reduce-historical/r38/packages/groebner/groebner.rlg

Tue Feb 10 12:28:13 2004 run on Linux
% Examples of use of Groebner code.

% In the Examples 1 - 3 the polynomial ring for the ideal operations
% (variable sequence, term order mode) is defined globally in advance.

% Example 1, Linz 85.

torder ({q1,q2,q3,q4,q5,q6},lex)$



groebner {q1,
          q2**2 + q3**2 + q4**2,
          q4*q3*q2,
          q3**2*q2**2 + q4**2*q2**2 + q4**2*q3**2,
          q6**2 + 1/3*q5**2,
          q6**3 - q5**2*q6,
          2*q2**2*q6 - q3**2*q6 - q4**2*q6 + q3**2*q5 - q4**2*q5,
          2*q2**2*q6**2 - q3**2*q6**2 - q4**2*q6**2 - 2*q3**2*q5*q6
          + 2*q4**2*q5*q6 - 2/3*q2**2*q5**2 + 1/3*q3**2*q5**2
          + 1/3*q4**2*q5**2,
          - q3**2*q2**2*q6 - q4**2*q2**2*q6 + 2*q4**2*q3**2*q6 -
          q3**2*q2**2*q5 + q4**2*q2**2*q5,
          - q3**2*q2**2*q6**2 - q4**2*q2**2*q6**2 + 2*q4**2*q3**2*q6**2
          + 2*q3**2*q2**2*q5*q6 - 2*q4**2*q2**2*q5*q6 + 1/3*q3**2*q2**2
          *q5**2 + 1/3*q4**2*q2**2*q5**2 - 2/3*q4**2*q3**2*q5**2,
          - 3*q3**2*q2**4*q5*q6**2 + 3*q4**2*q2**4*q5*q6**2
          + 3*q3**4*q2**2*q5*q6**2 - 3*q4**4*q2**2*q5*q6**2
          - 3*q4**2*q3**4*q5*q6**2 + 3*q4**4*q3**2*q5*q6**2
          + 1/3*q3**2*q2**4*q5**3 - 1/3*q4**2*q2**4*q5**3
          - 1/3*q3**4*q2**2*q5**3 + 1/3*q4**4*q2**2*q5**3 + 1/3*q4**2
            *q3**4*q5**3 - 1/3*q4**4*q3**2*q5**3};


{q1,

   2     2     2
 q2  + q3  + q4 ,

 q2*q3*q4,

      4
 q2*q4 *q6,

      3             3
 q2*q4 *q5 + 3*q2*q4 *q6,

      3   2
 q2*q4 *q6 ,

   4     2   2     4
 q3  + q3 *q4  + q4 ,

   3           3
 q3 *q4 + q3*q4 ,

   2   2
 q3 *q4 *q6,

   2          2        2          2
 q3 *q5 - 3*q3 *q6 - q4 *q5 - 3*q4 *q6,

   2   2     2   2
 q3 *q6  + q4 *q6 ,

      4
 q3*q4 *q6,

      3
 q3*q4 *q5,

      3   2
 q3*q4 *q6 ,

   5
 q4 ,

   4        4
 q4 *q5 + q4 *q6,

   4   2
 q4 *q6 ,

   2           2   2
 q4 *q5*q6 - q4 *q6 ,

   2       2
 q5  + 3*q6 ,

   3
 q6 }



% Example 2. (Little) Trinks problem with 7 polynomials in 6 variables.

trinkspolys:={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**2 + 33/50*b + 2673/10000}$



trinksvars :=  {w,p,z,t,s,b}$


torder(trinksvars,lex)$


switch varopt;

 off varopt;


groebner trinkspolys;


{60000*w + 9500*b + 3969,

 1800*p - 3100*b - 1377,

 18000*z + 24500*b + 10287,

 750*t - 1850*b + 81,

 200*s - 500*b - 9,

        2
 10000*b  + 6600*b + 2673}


groesolve ws;


     3*(4*sqrt(11)*i - 11)
{{b=-----------------------,
              100

     62*sqrt(11)*i + 59
  p=--------------------,
            300

     3*(5*sqrt(11)*i - 13)
  s=-----------------------,
              50

     148*sqrt(11)*i - 461
  t=----------------------,
             500

      - 190*sqrt(11)*i - 139
  w=-------------------------,
              10000

      - 490*sqrt(11)*i - 367
  z=-------------------------},
              3000

     3*( - 4*sqrt(11)*i - 11)
 {b=--------------------------,
               100

      - 62*sqrt(11)*i + 59
  p=-----------------------,
              300

     3*( - 5*sqrt(11)*i - 13)
  s=--------------------------,
                50

      - 148*sqrt(11)*i - 461
  t=-------------------------,
               500

     190*sqrt(11)*i - 139
  w=----------------------,
            10000

     490*sqrt(11)*i - 367
  z=----------------------}}
             3000

 

% Example 3. Hairer, Runge-Kutta 1, 6 polynomials 8 variables.
 
torder({c2,c3,b3,b2,b1,a21,a32,a31},lex);


{{w,p,z,t,s,b},lex}

groebnerf{c2 - a21,
          c3 - a31 - a32,
          b1 + b2 + b3 - 1,
          b2*c2 + b3*c3 - 1/2,
          b2*c2**2 + b3*c3**2 - 1/3,
          b3*a32*c2 - 1/6};


{{c2 - a21,

  c3 - a32 - a31,

  b3 + b2 + b1 - 1,

              2            2                            2    2            2    2
  96*b2*b1*a31  - 96*b2*a31  + 96*b2*a31 - 32*b2 - 72*b1 *a32 *a31 - 48*b1 *a32

           2        2         2                2    3             2
   - 144*b1 *a32*a31  - 144*b1 *a32*a31 - 72*b1 *a31  + 198*b1*a32 *a31

              2                 2                                          3
   + 60*b1*a32  + 396*b1*a32*a31  + 72*b1*a32*a31 - 144*b1*a32 + 198*b1*a31

               2                                                    2
   - 108*b1*a31  - 24*b1*a31 - 81*a21*a32*a31 + 54*a21*a32 - 126*a32 *a31

           2              2                                 3          2
   - 12*a32  - 252*a32*a31  + 126*a32*a31 + 36*a32 - 126*a31  + 162*a31

   - 30*a31 - 12,

                                2                                      2
  8*b2*a21 - 8*b2*a31 + 6*b1*a32  + 12*b1*a32*a31 + 4*b1*a32 + 6*b1*a31

                                 2                             2
   - 4*b1*a31 - 9*a21*a32 - 6*a32  - 12*a32*a31 + 8*a32 - 6*a31  + 10*a31 - 2,

                     2                                       2
  8*b2*a32 + 6*b1*a32  + 12*b1*a32*a31 + 12*b1*a32 + 6*b1*a31  + 4*b1*a31

                      2                     2
   - 9*a21*a32 - 6*a32  - 12*a32*a31 - 6*a31  + 2*a31 + 2,

                          2                           2                    2
  12*b1*a21*a32 - 6*b1*a32  - 12*b1*a32*a31 - 6*b1*a31  - 3*a21*a32 + 6*a32

                               2
   + 12*a32*a31 - 6*a32 + 6*a31  - 6*a31 + 2,

                         2                          2
  4*b1*a21*a31 + 2*b1*a32  + 4*b1*a32*a31 + 2*b1*a31  - 3*a21*a32 - 4*a21*a31

                  2                            2
   + 2*a21 - 2*a32  - 4*a32*a31 + 4*a32 - 2*a31  + 4*a31 - 2,

          3            2                    2           3            2
  6*b1*a32  + 18*b1*a32 *a31 + 18*b1*a32*a31  + 6*b1*a31  - 9*a21*a32

                                      3         2             2             2
   - 9*a21*a32*a31 + 6*a21*a32 - 6*a32  - 18*a32 *a31 + 12*a32  - 18*a32*a31

                               3        2
   + 18*a32*a31 - 6*a32 - 6*a31  + 6*a31  - 2*a31,

       2                                2                  2
  3*a21 *a32 - 3*a21*a32 - a21*a31 + a32  + 2*a32*a31 + a31 }}

 
        
% The examples 4 and 5 use automatic variable extraction.

% Example 4.
 
torder gradlex$


 
g4:= 
groebner{b + e + f - 1,
         c + d + 2*e - 3,
         b + d + 2*f - 1,
         a - b - c - d - e - f,
         d*e*a**2 - 1569/31250*b*c**3,
         c*f - 587/15625*b*d};


                                                   5
g4 := {144534461790680056924571742971580442350868*f

                                                      4
        - 644899801559202566371326081182412388593750*f

                                                         2
        - 5642454222593591361522253644740080176968509*e*f

                                                       3
        + 1026970650200404602876625225711718032483739*f

        + 60671378319336814425425106786936647125250*e*f

                                                        2
        + 12135463840178290842421221291430776956948795*f

        + 82342665293813692270756265387326300721851*e

        - 6546572608747272255841866021042619274525791*f

        - 455593441982762135422235490670177670637,

                              3                        4
       8282838608877853969*e*f  - 2667985333760708531*f

                                2                        3
        - 315490964385538173*e*f  - 8319462093247392142*f  - 25594942638053*e*f

                              2
        + 318993777538462620*f  + 33851175608089*e + 34163367871142*f

        - 8568425233089,

            2                      2
       587*e  - 46875*e*f + 15038*f  - 587*e + 47462*f,

       a + 2*e - 4,

       b + e + f - 1,

       c + 3*e - f - 3,

       d - e + f}

hilbertpolynomial g4;


8

glexconvert(g4,gvarslast,newvars={e},maxdeg=8);


                         8                            7
{8724935291855297898986*e  - 82886885272625330040367*e

                              6                             5
  + 304980377204235125220384*e  - 524915947547338451201596*e

                              4                            3
  + 362375013966993813907616*e  + 52719473339686639067952*e

                              2
  - 154986762992209058701440*e  + 27347344067139574366944*e + 430203494102932512

 }


% Example 5.

off varopt;


torder({u0,u2,u3,u1},lex)$


groesolve({u0**2 - u0 + 2*u1**2 + 2*u2**2 + 2*u3**2,
          2*u0*u1 + 2*u1*u2 + 2*u2*u3 - u1,
          2*u0*u2 + u1**2 + 2*u1*u3 - u2,
          u0 + 2*u1 + 2*u2 + 2*u3 - 1},
         {u0,u2,u3,u1});


{{u0=1,u2=0,u3=0,u1=0},

      1           1
 {u0=---,u2=0,u3=---,u1=0},
      3           3

                 5              4              3             2
 {u0=(85796172*u1  - 47481552*u1  - 10265256*u1  + 4828462*u1  + 414200*u1

       - 24707)/164805,

                  5              4              3             2
  u2=(490926744*u1  - 82790424*u1  - 46802952*u1  + 5425849*u1  + 1108070*u1

       - 83819)/164805,

  u3

                  5             4             3            2
     - 35588322*u1  + 7102080*u1  + 3462372*u1  - 522672*u1  - 98665*u1 + 11905
  =-----------------------------------------------------------------------------
                                       10987

  ,

                      6           5           4          3         2
  u1=root_of(24948*u1_  - 8424*u1_  - 1908*u1_  + 736*u1_  + 24*u1_  - 18*u1_

              + 1,u1_,tag_1)}}

 
% Example 6. (Big) Trinks problem with 6 polynomials in 6 variables.
 
torder(trinksvars,lex)$


btbas:=
 groebner{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*b*s + 3*b**2};


btbas := {17766149161458472422166115589155691471353640232570952361584640*w

                                                                               9
           + 3032932981764169411024286535087872715152793150994240000000000000*b

           + 11886822444254795859791802829918904596379497649520730600000000000

            8
          *b  + 

                                                                             7
          18842475008351431516615767365088235858572104823839818660000000000*b  +

                                                                              6
           18478618789454571665641479626067848900525899492180377333740000000*b  

                                                                               5
          + 11752365113063961011548983119538614396423298749092231098450400000*b

                                                                               4
           + 5110161259755495688253057699488605142801193206234091633443430000*b

                                                                               3
           + 1496961750963944475883560598484727796781670457510019079125319720*b

                                                                              2
           + 288690575257721822668492218552623049380964882774348400629792405*b

           + 36675221781192845731725910375461662443650512572339688148737880*b

           + 1576363174251807401047861085627012261518448811764870474808048,

          1079293561558602199646591522041208256884733644128685355966266880*p + 

                                                                               9
          3268477702530974927415861070452491173139572636038856000000000000000*b

           + 

          12885633343818230635528913313274512975854362843839764665000000000000

            8
          *b  + 

          20548731096300848092222002490748474767709483225818633322500000000000

            7
          *b  + 

          20182049540868333737979937480097593847242554499522522583343500000000

            6
          *b  + 

          12840592651209104850152262711039251760751322701157046861979660000000

            5
          *b  + 

                                                                               4
          5569707184558884260455460870514004047533638259197462099687709750000*b

           + 1626104523905067336734029117969017435050069455164231436772691393000

            3
          *b  + 

                                                                              2
          317837165064133808425156860561547977935248864650364953213370433325*b

           + 38814916107963233682867824475195786374043607759221055124383464600*b

           + 1271557117681971715777755868970298734422034654142333039426477936,

          79947671226563125899747520151200611621091381046569285627130880*z - 

                                                                              9
          207000360174268878618253807286221414267374039050881600000000000000*b  

          - 816930976846005632807581869594187232031930825060787069000000000000

            8
          *b  - 

                                                                               7
          1304191848597021137419209873493260430019068809677834324500000000000*b

           - 1281648951757969533154633755921969360988365079018184794999100000000

            6
          *b  - 

                                                                              5
          816111850476984294981540451378918253659030380648143145999676000000*b  

          - 354123157925898223808181474698490366723104830470028121053590350000

            4
          *b  - 

                                                                              3
          103524414072393919562685172085266423030522292688870620316927889800*b  

                                                                               2
          - 20314259597530323830287024948271996904872237353588201428371308545*b

           - 2537917907646239051588678539186026277776904294491429226344955896*b

           - 101754994043218022355542895254001231074817584410141704072917808,

          53964678077930109982329576102060412844236682206434267798313344*t - 

                                                                              9
          232158787821822686686268803096828213303267879649894080000000000000*b  

          - 914339994087255788035842922803409884324637299732580010200000000000

            8
          *b  - 

                                                                               7
          1456553024942306848445635398194494646048613632462079804220000000000*b

           - 1429773468085320579659912540829309032262384742022357855878580000000

            6
          *b  - 

                                                                              5
          908944691139155009098308941935669674404431611232759364790656800000*b  

          - 394123305458525780887811122985868682566594060374758630590008810000

            4
          *b  - 

                                                                              3
          114919063563435384108358931167592408356874179358918284670595993240*b  

                                                                               2
          - 22376181506466478409426169614162075694852682500804198791108921475*b

           - 2945714266609139709176973289117451707834537151497408879223183208*b

           - 127343046946408668687682889109197718306724189305639804298381200,

          23984301367968937769924256045360183486327414313970785688139264*s - 

                                                                             9
          93385077215170712211881744870071176375416361029681600000000000000*b  -

                                                                               8
           368160952680520875300826094664986085024410366966850419000000000000*b

           - 587106602751452802634914356878527850505985235023389523500000000000

            7
          *b  - 

                                                                              6
          576629986881952392513712499431359824206930128557786359524100000000*b  

          - 366874075748831567147207506029692907450037791461629910342276000000

            5
          *b  - 

                                                                              4
          159134490987396693155870310586114401358103950262784631419648850000*b  

                                                                               3
          - 46460129254430495335257974799114783858573413004692326764934039800*b

                                                                               2
           - 9081061858975251669290196016044227941007110418581855806096298095*b

           - 1222066452390803097568723620648006189979646603457892421797898376*b

           - 60999770483681527871286545331521866855137759127008037834271184,

                                10                          9
          43808000000000000000*b   + 189995300000000000000*b

                                    8                          7
           + 343169730200000000000*b  + 377900184178000000000*b

                                    6                          5
           + 277427432368460000000*b  + 141636786601439800000*b

                                   4                         3
           + 50921375336016834000*b  + 12792266529459977340*b

                                  2
           + 2215667232541084905*b  + 237653554658069880*b + 8984801833047216}

 
% The above system has dimension zero. Therefore its Hilbert polynomial
% is a constant which is the number of zero points (including complex
% zeros and multipliticities);

hilbertpolynomial ws;


10


% Example of Groebner with numerical postprocessing.

on rounded;

off varopt;


groesolve(trinkspolys,trinksvars);


{{b= - 0.397994974843*i - 0.33,

  p= - 0.685435790007*i + 0.196666666667,

  s= - 0.994987437107*i - 0.78,

  t= - 0.981720937945*i - 0.922,

  w=0.0630158710168*i - 0.0139,

  z=0.541715382425*i - 0.122333333333},

 {b=0.397994974843*i - 0.33,

  p=0.685435790007*i + 0.196666666667,

  s=0.994987437107*i - 0.78,

  t=0.981720937945*i - 0.922,

  w= - 0.0630158710168*i - 0.0139,

  z= - 0.541715382425*i - 0.122333333333}}

off rounded;



% Additional groebner operators.

% Reduce one polynomial wrt the basis of big Trinks. The result 0
% is a proof for the ideal membership of the polynomial.

torder(trinksvars,lex)$


preduce(45*p + 35*s - 165*b - 36,btbas);


0

 
% The following examples show how to work with the distributive
% form of polynomials.

torder({u0,u1,u2,u3},gradlex)$


gsplit(2*u0*u2 + u1**2 + 2*u1*u3 - u2,{u0,u1,u2,u3});


           2
{2*u0*u2,u1  + 2*u1*u3 - u2}


torder(trinksvars,lex)$


gsort trinkspolys;


                   3
{w*p + 2*z*t - 11*b ,

                    2
 99*w - 11*s*b + 3*b ,

  - 9*w + 15*p*t + 20*z*s,

                                    2
 15*w + 25*p*s + 30*z - 18*t - 165*b ,

 35*p + 40*z + 25*t - 27*s,

 45*p + 35*s - 165*b - 36,

  2    33       2673
 b  + ----*b + -------}
       50       10000

 
gspoly(first trinkspolys,second trinkspolys);


360*z + 225*t - 488*s + 1155*b + 252


gvars trinkspolys;


{w,p,z,t,s,b}


% Tagged basis and reduction trace. A tagged basis is a basis where
% each polynomial is equated to a linear combination of the input
% set. A tagged reduction shows how the result is computed by using
% the basis polynomials.

% First example for tagged polynomials: show how a polynomial is
% represented as linear combination of the basis polynomials.

  % First I set up an environment for the computation.

torder(trinksvars,lex)$



  % Then I compute an ordinary Groebner basis.

bas:=groebner trinkspolys$



  % Next I assign a tag to each basis polynomial.

taggedbas:=for i:=1:length bas collect mkid(p,i)=part(bas,i);


taggedbas := {p1=9500*b + 60000*w + 3969,

              p2= - 3100*b + 1800*p - 1377,

              p3=24500*b + 18000*z + 10287,

              p4= - 1850*b + 750*t + 81,

              p5= - 500*b + 200*s - 9,

                        2
              p6=10000*b  + 6600*b + 2673}


  % And finally I reduce a (tagged) polynomial wrt the tagged basis.

preducet(new=w*p + 2*z*t - 11*b**3,taggedbas);


                                                         3                    2
857375000000*p*w + 1714750000000*t*z + 2376000000000000*w  + 471517200000000*w

                                              2
 + 31190862780000*w + 687758524299=992750000*b *p1 - 6270000000*b*p1*w

                                                       2
 - 414760500*b*p1 + 857375000000*new + 39600000000*p1*w  + 5239080000*p1*w

 + 173282571*p1


% Second example for tagged polynomials: representing a Groebner basis 
% as a combination of the input polynomials, here in a simple geometric
% problem.

torder({x,y},lex)$


groebnert {circle=x**2 + y**2 - r**2,line=a*x + b*y};

 left 
------------------------------------------------------------------------------
 >> accum. cpu time : 0 ms 
 left 
------------------------------------------------------------------------------
 >> accum. cpu time : 0 ms 

{ - a*x - b*y= - line,

   2    2   2    2  2  2
 (a  + b )*y  - a *r =a *circle - a*line*x + b*line*y}


% In the third example I enter two polynomials that have no common zero.
% Consequently the basis is {1}. The tagged computation gives me a proof
% for the inconsistency of the system which is independent of the
% Groebner formalism.

groebnert {circle1=x**2 + y**2 - 10,circle2=x**2 + y**2 - 2};


     - circle1 + circle2
{1=----------------------}
             8


% Solve a special elimination task by using a blockwise elimination
% order defined by a matrix.  The equation set goes back to A.M.H.
% Levelt (Nijmegen).  The question is whether there is a member in the
% ideal which depends only on two variables.  Here we select x4 and y1.
% The existence of such a polynomial proves that the system has exactly
% one degree of freedom.

% The first two rows of the term order matrix define the groupwise
% elimination. The remaining lines define a secondary local
% lexicographical behavior which is needed to construct an admissible
% ordering.

f1:=y1^2 + z1^2 -1;


        2     2
f1 := y1  + z1  - 1

f2:=x2^2 + y2^2 + z2^2 -1;


        2     2     2
f2 := x2  + y2  + z2  - 1

f3:=x3^2 + y3^2 + z3^2 -1;


        2     2     2
f3 := x3  + y3  + z3  - 1

f4:=x4^2 + z4^2 -1;


        2     2
f4 := x4  + z4  - 1

f5:=y1*y2 + z1*z2;


f5 := y1*y2 + z1*z2

f6:=x2*x3 + y2*y3 + z2*z3;


f6 := x2*x3 + y2*y3 + z2*z3

f7:=x3*x4 + z3*z4;


f7 := x3*x4 + z3*z4

f8:=x2 + x3 + x4 + 1;


f8 := x2 + x3 + x4 + 1

f9:=y1 + y2 + y3 - 1;


f9 := y1 + y2 + y3 - 1

f10:=z1 + z2 + z3 + z4;


f10 := z1 + z2 + z3 + z4


eqns:={f1,f2,f3,f4,f5,f6,f7,f8,f9,f10}$


vars:={x2,x3,y2,y3,z1,z2,z3,z4,x4,y1}$



torder(vars,matrix,
mat((1,1,1,1,1,1,1,1,0,0),
    (0,0,0,0,0,0,0,0,1,1),
    (1,0,0,0,0,0,0,0,0,0),
    (0,1,0,0,0,0,0,0,0,0),
    (0,0,1,0,0,0,0,0,0,0),
    (0,0,0,1,0,0,0,0,0,0),
    (0,0,0,0,1,0,0,0,0,0),
    (0,0,0,0,0,1,0,0,0,0),
    (0,0,0,0,0,0,1,0,0,0),
    (0,0,0,0,0,0,0,0,1,0)));


{{x,y},lex}


first reverse groebner(eqns,vars);


  2   2       2                        2
x4 *y1  - 2*x4  + 2*x4*y1 - 2*x4 - 2*y1  + 2*y1


% For a faster execution we convert the matrix into a
% proper machine code routine.

on comp;


torder_compile(levelt,mat(
  (1,1,1,1,1,1,1,1,0,0),
  (0,0,0,0,0,0,0,0,1,1),
  (1,0,0,0,0,0,0,0,0,0),
  (0,1,0,0,0,0,0,0,0,0),
  (0,0,1,0,0,0,0,0,0,0),
  (0,0,0,1,0,0,0,0,0,0),
  (0,0,0,0,1,0,0,0,0,0),
  (0,0,0,0,0,1,0,0,0,0),
  (0,0,0,0,0,0,1,0,0,0),
  (0,0,0,0,0,0,0,0,1,0)));

+++ levelt compiled, 324 + 16 bytes

levelt

torder(vars,levelt)$


first reverse groebner(eqns,vars);


  2   2       2                        2
x4 *y1  - 2*x4  + 2*x4*y1 - 2*x4 - 2*y1  + 2*y1


% For a homogeneous polynomial set we compute a graded Groebner
% basis with grade limits. We use the graded term order with lex
% as following order. As the grade vector has no zeros, this ordering
% is functionally equivalent to a weighted ordering.

torder({x,y,z},graded,{1,1,2},lex);


{{x2,x3,y2,y3,z1,z2,z3,z4,x4,y1},levelt}

dd_groebner(0,10,{x^10*y + y*z^5, x*y^12 + y*z^6});


    12      6  10        5
{x*y   + y*z ,x  *y + y*z }
  
dd_groebner(0,50,{x^10*y + y*z^5, x*y^12 + y*z^6});


  7    18    34  5
{x *y*z   - y  *z ,

  8    12    23  5
 x *y*z   + y  *z ,

  9    6    12  5
 x *y*z  - y  *z ,

    12      6
 x*y   + y*z ,

  10        5
 x  *y + y*z }
  
dd_groebner(0,infinity,{x^10*y + y*z^5, x*y^12 + y*z^6});


  111  5      60
{y   *z  + y*z  ,

      54    100  5
 x*y*z   - y   *z ,

  2    48    89  5
 x *y*z   + y  *z ,

  3    42    78  5
 x *y*z   - y  *z ,

  4    36    67  5
 x *y*z   + y  *z ,

  5    30    56  5
 x *y*z   - y  *z ,

  6    24    45  5
 x *y*z   + y  *z ,

  7    18    34  5
 x *y*z   - y  *z ,

  8    12    23  5
 x *y*z   + y  *z ,

  9    6    12  5
 x *y*z  - y  *z ,

    12      6
 x*y   + y*z ,

  10        5
 x  *y + y*z }
  

% Test groebner_walk
trinkspolys := {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**2 + 33/50*b + 2673/10000}$


trinksvars :=  {w,p,z,t,s,b}$


torder(trinksvars,gradlex)$


gg:=groebner trinkspolys$


g:=groebner_walk gg$



on div$


g;


  2    33       2673
{b  + ----*b + -------,
       50       10000

  19            1323
 -----*b + w + -------,
  120           20000

     31           153
  - ----*b + p - -----,
     18           200

  49           1143
 ----*b + z + ------,
  36           2000

     37           27
  - ----*b + t + -----,
     15           250

     5            9
  - ---*b + s - -----}
     2           200


on varopt;


g1:=solve({first g},{b});


          3                 33
g1 := {b=----*sqrt(11)*i - -----,
          25                100

             3                 33
       b= - ----*sqrt(11)*i - -----}
             25                100

g0:=sub({first g1},g);


g0 := {0,

         19                      139
       ------*sqrt(11)*i + w + -------,
        1000                    10000

           31                     59
        - -----*sqrt(11)*i + p - -----,
           150                    300

        49                     367
       -----*sqrt(11)*i + z + ------,
        300                    3000

           37                     461
        - -----*sqrt(11)*i + t + -----,
           125                    500

           3                     39
        - ----*sqrt(11)*i + s + ----}
           10                    50

solve({ second g0},{w});


        19                  139
{w= - ------*sqrt(11)*i - -------}
       1000                10000

solve({third g0},{p});


    31                 59
{p=-----*sqrt(11)*i + -----}
    150                300

solve({part(g0,4)},{z});


       49                 367
{z= - -----*sqrt(11)*i - ------}
       300                3000

solve({part(g0,5)},{t});


    37                 461
{t=-----*sqrt(11)*i - -----}
    125                500

solve({part(g0,6)},{s});


    3                 39
{s=----*sqrt(11)*i - ----}
    10                50


g0:=sub({second g1},g);


g0 := {0,

            19                      139
        - ------*sqrt(11)*i + w + -------,
           1000                    10000

        31                     59
       -----*sqrt(11)*i + p - -----,
        150                    300

           49                     367
        - -----*sqrt(11)*i + z + ------,
           300                    3000

        37                     461
       -----*sqrt(11)*i + t + -----,
        125                    500

        3                     39
       ----*sqrt(11)*i + s + ----}
        10                    50

solve({second g0},{w});


     19                  139
{w=------*sqrt(11)*i - -------}
    1000                10000

solve({third g0},{p});


       31                 59
{p= - -----*sqrt(11)*i + -----}
       150                300

solve({part(g0,4)},{z});


    49                 367
{z=-----*sqrt(11)*i - ------}
    300                3000

solve({part(g0,5)},{t});


       37                 461
{t= - -----*sqrt(11)*i - -----}
       125                500

solve({part(g0,6)},{s});


       3                 39
{s= - ----*sqrt(11)*i - ----}
       10                50


% Example after the book "David Cox, John Little, Donal O'Shea:
% "Ideals, Varieties and Algorithms", chapter 2, paragraph 8, example 3.
% This example was given by Shigetoshi Katsura (Japan).
 
off groebopt;

torder({x,y,z,l},lex);


{{w,p,z,t,s,b},gradlex,1,0,0,0,0,0}

g:=groebner{3*x^2+2*y*z-2*x*l,2*x*z-2*y*l,2*x*y-2*z-2*z*l,x^2+y^2+z^2-1}$


gdimension g;


0

gindependent_sets g;


{{}}


clear g, gg, trinkspolys, trinksvars$



end;


Time for test: 510 ms, plus GC time: 10 ms