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Re: GroebnerBasis


Here are a few data points. The first is an example test from the text
by Cox, Little, and O'Shea. The second comes from an article by Steve
Czapor. The third is from the "sugar-cube" article re pair selection
stragegies for Groebner basis computation (ISSAC '91). The fourth is
some flavor of the Trinks problem, now fairly standard in the
literature. I ran all these on the same machine, a pentium pro (200mz).

In[2]:= GroebnerBasis[{x^5+y^4+z^3-1, x^3+y^2+z^2-1},{x,y,z}];//Timing
Out[2]= {0.12 Second, Null}

In[3]:= $Version
Out[3]= Linux 3.0 (October 4, 1996)            

In[6]:= GroebnerBasis[{8*x^2
-2*x*y-6*x*z+3*x+3*y^2-7*y*z+10*y+10*z^2-8*z-4,
        10*x^2-2*x*y+6*x*z-6*x+9*y^2-y*z-4*y-2*z^2+5*z-9,
        5*x^2+8*x*y+4*x*z+8*x+9*y^2-6*y*z+2*y-z^2-7*z+5}, {x,y,z}]; //
Timing
Out[6]= {0.1 Second, Null}
 
In[8]:= GroebnerBasis[{x^31 - x^6 - x - y, x^8 - z, x^10 - t},
{x,y,z,t}]; // Timing
Out[8]= {0.32 Second, Null}

In[9]:= GroebnerBasis[{45*p+35*s-165*b-36, 35*p+40*z+25*t-27*s,
    15*w+25*p*s+30*z-18*t-165*b^2, -9*w+15*p*t+20*z*s,
  w*p+2*z*t-11*b^3, 99*w-11*s*b+3*b^2}, {b,t,s,w,p,z}]; // Timing
Out[9]= {8.05 Second, Null}

vs.

In[3]:= GroebnerBasis[{x^5+y^4+z^3-1, x^3+y^2+z^2-1},{x,y,z}];//Timing
Out[3]= {3.72 Second, Null}

In[4]:= $Version
Out[4]= Linux 2.2 (June 3, 1995)
 
In[5]:= GroebnerBasis[{8*x^2
-2*x*y-6*x*z+3*x+3*y^2-7*y*z+10*y+10*z^2-8*z-4,
        10*x^2-2*x*y+6*x*z-6*x+9*y^2-y*z-4*y-2*z^2+5*z-9,
        5*x^2+8*x*y+4*x*z+8*x+9*y^2-6*y*z+2*y-z^2-7*z+5},
{x,y,z}]//Timing
Interrupt> a
Out[5]= $Aborted

(* it cranked a few minutes at least *)

In[8]:= GroebnerBasis[{x^31 - x^6 - x - y, x^8 - z, x^10 - t},
{x,y,z,t}]; // Timing
Interrupt> a
Out[8]= $Aborted                 
(* also hung *)

(* I won't even try the last example *)


For more 3.0 timings and other pertinent info, you might check the
"GroebnerBasis in Mathematica 3.0" article in the Fall '96 issue of The
Mathematica Journal. Other articles explain many of the other various
improvements to version 3.0.


Daniel Lichtblau
Wolfram Research
danl at wolfram.com


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