52]

comment:22

In
def _square_root(self):

• replaced parent = self.parent() by ring = self.parent()
• corrected a bug in the computation of Artin-Schreier equation (carac 2). Pushed the changes on trac.

[xcaruso]

Author: Xavier Caruso

[xcaruso]

Reviewer: David Lubicz

[fchapoton]

comment:25

patchbot reports failing doctests..

[roed314]

comment:26

It looks like many of the failures result from different choices of square root, but I agree that they need to be fixed.

[xcaruso]

Changed branch from u/lubicz/23344/2adic_sqrt to u/caruso/23344/2adic_sqrt

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

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

44ae009	Better ordering of square roots
3085538	Redo David's changes (sorry: forgot to pull)

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 332fe2c to 3085538

[xcaruso]

comment:29

As David pointed out, the issue was due to the choice/ordering of square roots (which was not consistent when precision varies). I fixed it. Let's see what the patchbot says now.

[xcaruso]

Description changed:

--- 
+++ 
@@ -1 +1 @@
-This ticket will implement the square root over unramified entensions of Z_p
+This ticket implements square root over p-adic fields (i.e. finite extension of Qp)

[fchapoton]

comment:31

bot is green

[xcaruso]

comment:32

OK. So, I give a positive review to this ticket again. Please change this if you disagree.

[vbraun]

Changed branch from u/caruso/23344/2adic_sqrt to 3085538