behaviour of the norm function in the p-adic ring #5105

sagetrac-ljpk · 2009-01-26T18:16:35Z

The p-adic norm seems to be defined differently in SAGE to the standard textbook definition, in which it is usually normalized so that $|p|=1/p$, but this is what SAGE does:

sage: Q11=pAdicField(11)
sage: n=Q11(11)
sage: n.norm()
11 + O(11^21)

Would it be possible to swap it round so that the norm of 11 is given as 1/11?

Component: padics

Author: David Roe

Reviewer: Robert Bradshaw

Merged: 4.0.alpha0

Issue created by migration from https://trac.sagemath.org/ticket/5105

The text was updated successfully, but these errors were encountered:

sagetrac-mabshoff · 2009-02-06T23:02:32Z

comment:1

3.4 is for ReST tickets only.

Cheers,

Michael

sagetrac-fwclarke · 2009-02-10T18:37:54Z

comment:2

There is a confusion of terminology here. It's the "field norm" that's defined for p-adics. Thus

sage: Q11 = pAdicField(11, 6)
sage: F.<a> = Q11.ext(x^2 - 2)
sage: (2 + 3*a).norm()
8 + 9*11 + 10*11^2 + 10*11^3 + 10*11^4 + 10*11^5 + O(11^6)
sage: (2 + 3*a)*(2 - 3*a)
8 + 9*11 + 10*11^2 + 10*11^3 + 10*11^4 + 10*11^5 + O(11^6)

So

sage: Q11(22).norm()
2*11 + O(11^7)

is correct, as is

sage: QQ(-163).norm()
-163

What you're wanting is usually called the p-adic absolute value (it's a norm in the functional analysis sense). It would be
good if one could do

sage: Q11(22).abs()

and get 1/11. This isn't currently defined, if z is an element of a padic field, the absolute value can be obtained as

z.parent().prime()^(-z.ordp())

JohnCremona · 2009-04-09T16:07:32Z

comment:4

I thought this looked good, and it applied ok to 3.4.1.rc1, but I got a whole lot of doctest failures in sage/rings/padics:

The following tests failed:


        sage -t  "devel/sage-5105/sage/rings/padics/padic_generic.py"
        sage -t  "devel/sage-5105/sage/rings/padics/padic_ZZ_pX_CA_element.pyx"
        sage -t  "devel/sage-5105/sage/rings/padics/padic_ZZ_pX_element.pyx"
        sage -t  "devel/sage-5105/sage/rings/padics/pow_computer_ext.pyx"
        sage -t  "devel/sage-5105/sage/rings/padics/padic_ZZ_pX_FM_element.pyx"
        sage -t  "devel/sage-5105/sage/rings/padics/padic_ZZ_pX_CR_element.pyx"
        sage -t  "devel/sage-5105/sage/rings/padics/padic_printing.pyx"

Most look like this:

sage -t  "devel/sage-5105/sage/rings/padics/padic_generic.py"
**********************************************************************
File "/home/jec/sage-3.4.1.rc1/devel/sage-5105/sage/rings/padics/padic_generic.py", line 304:
    sage: y = W.teichmuller(3); y
Exception raised:
    Traceback (most recent call last):
      File "/home/jec/sage-3.4.1.rc1/local/bin/ncadoctest.py", line 1231, in run_one_test
        self.run_one_example(test, example, filename, compileflags)
      File "/home/jec/sage-3.4.1.rc1/local/bin/sagedoctest.py", line 38, in run_one_example
        OrigDocTestRunner.run_one_example(self, test, example, filename, compileflags)
      File "/home/jec/sage-3.4.1.rc1/local/bin/ncadoctest.py", line 1172, in run_one_example
        compileflags, 1) in test.globs
      File "<doctest __main__.example_11[14]>", line 1, in <module>
        y = W.teichmuller(Integer(3)); y###line 304:
    sage: y = W.teichmuller(3); y
      File "sage_object.pyx", line 98, in sage.structure.sage_object.SageObject.__repr__ (sage/structure/sage_object.c:1342)
      File "padic_generic_element.pyx", line 212, in sage.rings.padics.padic_generic_element.pAdicGenericElement._repr_ (sage/rings/padics/padic_generic_element.c:6185)
      File "padic_printing.pyx", line 373, in sage.rings.padics.padic_printing.pAdicPrinter_class.repr_gen (sage/rings/padics/padic_printing.cpp:8469)
      File "padic_printing.pyx", line 460, in sage.rings.padics.padic_printing.pAdicPrinter_class._repr_gen (sage/rings/padics/padic_printing.cpp:9866)
      File "padic_printing.pyx", line 580, in sage.rings.padics.padic_printing.pAdicPrinter_class._repr_spec (sage/rings/padics/padic_printing.cpp:12415)
      File "padic_ext_element.pyx", line 171, in sage.rings.padics.padic_ext_element.pAdicExtElement._ext_p_list (sage/rings/padics/padic_ext_element.cpp:3643)
      File "padic_ext_element.pyx", line 162, in sage.rings.padics.padic_ext_element.pAdicExtElement.ext_p_list (sage/rings/padics/padic_ext_element.cpp:3540)
    NotImplementedError
**********************************************************************
File "/home/jec/sage-3.4.1.rc1/devel/sage-5105/sage/rings/padics/padic_generic.py", line 310:
    sage: b = A.teichmuller(1 + 2*a - a^2); b
Exception raised:
    Traceback (most recent call last):
      File "/home/jec/sage-3.4.1.rc1/local/bin/ncadoctest.py", line 1231, in run_one_test
        self.run_one_example(test, example, filename, compileflags)
      File "/home/jec/sage-3.4.1.rc1/local/bin/sagedoctest.py", line 38, in run_one_example
        OrigDocTestRunner.run_one_example(self, test, example, filename, compileflags)
      File "/home/jec/sage-3.4.1.rc1/local/bin/ncadoctest.py", line 1172, in run_one_example
        compileflags, 1) in test.globs
      File "<doctest __main__.example_11[18]>", line 1, in <module>
        b = A.teichmuller(Integer(1) + Integer(2)*a - a**Integer(2)); b###line 310:
    sage: b = A.teichmuller(1 + 2*a - a^2); b
      File "sage_object.pyx", line 98, in sage.structure.sage_object.SageObject.__repr__ (sage/structure/sage_object.c:1342)
      File "padic_generic_element.pyx", line 212, in sage.rings.padics.padic_generic_element.pAdicGenericElement._repr_ (sage/rings/padics/padic_generic_element.c:6185)
      File "padic_printing.pyx", line 373, in sage.rings.padics.padic_printing.pAdicPrinter_class.repr_gen (sage/rings/padics/padic_printing.cpp:8469)
      File "padic_printing.pyx", line 460, in sage.rings.padics.padic_printing.pAdicPrinter_class._repr_gen (sage/rings/padics/padic_printing.cpp:9866)
      File "padic_printing.pyx", line 580, in sage.rings.padics.padic_printing.pAdicPrinter_class._repr_spec (sage/rings/padics/padic_printing.cpp:12415)
      File "padic_ext_element.pyx", line 171, in sage.rings.padics.padic_ext_element.pAdicExtElement._ext_p_list (sage/rings/padics/padic_ext_element.cpp:3643)
      File "padic_ext_element.pyx", line 162, in sage.rings.padics.padic_ext_element.pAdicExtElement.ext_p_list (sage/rings/padics/padic_ext_element.cpp:3540)
    NotImplementedError
**********************************************************************
1 items had failures:
   2 of  20 in __main__.example_11

while there also some simpler ones:

File "/home/jec/sage-3.4.1.rc1/devel/sage-5105/sage/rings/padics/padic_ZZ_pX_CR_element.pyx", line 80:
    sage: y.precision_relative()
Expected:
    20
Got:
    2
**********************************************************************
File "/home/jec/sage-3.4.1.rc1/devel/sage-5105/sage/rings/padics/padic_ZZ_pX_CR_element.pyx", line 82:
    sage: y.precision_absolute()
Expected:
    24
Got:
    6

and

sage -t  "devel/sage-5105/sage/rings/padics/padic_ZZ_pX_FM_element.pyx"
**********************************************************************
File "/home/jec/sage-3.4.1.rc1/devel/sage-5105/sage/rings/padics/padic_ZZ_pX_FM_element.pyx", line 1214:
    sage: ((1+2*w)^5).norm()
Expected:
    1 + 5^2 + O(5^5)
Got:
    1 + O(5^2)
**********************************************************************
File "/home/jec/sage-3.4.1.rc1/devel/sage-5105/sage/rings/padics/padic_ZZ_pX_FM_element.pyx", line 1216:
    sage: ((1+2*w)).norm()^5
Expected:
    1 + 5^2 + O(5^5)
Got:
    1 + O(5^2)
**********************************************************************
File "/home/jec/sage-3.4.1.rc1/devel/sage-5105/sage/rings/padics/padic_ZZ_pX_FM_element.pyx", line 1246:
    sage: a.trace()
Expected:
    3*5 + 2*5^2 + 3*5^3 + 2*5^4 + O(5^5)
Got:
    O(5)
**********************************************************************
File "/home/jec/sage-3.4.1.rc1/devel/sage-5105/sage/rings/padics/padic_ZZ_pX_FM_element.pyx", line 1248:
    sage: a.trace() + b.trace()
Expected:
    4*5 + 5^2 + 5^3 + 2*5^4 + O(5^5)
Got:
    O(5)
**********************************************************************
File "/home/jec/sage-3.4.1.rc1/devel/sage-5105/sage/rings/padics/padic_ZZ_pX_FM_element.pyx", line 1250:
    sage: (a+b).trace()
Expected:
    4*5 + 5^2 + 5^3 + 2*5^4 + O(5^5)
Got:
    O(5)
**********************************************************************
File "/home/jec/sage-3.4.1.rc1/devel/sage-5105/sage/rings/padics/padic_ZZ_pX_FM_element.pyx", line 1277:
    sage: c._ntl_rep()
Expected:
    [89 9 4 1]
Got:
    [4 4 4]
**********************************************************************
3 items had failures:
   2 of   8 in __main__.example_29
   3 of  11 in __main__.example_30
   1 of   8 in __main__.example_31
***Test Failed*** 6 failures.
For whitespace errors, see the file /home/jec/sage-3.4.1.rc1/tmp/.doctest_padic_ZZ_pX_FM_element.py
         [1.4 s]

I have absolutely no idea what in the patch has caused this, but it needs to be looked at!

sagetrac-mabshoff · 2009-05-11T09:09:46Z

comment:6

I guess this is reviewed by #5778 and the issues reported here due to doctest failures have been fixed there.

Cheers,

Michael

roed314 · 2009-05-11T10:10:35Z

Implements abs() and exlains the difference between it and norm()

sagetrac-mabshoff · 2009-05-11T10:33:04Z

comment:7

Attachment: trac_5105.patch.gz

Positive review due to #5778 - credit goes to RobertWB.

Cheers,

Michael

sagetrac-mabshoff · 2009-05-11T10:33:37Z

comment:8

Merged in Sage 4.0.alpha0.

Cheers,

Michael

loefflerd · 2009-06-09T08:39:52Z

Reviewer: Robert Bradshaw

loefflerd · 2009-06-09T08:39:52Z

Merged: 4.0.alpha0

loefflerd · 2009-06-09T08:39:52Z

Author: David Roe

sagetrac-ljpk mannequin added this to the sage-3.4 milestone Jan 26, 2009

sagetrac-ljpk mannequin added c: number theory labels Jan 26, 2009

sagetrac-ljpk mannequin assigned williamstein Jan 26, 2009

sagetrac-mabshoff mannequin modified the milestones: sage-3.4, sage-3.4.1 Feb 6, 2009

sagetrac-fwclarke mannequin added t: enhancement and removed t: bug labels Feb 10, 2009

roed314 added the s: needs review label Mar 18, 2009

JohnCremona changed the title ~~behaviour of the norm function in the p-adic ring~~ [with review] behaviour of the norm function in the p-adic ring Apr 9, 2009

JohnCremona added s: needs work and removed s: needs review labels Apr 9, 2009

sagetrac-mabshoff mannequin assigned roed314 and unassigned williamstein Apr 26, 2009

sagetrac-mabshoff mannequin added c: padics and removed c: number theory labels Apr 26, 2009

sagetrac-mabshoff mannequin changed the title ~~[with review] behaviour of the norm function in the p-adic ring~~ behaviour of the norm function in the p-adic ring May 11, 2009

sagetrac-mabshoff mannequin added s: positive review and removed s: needs work labels May 11, 2009

sagetrac-mabshoff mannequin removed the s: positive review label May 11, 2009

sagetrac-mabshoff mannequin closed this as completed May 11, 2009

roed314 mentioned this issue Jun 9, 2009

Get p-adic doctest coverage to 100% (depends on #5105) #5778

Closed

Provide feedback

Saved searches

Use saved searches to filter your results more quickly

behaviour of the norm function in the p-adic ring #5105

behaviour of the norm function in the p-adic ring #5105

sagetrac-ljpk mannequin commented Jan 26, 2009

sagetrac-mabshoff mannequin commented Feb 6, 2009

sagetrac-fwclarke mannequin commented Feb 10, 2009

JohnCremona commented Apr 9, 2009

sagetrac-mabshoff mannequin commented May 11, 2009

roed314 commented May 11, 2009

sagetrac-mabshoff mannequin commented May 11, 2009

sagetrac-mabshoff mannequin commented May 11, 2009

loefflerd mannequin commented Jun 9, 2009

loefflerd mannequin commented Jun 9, 2009

loefflerd mannequin commented Jun 9, 2009

behaviour of the norm function in the p-adic ring #5105

behaviour of the norm function in the p-adic ring #5105

Comments

sagetrac-ljpk mannequin commented Jan 26, 2009

sagetrac-mabshoff mannequin commented Feb 6, 2009

sagetrac-fwclarke mannequin commented Feb 10, 2009

JohnCremona commented Apr 9, 2009

sagetrac-mabshoff mannequin commented May 11, 2009

roed314 commented May 11, 2009

sagetrac-mabshoff mannequin commented May 11, 2009

sagetrac-mabshoff mannequin commented May 11, 2009

loefflerd mannequin commented Jun 9, 2009

loefflerd mannequin commented Jun 9, 2009

loefflerd mannequin commented Jun 9, 2009