

<p>Times are on a 2 core 2012 notebook:</p>
<p style="text-align: center;">Intel(R) Core(TM) i5-2435M CPU @ 2.40GHz</p>
<p style="text-align: left;">&nbsp;</p>

{{{id=1|
n=6;R=PolynomialRing(QQ,n,'x');
I = sage.rings.ideal.Cyclic(R,n);
I.gens()
///
[x0 + x1 + x2 + x3 + x4 + x5, x0*x1 + x1*x2 + x2*x3 + x3*x4 + x0*x5 + x4*x5, x0*x1*x2 + x1*x2*x3 + x2*x3*x4 + x0*x1*x5 + x0*x4*x5 + x3*x4*x5, x0*x1*x2*x3 + x1*x2*x3*x4 + x0*x1*x2*x5 + x0*x1*x4*x5 + x0*x3*x4*x5 + x2*x3*x4*x5, x0*x1*x2*x3*x4 + x0*x1*x2*x3*x5 + x0*x1*x2*x4*x5 + x0*x1*x3*x4*x5 + x0*x2*x3*x4*x5 + x1*x2*x3*x4*x5, x0*x1*x2*x3*x4*x5 - 1]
}}}

{{{id=4|
time B = I.groebner_basis("libsingular:std")
///
Time: CPU 0.14 s, Wall: 0.15 s
}}}

{{{id=59|
import giacpy
from giacpy import *
///
// Giac share root-directory:/home/fred/dev/sage/git-trac-command/local/share/giac/
// Unable to find keyword file /home/fred/dev/sage/git-trac-command/local/share/giac/doc/fr/keywords
// Giac share root-directory:/home/fred/dev/sage/git-trac-command/local/share/giac/
Help file /home/fred/dev/sage/git-trac-command/local/share/giac/doc/fr/aide_cas not found
Added 0 synonyms
}}}

{{{id=8|
I1=libgiac(I.gens())
///
}}}

<p>In&nbsp; giac 1.1 there is a new code for grobner basis with revlex.</p>
<p>It is much faster than the old ones, but only with revlex and doesn't use cocoa.</p>

{{{id=14|
time B1=I1.gbasis([R.gens()],'revlex')
///
Time: CPU 0.09 s, Wall: 0.09 s
}}}

{{{id=9|
time B1=I1.gbasis([R.gens()],'revlex,cocoa=false')
///
Time: CPU 0.09 s, Wall: 0.09 s
}}}

{{{id=22|
[(B[-i-1]/B1[i]).simplify()  for i in range(len(B))] #singular donne des fractions. les degres ne sont pas dans le meme ordre
///
[1,
 1,
 1/14,
 1,
 1/2,
 1/7,
 1/2,
 1/903,
 1/1806,
 1/258,
 1/903,
 1/903,
 1/1806,
 1/903,
 1/1806,
 1/1806,
 1/7,
 1/602,
 1/258,
 1/1806,
 1/18,
 1/274400,
 1/54880,
 1/109760,
 1/164640,
 1/86596,
 1/43298,
 1/86596,
 1/35739035160,
 1/2978252930,
 1/1191301172,
 1/2382602344,
 1/3573903516,
 1/35739035160,
 1/14,
 1/24990,
 1/42,
 1/24990,
 1/99960,
 1/6664,
 1/99960,
 1/19992,
 1/99960,
 1/2040,
 1/16660]
}}}

{{{id=13|
n=8;R=PolynomialRing(QQ,n,'x');
I = sage.rings.ideal.Cyclic(R,n);I
///
Ideal (x0 + x1 + x2 + x3 + x4 + x5 + x6 + x7, x0*x1 + x1*x2 + x2*x3 + x3*x4 + x4*x5 + x5*x6 + x0*x7 + x6*x7, x0*x1*x2 + x1*x2*x3 + x2*x3*x4 + x3*x4*x5 + x4*x5*x6 + x0*x1*x7 + x0*x6*x7 + x5*x6*x7, x0*x1*x2*x3 + x1*x2*x3*x4 + x2*x3*x4*x5 + x3*x4*x5*x6 + x0*x1*x2*x7 + x0*x1*x6*x7 + x0*x5*x6*x7 + x4*x5*x6*x7, x0*x1*x2*x3*x4 + x1*x2*x3*x4*x5 + x2*x3*x4*x5*x6 + x0*x1*x2*x3*x7 + x0*x1*x2*x6*x7 + x0*x1*x5*x6*x7 + x0*x4*x5*x6*x7 + x3*x4*x5*x6*x7, x0*x1*x2*x3*x4*x5 + x1*x2*x3*x4*x5*x6 + x0*x1*x2*x3*x4*x7 + x0*x1*x2*x3*x6*x7 + x0*x1*x2*x5*x6*x7 + x0*x1*x4*x5*x6*x7 + x0*x3*x4*x5*x6*x7 + x2*x3*x4*x5*x6*x7, x0*x1*x2*x3*x4*x5*x6 + x0*x1*x2*x3*x4*x5*x7 + x0*x1*x2*x3*x4*x6*x7 + x0*x1*x2*x3*x5*x6*x7 + x0*x1*x2*x4*x5*x6*x7 + x0*x1*x3*x4*x5*x6*x7 + x0*x2*x3*x4*x5*x6*x7 + x1*x2*x3*x4*x5*x6*x7, x0*x1*x2*x3*x4*x5*x6*x7 - 1) of Multivariate Polynomial Ring in x0, x1, x2, x3, x4, x5, x6, x7 over Rational Field
}}}

{{{id=15|
#time B = I.groebner_basis("libsingular:std")
///
}}}

{{{id=54|
I1=libgiac(I.gens())
///
}}}

{{{id=16|
# cyclic 8 over QQ (with probabilistic algorithms)
time B1=libgiac(I.gens()).gbasis([R.gens()],'revlex') 
///
Running a probabilistic check for the reconstructed Groebner basis. If successfull, error probability is less than 1e-15 and is estimated to be less than 10^-105. Use proba_epsilon:=0 to certify (this takes more time).
Time: CPU 274.18 s, Wall: 84.38 s
}}}

{{{id=81|
p=65521
n=8;R=PolynomialRing(GF(p),n,'x');
I = sage.rings.ideal.Cyclic(R,n);
///
}}}

{{{id=82|
# cyclic 8 over GF(65521)
time B = I.groebner_basis("libsingular:std") 
///
Time: CPU 47.90 s, Wall: 47.91 s
}}}

{{{id=84|
Ip=libgiac(I.gens()) % p;
///
}}}

{{{id=83|
# cyclic 8 over GF(65521)
time BG = Ip.gbasis([R.gens()])
///
Time: CPU 10.84 s, Wall: 10.84 s
}}}

{{{id=85|
Ip=libgiac(I.gens()) % 33554393;
///
}}}

{{{id=86|
# cyclic 8 over GF(33554393)
time BG = Ip.gbasis([R.gens()])
///
Time: CPU 13.00 s, Wall: 13.00 s
}}}

<p style="text-align: center;"><strong>Two different way to convert the giac answer to sage.</strong></p>

{{{id=60|
time rep1=vector(B1.revlist(),R).list();
time rep2=[R(j) for j in B1.revlist()];
rep1==rep2
///
Time: CPU 2.02 s, Wall: 2.83 s
Time: CPU 1.90 s, Wall: 1.90 s
True
}}}

{{{id=65|
B1.savegen('c8.gen')
///
}}}

{{{id=17|
giacsettings.proba_epsilon=0
///
}}}

{{{id=19|
# cyclic 8 over QQ with giac without proba algo
time B2=I1.gbasis([R.gens()],'revlex')
///
Time: CPU 363.17 s, Wall: 165.87 s
}}}

{{{id=21|
giacsettings.proba_epsilon=1e-15
///
}}}

{{{id=42|
n=9;R=PolynomialRing(QQ,n,'x');
I = sage.rings.ideal.Katsura(R,n);
I.gens()
///
[x0 + 2*x1 + 2*x2 + 2*x3 + 2*x4 + 2*x5 + 2*x6 + 2*x7 + 2*x8 - 1, x0^2 + 2*x1^2 + 2*x2^2 + 2*x3^2 + 2*x4^2 + 2*x5^2 + 2*x6^2 + 2*x7^2 + 2*x8^2 - x0, 2*x0*x1 + 2*x1*x2 + 2*x2*x3 + 2*x3*x4 + 2*x4*x5 + 2*x5*x6 + 2*x6*x7 + 2*x7*x8 - x1, x1^2 + 2*x0*x2 + 2*x1*x3 + 2*x2*x4 + 2*x3*x5 + 2*x4*x6 + 2*x5*x7 + 2*x6*x8 - x2, 2*x1*x2 + 2*x0*x3 + 2*x1*x4 + 2*x2*x5 + 2*x3*x6 + 2*x4*x7 + 2*x5*x8 - x3, x2^2 + 2*x1*x3 + 2*x0*x4 + 2*x1*x5 + 2*x2*x6 + 2*x3*x7 + 2*x4*x8 - x4, 2*x2*x3 + 2*x1*x4 + 2*x0*x5 + 2*x1*x6 + 2*x2*x7 + 2*x3*x8 - x5, x3^2 + 2*x2*x4 + 2*x1*x5 + 2*x0*x6 + 2*x1*x7 + 2*x2*x8 - x6, 2*x3*x4 + 2*x2*x5 + 2*x1*x6 + 2*x0*x7 + 2*x1*x8 - x7]
}}}

{{{id=45|
time B = I.groebner_basis("libsingular:std")
///
Time: CPU 20.70 s, Wall: 20.70 s
}}}

{{{id=46|
giacsettings.proba_epsilon=1e-15;
Igiac=libgiac(I.gens());
time Bgiac=Igiac.gbasis([R.gens()],'revlex')
///
Running a probabilistic check for the reconstructed Groebner basis. If successfull, error probability is less than 1e-15 and is estimated to be less than 10^-133. Use proba_epsilon:=0 to certify (this takes more time).
Time: CPU 4.74 s, Wall: 3.43 s
}}}

{{{id=56|
test=[(Bgiac[i]/B[-i-1]).simplify()  for i in range(len(B))] #singular donne des coeffs ds Q et giac dans Z. les degres ne sont pas dans le meme ordre.
test[123]
///
2345414220966791531953989150167779837563063168000000000000000
}}}

{{{id=77|

///
}}}

{{{id=78|
p=65521
n=12;R=PolynomialRing(GF(p),n,'x');
I = sage.rings.ideal.Katsura(R,n);
I.gens()
///
[x0 + 2*x1 + 2*x2 + 2*x3 + 2*x4 + 2*x5 + 2*x6 + 2*x7 + 2*x8 + 2*x9 + 2*x10 + 2*x11 - 1, x0^2 + 2*x1^2 + 2*x2^2 + 2*x3^2 + 2*x4^2 + 2*x5^2 + 2*x6^2 + 2*x7^2 + 2*x8^2 + 2*x9^2 + 2*x10^2 + 2*x11^2 - x0, 2*x0*x1 + 2*x1*x2 + 2*x2*x3 + 2*x3*x4 + 2*x4*x5 + 2*x5*x6 + 2*x6*x7 + 2*x7*x8 + 2*x8*x9 + 2*x9*x10 + 2*x10*x11 - x1, x1^2 + 2*x0*x2 + 2*x1*x3 + 2*x2*x4 + 2*x3*x5 + 2*x4*x6 + 2*x5*x7 + 2*x6*x8 + 2*x7*x9 + 2*x8*x10 + 2*x9*x11 - x2, 2*x1*x2 + 2*x0*x3 + 2*x1*x4 + 2*x2*x5 + 2*x3*x6 + 2*x4*x7 + 2*x5*x8 + 2*x6*x9 + 2*x7*x10 + 2*x8*x11 - x3, x2^2 + 2*x1*x3 + 2*x0*x4 + 2*x1*x5 + 2*x2*x6 + 2*x3*x7 + 2*x4*x8 + 2*x5*x9 + 2*x6*x10 + 2*x7*x11 - x4, 2*x2*x3 + 2*x1*x4 + 2*x0*x5 + 2*x1*x6 + 2*x2*x7 + 2*x3*x8 + 2*x4*x9 + 2*x5*x10 + 2*x6*x11 - x5, x3^2 + 2*x2*x4 + 2*x1*x5 + 2*x0*x6 + 2*x1*x7 + 2*x2*x8 + 2*x3*x9 + 2*x4*x10 + 2*x5*x11 - x6, 2*x3*x4 + 2*x2*x5 + 2*x1*x6 + 2*x0*x7 + 2*x1*x8 + 2*x2*x9 + 2*x3*x10 + 2*x4*x11 - x7, x4^2 + 2*x3*x5 + 2*x2*x6 + 2*x1*x7 + 2*x0*x8 + 2*x1*x9 + 2*x2*x10 + 2*x3*x11 - x8, 2*x4*x5 + 2*x3*x6 + 2*x2*x7 + 2*x1*x8 + 2*x0*x9 + 2*x1*x10 + 2*x2*x11 - x9, x5^2 + 2*x4*x6 + 2*x3*x7 + 2*x2*x8 + 2*x1*x9 + 2*x0*x10 + 2*x1*x11 - x10]
}}}

{{{id=79|
# Katsura 12 over GF(65521)
time B = I.groebner_basis("libsingular:std")
///
Time: CPU 814.13 s, Wall: 814.09 s
}}}

{{{id=80|
Ip=libgiac(I.gens()) % p
///
}}}

{{{id=89|
# Katsura 12 over GF(65521)
time BG = Ip.gbasis([R.gens()])
///
Time: CPU 90.45 s, Wall: 90.45 s
}}}

{{{id=90|
# conversion to sage
time V=vector(BG.revlist(),R)
///
Time: CPU 42.91 s, Wall: 42.90 s
}}}

{{{id=96|
type(V[0])
///
<type 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular'>
}}}

{{{id=95|

///
}}}