Opened 13 years ago

Closed 13 years ago

#7517 closed defect (fixed)

improve documentation of xgcd command

Reported by: William Stein Owned by: Alex Ghitza
Priority: minor Milestone: sage-4.3
Component: basic arithmetic Keywords: xgcd docstring
Cc: Merged in: sage-4.3.rc0
Authors: William Stein Reviewers: Alex Ghitza
Report Upstream: N/A Work issues:
Branch: Commit:
Dependencies: Stopgaps:

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On Sun, Nov 22, 2009 at 4:57 PM, Ricky Farr <> wrote:
> Dear All,
> I'd like to sincerely thank you for your help before hand.  I'm having
> some issues that need to be straightened out.  I was under the
> impression that xgcd(a,b) returned (g,s,t) so that g = s*a + t*b,
> where g=gcd(a,b).  Please review the following code, and tell me why
> this happens:
> sage: Q.<x> = PolynomialRing(ZZ);
> sage: gcd(x-2,x^3+2*x^2);
> 1
> sage: g,s,t = xgcd(x-2,x^3+2*x^2);
> sage: g
> 16
> sage: s*(x-2)+t*(x^3+2*x^2)
> 16
> I was under the impression, like I said that g would have been equal
> to 1.  Why is g, 16?

The ring ZZ[x] is not a principal ideal domain (e.g., the ideal (2, x) isn't principal), so xgcd *can't* in general return polynomials s, t such that g = s*a+t*b.    A simple example is a=2*x and b=x^2. Then x is the gcd, but you can't write x as a ZZ[x] linear combination of 2*x and x^2, since the linear term of s*(2*x) + t*x^2 is even. 

What it does return is the next best thing, which is s, t such that 

   s*a + t*b = resultant(a,b), 

assuming a, b are coprime (if they aren't, rescale so they are, do the above, then multiply through). 

Note that Sage just calls the FLINT library, and this behavior of xgcd is documented there. 

I did just maybe (?) find a bug in FLINT though (and certainly one in sage):
sage: gcd(Q(2),x^2)
sage: xgcd(Q(2),x^2)
<hang forever> 

Doing the same using NTL works fine:
sage: Q.<x> = PolynomialRing(ZZ,implementation="NTL")
sage: type(x)
<type 'sage.rings.polynomial.polynomial_integer_dense_ntl.Polynomial_integer_dense_ntl'>
sage: gcd(Q(2),x^2)
sage: xgcd(Q(2),x^2)
(4, 2, 0)
sage: xgcd(x-2, x^3+2*x^2)
(16, -x^2 - 4*x - 8, 1)


So, the docs in Sage need to change to correctly define xgcd over a non-PID.  Also, there is maybe a serious bug in FLINT. 

 -- William

Attachments (1)

sagelib_7517.patch (4.1 KB) - added by William Stein 13 years ago.

Download all attachments as: .zip

Change History (4)

comment:1 Changed 13 years ago by William Stein

Status: newneeds_review

I'm making the xgcd *hang* another ticket: #7518

Changed 13 years ago by William Stein

Attachment: sagelib_7517.patch added

comment:2 Changed 13 years ago by Alex Ghitza

Authors: William Stein
Keywords: xgcd docstring added
Reviewers: Alex Ghitza
Status: needs_reviewpositive_review

comment:3 Changed 13 years ago by Mike Hansen

Merged in: sage-4.3.rc0
Resolution: fixed
Status: positive_reviewclosed
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