REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ...
% ***** Example 1 *****
g:=invbase{4*x^2 + x*y^2 - z +1/4,
2*x + y^2*z + 1/2,
x^2*z - 1/2*x - y^2};
3 2 3 2
g := {8*x*y*z - 2*x*y*z + 4*y - 4*y*z + 16*x*y + 17*y*z - 4*y,
4 2 2 2
8*y - 8*x*z - 256*y + 2*x*z + 64*z - 96*x + 20*z - 9,
3
2*y *z + 4*x*y + y,
3 2 2 2
8*x*z - 2*x*z + 4*y - 4*z + 16*x + 17*z - 4,
3 3 2
- 4*y*z - 8*y + 6*x*y*z + y*z - 36*x*y - 8*y,
2 2 2
4*x*y + 32*y - 8*z + 12*x - 2*z + 1,
2
2*y *z + 4*x + 1,
3 2 2
- 4*z - 8*y + 6*x*z + z - 36*x - 8,
2 2 2
8*x - 16*y + 4*z - 6*x - z}
h:=invlex g;
6 5 4 3 2
h := {3976*x + 37104*z - 600*z + 2111*z + 122062*z + 232833*z - 680336*z
+ 288814,
2 6 5 4 3 2
1988*y - 76752*z + 1272*z - 4197*z - 251555*z - 481837*z + 1407741*z
- 595666,
7 6 5 4 3 2
16*z - 8*z + z + 52*z + 75*z - 342*z + 266*z - 60}
% ***** Example 2 *****
on trinvbase$
invtorder revgradlex,{x,y,z}$
g:=invbase{x^3 + y^2 + z - 3,
y^3 + z^2 + x - 3,
z^3 + x^2 + y - 3};
---------- ORDER = 3 ----------
---------- ORDER = 4 ----------
---------- ORDER = 5 ----------
---------- ORDER = 6 ----------
---------- ORDER = 7 ----------
reductions = 77 zeros = 11 maxord = 7 order = 7 length = 13
D i m e n s i o n = 0
N u m b e r o f s o l u t i o n s = 27
2 2 3 2 2 2 2 2 2 2 2 2
g := {x *y *z - 3*x *y - x*y *z - x *z + x*y*z + x *y + 3*x*y + 3*x
2
- 3*x*y + y + z - 3,
2 3 2 2 2 2 2
x *y*z + x *y - 3*x *y - x*y*z + x*z + x + 3*x*y - 3*x,
2 3 2 2 2 2 2 2
x*y *z - 3*x*y - y *z - x*z + y*z - x + x*y + 3*y + 3*x - 3*y,
2 3 2 2 2 2
x *y + x *z - 3*x - y - z + 3,
2 3 2 2 2
x *z + x *y - x*y - 3*x - x*z + 3*x,
3 2 2
x*y*z + x*y - 3*x*y - y*z + z + x + 3*y - 3,
2 3 2 2 2 2
y *z + x *y - 3*y - z - x + 3,
3 2 2
x*y + x*z + x - 3*x,
3 2
x*z + x*y - y - 3*x - z + 3,
3 2 2
y*z + x *y + y - 3*y,
3 2
x + y + z - 3,
3 2
y + z + x - 3,
3 2
z + x + y - 3}
h:=invlex g;
h := { - 412373224241856640945111992285148*x
26
- 1449641911307232269543863070491*z
25
- 2168612583844782211565651535007*z
24
- 2847785553349083352614138977565*z
23
+ 35576725674692081471990149502410*z
22
+ 54428253744724168431241789131696*z
21
+ 72399213723404842594731673129040*z
20
- 367271934803243933721304377312611*z
19
- 577412401939211224792461395441215*z
18
- 752437808233499373488146484648759*z
17
+ 2023265153056028087298524971059780*z
16
+ 3362763223678472034221124579531852*z
15
+ 4206754352383617663824252489277347*z
14
- 6294684651757967009725536832231313*z
13
- 11645937803380007452970955449190202*z
12
- 13912359441969785881761771576274650*z
11
+ 10813944944367254864931915957111635*z
10
+ 24146769890624467199683669920316403*z
9
+ 28253894162862384778437975597863994*z
8
- 9413195341759783675090699662838024*z
7
- 28732526014615244592092156992897700*z
6
- 34274801170918929476253738727746640*z
5
+ 3129736563440111416048255862484824*z
4
+ 17956474721641990844572020234799903*z
3
+ 21526113174342847360723274047268152*z
2
+ 795762450545743140366490379212137*z
- 6078501600786528783018721470971548*z
- 3909915395631179340911139035268300,
- 412373224241856640945111992285148*y
26
+ 3680069960199680647552580014011*z
25
+ 4946533576928304373640222248439*z
24
+ 6522058320833813074018729716109*z
23
- 91123955793021263648983859056246*z
22
- 122860148727246593163920662895892*z
21
- 161652285275223157884596590612424*z
20
+ 962753147411097965886678769071203*z
19
+ 1303906344577106971108666976068347*z
18
+ 1646174502798616879170351863301227*z
17
- 5539016636709239326199213127901604*z
16
- 7732787650045519336370661934943044*z
15
- 9110016144563661988538140239320223*z
14
+ 18612337918090152097453612706753413*z
13
+ 27965492180063505085033283788513066*z
12
+ 30440317356106139389125602029763822*z
11
- 36863224004805998790098755360970471*z
10
- 62542906673581589636380853858043447*z
9
- 64689461678563738668073440578715518*z
8
+ 42623160090556250860454187465583768*z
7
+ 83548043234053149543179359124170180*z
6
+ 85865493477306743665317502795142584*z
5
- 27434780477528021937653276615015928*z
4
- 61602505785524913541319871156904287*z
3
- 62515628463318116801915981996829328*z
2
+ 5925778048881538700551831705942583*z
+ 24088990130824351149845277501309728*z
+ 15820742036151533576971241715895080,
27 24 21 19 18 17 16
- z + 27*z - 317*z + 18*z + 2067*z + 50*z - 279*z
15 14 13 12 11 10
- 8156*z - 645*z + 1674*z + 20359*z + 3044*z - 4645*z
9 8 7 6 5 4 3
- 33644*z - 6288*z + 6388*z + 36936*z + 5925*z - 4957*z - 23187*z
2
- 4063*z + 4342*z + 5352}
% ***** Example 3 (limited by the degree bound) *****
invtorder revgradlex,{x,z,y,t}$
k:=5$
on errcont$
invbase{x^(k+1)-y^(k-1)*z*t,
x*z^(k-1)-y**k,
x^k*y-z^k*t};
---------- ORDER = 6 ----------
---------- ORDER = 7 ----------
---------- ORDER = 8 ----------
---------- ORDER = 9 ----------
---------- ORDER = 10 ----------
---------- ORDER = 11 ----------
---------- ORDER = 12 ----------
---------- ORDER = 13 ----------
---------- ORDER = 14 ----------
---------- ORDER = 15 ----------
---------- ORDER = 16 ----------
---------- ORDER = 17 ----------
---------- ORDER = 18 ----------
---------- ORDER = 19 ----------
---------- ORDER = 20 ----------
---------- ORDER = 21 ----------
***** Maximum degree bound exceeded.
invtempbasis;
17 2 16
{ - t*z + x *y ,
13 3 11
- t*z + x *y ,
9 4 6
- t*z + x *y ,
4 6
- t*y *z + x ,
5 5
- t*z + x *y,
4 5
x*z - y }
end$
(TIME: invbase 1020 1020)