Roswitha Rissner found a 25x26 matrix over GF(3)[x] such that calling
smith_form() leads to a crash:
*** Error in `python': free(): invalid next size (normal): 0x0000000003ad4fe0
***
Input file as well as the full output are below.
The mathematical problem itself does seem to be solvable: pari can handle it.
A1 = A.transpose().augment(vector(R, 26*[0])).transpose()
S = pari(A1).matsnf(3)
def to_sage(gen):
return matrix(R, [[R(entry) for entry in column] for column in
gen]).transpose()
U = to_sage(S[0])
V = to_sage(S[1])
D = to_sage(S[2])
assert U*A1*V == D
assert U.det() == 1
assert V.det() == 1
assert matrix.diagonal(D.diagonal()) == D
Regards, Clemens
========================
R.<x> = GF(3)[]
A = matrix([[x^4 + x^3 + 2*x + 2, 2*x^5 + x^4 + 2*x^3 + x^2 + 2, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[2*x^3 + x + 1, 0, 2*x^5 + x^4 + 2*x^3 + x^2 + 2, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[2*x^3 + 2*x + 2, 0, 0, 2*x^5 + x^4 + 2*x^3 + x^2 + 2, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[x^3 + x^2, 0, 0, 0, 2*x^5 + x^4 + 2*x^3 + x^2 + 2, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[x^3 + 2, 0, 0, 0, 0, 2*x^5 + x^4 + 2*x^3 + x^2 + 2, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[2*x^3 + 2*x^2, 0, 0, 0, 0, 0, 2*x^5 + x^4 + 2*x^3 + x^2 + 2, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[x^4 + x^3 + x^2 + x + 2, 0, 0, 0, 0, 0, 0, 2*x^5 + x^4 + 2*x^3 + x^2 + 2, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[2*x^3 + x + 2, 0, 0, 0, 0, 0, 0, 0, 2*x^5 + x^4 + 2*x^3 + x^2 + 2, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[2*x^3 + x, 0, 0, 0, 0, 0, 0, 0, 0, 2*x^5 + x^4 + 2*x^3 + x^2 + 2, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[2*x^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2*x^5 + x^4 + 2*x^3 + x^2 + 2, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[x^3 + 2*x, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2*x^5 + x^4 + 2*x^3 + x^2 + 2, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[2*x^3 + x^2 + 2*x + 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2*x^5 + x^4 + 2*x^3 +
x^2 + 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[x^4 + x + 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2*x^5 + x^4 + 2*x^3 + x^2 +
2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[x^3 + 2*x^2 + 2*x + 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2*x^5 + x^4 +
2*x^3 + x^2 + 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[x^2 + 2*x, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2*x^5 + x^4 + 2*x^3 +
x^2 + 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[x^2 + 2*x + 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2*x^5 + x^4 +
2*x^3 + x^2 + 2, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[x^3 + 2*x^2 + 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2*x^5 + x^4
+ 2*x^3 + x^2 + 2, 0, 0, 0, 0, 0, 0, 0, 0],
[2*x^3 + 2*x^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2*x^5 + x^4
+ 2*x^3 + x^2 + 2, 0, 0, 0, 0, 0, 0, 0],
[x^4 + 2*x^3 + 2*x^2 + 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 2*x^5 + x^4 + 2*x^3 + x^2 + 2, 0, 0, 0, 0, 0, 0],
[x + 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2*x^5 + x^4 +
2*x^3 + x^2 + 2, 0, 0, 0, 0, 0],
[x^2 + x, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2*x^5 +
x^4 + 2*x^3 + x^2 + 2, 0, 0, 0, 0],
[x^3 + x^2 + 2*x, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 2*x^5 + x^4 + 2*x^3 + x^2 + 2, 0, 0, 0],
[2*x^3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
2*x^5 + x^4 + 2*x^3 + x^2 + 2, 0, 0],
[x^2 + 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
2*x^5 + x^4 + 2*x^3 + x^2 + 2, 0],
[x^4 + x^3 + 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 2*x^5 + x^4 + 2*x^3 + x^2 + 2]])
A.smith_form()
============
*** Error in `python': free(): invalid pointer: 0x000000000485b630 ***
======= Backtrace: =========
/lib/x86_64-linux-gnu/libc.so.6(+0x777e5)[0x7fb0d65da7e5]
/lib/x86_64-linux-gnu/libc.so.6(+0x7fe0a)[0x7fb0d65e2e0a]
/lib/x86_64-linux-gnu/libc.so.6(cfree+0x4c)[0x7fb0d65e698c]
/local/sage/sage-7.6.beta4/local/lib/libflint.so.13(_nmod_poly_xgcd_hgcd+0x1baa)[0x7fb0c836787a]
/local/sage/sage-7.6.beta4/local/lib/libflint.so.13(nmod_poly_xgcd+0x14a)[0x7fb0c83604aa]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/rings/polynomial/polynomial_zmod_flint.so(+0xf147)[0x7fafa7c69147]
/local/sage/sage-7.6.beta4/local/lib/libpython2.7.so.1.0(PyObject_Call+0x43)[0x7fb0d72ad483]
/local/sage/sage-7.6.beta4/local/lib/libpython2.7.so.1.0(PyEval_CallObjectWithKeywords+0x47)[0x7fb0d735c9f7]
/local/sage/sage-7.6.beta4/local/lib/libpython2.7.so.1.0(+0x72ad8)[0x7fb0d72cbad8]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/structure/element.so(+0xb920)[0x7fb0cc57f920]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/structure/element.so(+0x13697)[0x7fb0cc587697]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/rings/polynomial/polynomial_element.so(+0x143c1)[0x7fafd0bb23c1]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/rings/polynomial/polynomial_element.so(+0x144157)[0x7fafd0ce2157]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x1c7b4)[0x7fafca43f7b4]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x15f487)[0x7fafca582487]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x1c7b4)[0x7fafca43f7b4]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x1b1634)[0x7fafca5d4634]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x1c7b4)[0x7fafca43f7b4]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x14975a)[0x7fafca56c75a]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x1c602)[0x7fafca43f602]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x14ae7d)[0x7fafca56de7d]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x1c602)[0x7fafca43f602]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x14ae7d)[0x7fafca56de7d]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x1c602)[0x7fafca43f602]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x14ae7d)[0x7fafca56de7d]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x1c602)[0x7fafca43f602]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x14ae7d)[0x7fafca56de7d]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x1c602)[0x7fafca43f602]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x14ae7d)[0x7fafca56de7d]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x1c602)[0x7fafca43f602]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x14ae7d)[0x7fafca56de7d]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x1c602)[0x7fafca43f602]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x14ae7d)[0x7fafca56de7d]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x1c602)[0x7fafca43f602]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x14ae7d)[0x7fafca56de7d]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x1c602)[0x7fafca43f602]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x14ae7d)[0x7fafca56de7d]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x1c602)[0x7fafca43f602]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x14ae7d)[0x7fafca56de7d]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x1c602)[0x7fafca43f602]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x14ae7d)[0x7fafca56de7d]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x1c602)[0x7fafca43f602]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x14ae7d)[0x7fafca56de7d]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x1c602)[0x7fafca43f602]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x14ae7d)[0x7fafca56de7d]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x1c602)[0x7fafca43f602]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x14ae7d)[0x7fafca56de7d]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x1c602)[0x7fafca43f602]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x14ae7d)[0x7fafca56de7d]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x1c602)[0x7fafca43f602]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x14ae7d)[0x7fafca56de7d]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x1c602)[0x7fafca43f602]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x14ae7d)[0x7fafca56de7d]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x1c602)[0x7fafca43f602]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x14ae7d)[0x7fafca56de7d]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x1c602)[0x7fafca43f602]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x14ae7d)[0x7fafca56de7d]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x1c602)[0x7fafca43f602]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x14ae7d)[0x7fafca56de7d]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x1c602)[0x7fafca43f602]
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/matrix2.so(+0x14ae7d)[0x7fafca56de7d]
/local/sage/sage-7.6.beta4/local/lib/libpython2.7.so.1.0(PyEval_EvalFrameEx+0x79f7)[0x7fb0d73649e7]
/local/sage/sage-7.6.beta4/local/lib/libpython2.7.so.1.0(PyEval_EvalCodeEx+0x81c)[0x7fb0d736688c]
======= Memory map: ========
00400000-00401000 r-xp 00000000 08:02 9728105
/local/sage/sage-7.6.beta4/local/bin/python2.7
00600000-00601000 rw-p 00000000 08:02 9728105
/local/sage/sage-7.6.beta4/local/bin/python2.7
01a53000-04891000 rw-p 00000000 00:00 0 [heap]
7fafa0000000-7fafa0021000 rw-p 00000000 00:00 0
7fafa0021000-7fafa4000000 ---p 00000000 00:00 0
7fafa54f0000-7fafa5830000 rw-p 00000000 00:00 0
7fafa5830000-7fafa584b000 r-xp 00000000 08:02 11405834
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7fafa584b000-7fafa5a4a000 ---p 0001b000 08:02 11405834
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/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/matrix/action.so
7fafa5a4f000-7fafa5a8f000 rw-p 00000000 00:00 0
7fafa5a8f000-7fafa5ac6000 r-xp 00000000 08:02 7113656
/local/sage/sage-7.6.beta4/local/lib/python2.7/site-packages/sage/rings/fraction_field_FpT.so
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7fafa5cd0000-7fafa5cd1000 rw-p 00000000 00:00 0
7fafa5cd1000-7fafa5cdd000 r-xp 00000000 08:02 11406062
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7fafb97e4000-7fafb9824000 rw-p 00000000 00:00 0
------------------------------------------------------------------------
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/lib/x86_64-linux-gnu/libc.so.6(__libc_start_main+0xf0)[0x7fb0d6583830]
python(_start+0x29)[0x400719]
------------------------------------------------------------------------
Attaching gdb to process id 16272.
Saved trace to /home/sci/cheuberg/.sage/crash_logs/crash_b2XJWj.log
------------------------------------------------------------------------
Unhandled SIGABRT: An abort() occurred.
This probably occurred because a *compiled* module has a bug
in it and is not properly wrapped with sig_on(), sig_off().
Python will now terminate.
------------------------------------------------------------------------
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