Mon Jan 4 00:02:12 MET 1999
REDUCE 3.7, 15-Jan-99 ...
1: 1:
2: 2: 2: 2: 2: 2: 2: 2: 2:
3: 3: % 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)$
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.
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});
1 1
{{u3=---,u0=---,u2=0,u1=0},
3 3
{u3=0,u0=1,u2=0,u1=0},
{u3
5 4 3 2
- 35588322*u1 + 7102080*u1 + 3462372*u1 - 522672*u1 - 98665*u1 + 11905
=-----------------------------------------------------------------------------
10987
,
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,
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;
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};
{ - 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. This step can be taken only
% if there is access to a compiler.
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): base 16574010, length 920 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
clear g, gg, trinkspolys, trinksvars$
end;
4: 4: 4: 4: 4: 4: 4: 4: 4:
Time for test: 2980 ms, plus GC time: 190 ms
5: 5:
Quitting
Mon Jan 4 00:02:39 MET 1999