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| solving multivariate equations over finite field https://www.singular.uni-kl.de/forum/viewtopic.php?f=10&t=1949 |
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| Author: | borisz [ Sat Jul 30, 2011 2:10 pm ] |
| Post subject: | solving multivariate equations over finite field |
As part of my master's thesis I started to write a solver for systems of multivariate polynomial equations over finite fields. I am curious if there is any work in progress in this field or in finding roots of univariate polynomials with coefficients in GF(p^n)? |
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| Author: | gorzel [ Mon Aug 01, 2011 3:20 pm ] |
| Post subject: | Re: solving multivariate equations over finite field |
See the discussion in the forum: Solving systems of polynomials over GF(2) viewtopic.php?f=10&t=1908 combined with ...declaration of ring or galois field viewtopic.php?f=10&t=1823 |
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| Author: | gorzel [ Mon Aug 01, 2011 3:41 pm ] |
| Post subject: | Re: solving multivariate equations over finite field |
http://www.singular.uni-kl.de/Manual/la ... htm#SEC766 Edit: see the Note http://www.singular.uni-kl.de/Manual/la ... htm#SEC256 Singular can (at the moment) not factorize over GF(p^n) for baserings defined as ring r =(p^n,a),.... So you have to choose the fir st form by setting the minpoly yourself. Code: > ring ra5 = (25,a),(x,y,z),dp; > poly f = (x+y)*(z-x); > factorize(f); ? not implemented ? error occurred in or before STDIN line 9: `factorize(f);` > basering; // # ground field : 25 // primitive element : a // minpoly : 1*a^2+4*a^1+2*a^0 // number of vars : 3 // block 1 : ordering dp // : names x y z // block 2 : ordering C > ring ra25 = (5,a),(x,y,z),lp; > minpoly = 1*a^2+4*a^1+2*a^0; > poly f = (x+y)*(z-x); > factorize(f); [1]: _[1]=1 _[2]=x+y _[3]=-x+z [2]: 1,1,1 Code: > LIB "general.lib"; > cyclic(3); _[1]=x+y+z _[2]=xy+xz+yz _[3]=xyz-1 > ideal I = _; > option(redSB); > facstd(I); [1]: _[1]=z+(-2a-1) _[2]=y+(2a+2) _[3]=x-1 [2]: _[1]=z-1 _[2]=y+(-2a-1) _[3]=x+(2a+2) [3]: _[1]=z+(2a+2) _[2]=y-1 _[3]=x+(-2a-1) [4]: _[1]=z-1 _[2]=y+(2a+2) _[3]=x+(-2a-1) [5]: _[1]=z+(2a+2) _[2]=y+(-2a-1) _[3]=x-1 [6]: _[1]=z+(-2a-1) _[2]=y-1 _[3]=x+(2a+2) Then check also triang.lib http://www.singular.uni-kl.de/Manual/la ... tm#SEC1556 Code: > triangLfak(std(I));
[1]: _[1]=z-1 _[2]=y+(-2a-1) _[3]=x+(2a+2) [2]: _[1]=z-1 _[2]=y+(2a+2) _[3]=x+(-2a-1) [3]: _[1]=z+(-2a-1) _[2]=y-1 _[3]=x+(2a+2) [4]: _[1]=z+(-2a-1) _[2]=y+(2a+2) _[3]=x-1 [5]: _[1]=z+(2a+2) _[2]=y+(-2a-1) _[3]=x-1 [6]: _[1]=z+(2a+2) _[2]=y-1 _[3]=x+(-2a-1) |
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