Opened 13 years ago
Last modified 10 years ago
#362 new bug
problem with syzygies over Z/nZ in the presence of zerodivisors
Reported by: | Owned by: | wienand | |
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Priority: | minor | Milestone: | 3-1-4 and higher |
Component: | singular-kernel | Version: | 3-1-3 |
Keywords: | syzygies over rings with zerodivisors | Cc: |
Description
Bug originally found by Eva Zerz: In the following example over Z/4Z, 'syz' does not return the generator of the syzygy module, but only 2*generator: SINGULAR / A Computer Algebra System for Polynomial Computations / version 3-1-3 0< by: W. Decker, G.-M. Greuel, G. Pfister, H. Schoenemann \ March 2011 FB Mathematik der Universitaet, D-67653 Kaiserslautern \ > ring r=(integer,4),(x,y,z),lp; > module m=[x,0],[2*y,1],[y,z]; > syz(m); _[1]=2xz*gen(2)+2x*gen(3)+2y*gen(1) The correct generator can be computed using: > ring r=integer,(x,y,z),lp; > module m=[x,0],[2*y,1],[y,z],[4,0],[0,4]; > syz(m); _[1]=y*gen(1)+(2xz2+xz)*gen(2)-(2xz+x)*gen(3)
Change History (3)
comment:1 Changed 12 years ago by
Owner: | changed from somebody to wienand |
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comment:2 Changed 10 years ago by
when looking at this ticked again, I observe that the output for recent Singular version differs significantly. Which output is correct (or none?)
ring r=(integer,4),(x,y,z),lp; > module m=[x,0],[2*y,1],[y,z]; > syz(m); _[1]=2xz*gen(2)+2x*gen(3)+2y*gen(1) _[2]=2xz2*gen(2)+2xz*gen(3)+xz*gen(2)+3x*gen(3)+y*gen(1) > . . ring r=integer,(x,y,z),lp; // ** redefining r ** > module m=[x,0],[2*y,1],[y,z],[4,0],[0,4]; > syz(m); _[1]=2z*gen(5)-gen(5)-8*gen(3)+4*gen(2) _[2]=y*gen(4)+z*gen(5)-4*gen(3) _[3]=x*gen(4)-4*gen(1) _[4]=xz3*gen(5)-4xz2*gen(3)+2xz2*gen(2)-2xz*gen(3)+xz*gen(2)-x*gen(3)+y*gen(1)
comment:3 Changed 10 years ago by
Both sets of generators for the respective syzygy modules are correct.
_[1] of the first module is 2*_[2] modulo 4.
_[1],_[2] and _[3] of the second module are trivial syzygies in the sense that only gen(4) and gen(5) do not contain a factor 4 in their coefficient. _[4] is the syzygy _[2] of the first module.
Hence the two modules are the same (up to the different way of writing 'modulo 4'), just the set of generators differs.
Anne
Please, have a look and try to help us if you have time.