Singular

LIB "gradedModules.lib";
//  if( defined(assumeLevel) ){ int assumeLevel0 = assumeLevel; } else { int assumeLevel; export(assumeLevel); }; assumeLevel = 5; // store the state of aL
// note: data from random generation 2
string Name = "castelnuovo"; int @p=31991; ring R = (@p),(x,y,z,u,v), dp;ideal I = 5153xy2-98/23y3-101/51xyz+33/41y2z+99/79xz2+7136yz2-106/111z3+119/53xyu+34/57y2u-77/92xzu+84/73yzu-109/78z2u-27/56xu2+10023yu2+82/103zu2-34/25u3+3/2xyv-68/25y2v+12721xzv+4/63yzv-73/21z2v-7291xuv-91/53yuv-4/79zuv-34/91u2v-122/53xv2+123/70yv2-64/73zv2+44/65uv2+14/31v3,xy2-15202y3+10613xyz+13640y2z-107/103xz2+5292yz2+19/119z3-10042xyu+2770y2u+7957xzu+14008yzu+92/121z2u-92/51xu2+1178yu2+1/117zu2-12726u3+82/101xyv-92/17y2v-107/56xzv+14233yzv+79/28z2v+51/50xuv-31/5yuv+95/91zuv+19/108u2v+12151xv2-69/110yv2+37/89zv2-63/116uv2-88/23v3,-5153x2+37/23xy+8706y2-13160xz+68/115yz+5548z2-22/61xu-113/98yu+11818zu+2114u2-101/97xv+89/22yv-3355zv-113/5uv-5521v2;TestGRRes(Name, I); kill R, Name, @p;  "";
==> ==============================================
==> 
==> === Example: [ castelnuovo ]
==>  = Ring:  (ZZ/31991),(x,y,z,u,v),(dp(5),C,L(1048575))
==>  ! Resolution via 'grres': 
==>  = Non-minimal betti numbers: 
==>            0     1     2
==> ------------------------
==>     0:     1     -     -
==>     1:     -     1     -
==>     2:     -     2     2
==> ------------------------
==> total:     1     3     2
==> 
==>  = Degrees of (ordered) maps: 
==> Graded resolution: 
==> R <-- d_1 --
==> R(-2) + R(-3)^2 <-- d_2 --
==> R(-4)^2, given by maps: 
==> d_1 :
==> Graded homomorphism: R <- R(-2) + R(-3)^2, given by a matrix, with degree\
   s: 
==>     .1 .2 .3 ...
==>     -- -- -- +..
==>  0 : 2  3  3 |.1
==>     == == ==    
==>      2  3  3    
==> d_2 :
==> Graded homomorphism: R(-2) + R(-3)^2 <- R(-4)^2, given by a matrix, with \
   degrees: 
==>     .1 .2 ...
==>     -- -- +..
==>  2 : 2  2 |.1
==>  3 : 1  1 |.2
==>  3 : 1  1 |.3
==>     == ==    
==>      4  4    
==>  = TRANSPOSE'd complex: %%%%%%%%%%%%%%
==>            0     1     2
==> ------------------------
==>    -4:     2     2     -
==>    -3:     -     1     -
==>    -2:     -     -     1
==> ------------------------
==> total:     2     3     1
==> 
==> Graded resolution: 
==> R(4)^2 <-- d_1 --
==> R(3)^2 + R(2) <-- d_2 --
==> R, given by maps: 
==> d_1 :
==> Graded homomorphism: R(4)^2 <- R(3)^2 + R(2), given by a matrix, with deg\
   rees: 
==>      ..1 ..2 ..3 ....
==>      --- --- --- +...
==>  -4 :  1   1   2 |..1
==>  -4 :  1   1   2 |..2
==>      === === ===     
==>       -3  -3  -2     
==> d_2 :
==> Graded homomorphism: R(3)^2 + R(2) <- R, given by a matrix, with degrees: 
==>      ..1 ....
==>      --- +...
==>  -3 :  3 |..1
==>  -3 :  3 |..2
==>  -2 :  2 |..3
==>      ===     
==>        0     
==> 
==> 
string Name = "ell.d8.g7"; int @p=31991; ring R = (@p),(x,y,z,u,v), dp;ideal I = x2y2-47/69xy3+6059y4+78/85x2yz+55/124xy2z+13641y3z+8/17x2z2+7817xyz2-2746y2z2+85/124xz3+87yz3+13182z4+109/93x2yu-69/17xy2u+12089y3u+8769x2zu-53/36xyzu-14834y2zu+123/23xz2u+103/77yz2u-2344z3u-43/104x2u2-6198xyu2+47/115y2u2-39/19xzu2-29/24yzu2+51/89z2u2-65/37xu3-95/94yu3+11302zu3-53/57u4-2874x2yv+4347xy2v-25/77y3v+13819x2zv+29/34xyzv+474y2zv+33/107xz2v-3517yz2v+10617z3v+1834x2uv+54/113xyuv-8751y2uv+111/70xzuv-66/61yzuv+9195z2uv-14289xu2v-13/110yu2v+103/9zu2v+5113u3v+116/89x2v2+15142xyv2+13078y2v2-38/41xzv2-13/113yzv2-12824z2v2-57/11xuv2-114/17yuv2-125/31zuv2+11939u2v2+44/13xv3+56/69yv3+12/125zv3+643uv3+3530v4,-3454x2y-1285xy2-6182y3-8/69x2z+9/19xyz+64/49y2z+98/67xz2-13809yz2+21/44z3+77/47x2u+748xyu-41/77y2u+7318xzu+4217yzu+12562z2u-98/69xu2-14/85yu2+119/46zu2-61/121u3+5582x2v+108/77xyv-93/4y2v-65/49xzv-4135yzv+2477z2v+11114xuv+85/14yuv+51/125zuv-7572u2v-115/52xv2-7647yv2+4647zv2-5684uv2-1/55v3,3454x3-6645x2y-43/34xy2+14590y3+8/11x2z-117/112xyz+109/54y2z+6566xz2+23/57yz2-13078z3+95/61x2u+67/40xyu-4544y2u-95/72xzu-8/103yzu+100/77z2u+23/63xu2+69/61yu2-94/105zu2+8619u3+68/123x2v+8/117xyv+101/77y2v+124/125xzv+17/84yzv+23/67z2v+18/59xuv+3216yuv-77/59zuv-9/50u2v+96/109xv2-2491yv2+14089zv2+14067uv2-56/113v3;TestGRRes(Name, I); kill R, Name, @p;  "";
==> ==============================================
==> 
==> === Example: [ ell.d8.g7 ]
==>  = Ring:  (ZZ/31991),(x,y,z,u,v),(dp(5),C,L(1048575))
==>  ! Resolution via 'grres': 
==>  = Non-minimal betti numbers: 
==>            0     1     2
==> ------------------------
==>     0:     1     -     -
==>     1:     -     -     -
==>     2:     -     2     -
==>     3:     -     1     2
==> ------------------------
==> total:     1     3     2
==> 
==>  = Degrees of (ordered) maps: 
==> Graded resolution: 
==> R <-- d_1 --
==> R(-3)^2 + R(-4) <-- d_2 --
==> R(-5)^2, given by maps: 
==> d_1 :
==> Graded homomorphism: R <- R(-3)^2 + R(-4), given by a matrix, with degree\
   s: 
==>     .1 .2 .3 ...
==>     -- -- -- +..
==>  0 : 3  3  4 |.1
==>     == == ==    
==>      3  3  4    
==> d_2 :
==> Graded homomorphism: R(-3)^2 + R(-4) <- R(-5)^2, given by a matrix, with \
   degrees: 
==>     .1 .2 ...
==>     -- -- +..
==>  3 : 2  2 |.1
==>  3 : 2  2 |.2
==>  4 : 1  1 |.3
==>     == ==    
==>      5  5    
==>  = TRANSPOSE'd complex: %%%%%%%%%%%%%%
==>            0     1     2
==> ------------------------
==>    -5:     2     1     -
==>    -4:     -     2     -
==>    -3:     -     -     -
==>    -2:     -     -     1
==> ------------------------
==> total:     2     3     1
==> 
==> Graded resolution: 
==> R(5)^2 <-- d_1 --
==> R(4) + R(3)^2 <-- d_2 --
==> R, given by maps: 
==> d_1 :
==> Graded homomorphism: R(5)^2 <- R(4) + R(3)^2, given by a matrix, with deg\
   rees: 
==>      ..1 ..2 ..3 ....
==>      --- --- --- +...
==>  -5 :  1   2   2 |..1
==>  -5 :  1   2   2 |..2
==>      === === ===     
==>       -4  -3  -3     
==> d_2 :
==> Graded homomorphism: R(4) + R(3)^2 <- R, given by a matrix, with degrees: 
==>      ..1 ....
==>      --- +...
==>  -4 :  4 |..1
==>  -3 :  3 |..2
==>  -3 :  3 |..3
==>      ===     
==>        0     
==> 
==> 
string Name = "ell.d7.g6"; int @p=31991; ring R = (@p),(x,y,z,u,v), dp;ideal I = 4971xy3+3/101y4-12318xy2z-12835y3z+97/98xyz2+63y2z2-8056xz3+23/91yz3-9662z4-7398xy2u+69/71y3u-53/68xyzu-49/67y2zu-113/122xz2u-9/61yz2u+71/88z3u+11358xyu2-38/29y2u2-10232xzu2+14490yzu2+2274z2u2+3501xu3+10427yu3-109/38zu3-99/5u4-6605xy2v-1555y3v-648xyzv-2083y2zv-61/41xz2v+75/17yz2v-69/55z3v-6104xyuv-9582y2uv+69/2xzuv-12551yzuv+47/49z2uv-118/13xu2v+34/105yu2v+105/41zu2v+6533u3v+122/25xyv2+2/43y2v2+16/61xzv2+11524yzv2+113/99z2v2-71/26xuv2+7809yuv2-4865zuv2-2122u2v2+53/118xv3-13209yv3-11106zv3-49/79uv3+3006v4,xy3+15492y4-13742xy2z+112/117y3z+6/47xyz2+28/41y2z2+71/111xz3+49/57yz3-61/44z4-11759xy2u+4242y3u-109/18xyzu+2260y2zu-6873xz2u-41/112yz2u+12574z3u-10939xyu2+119/38y2u2-62/33xzu2-3699yzu2+2651z2u2-13194xu3-15185yu3-11/116zu3-61/83u4-10094xy2v+13/4y3v-74/73xyzv+43/20y2zv-11547xz2v+53/43yz2v-92/93z3v+32/41xyuv+118/33y2uv-121/39xzuv-15913yzuv+53/11z2uv+97/76xu2v+85/29yu2v-5183zu2v+8520u3v+121/28xyv2+64/51y2v2-15810xzv2+1/43yzv2-6160z2v2+13988xuv2+9/40yuv2+123/4zuv2+15024u2v2+73/95xv3+80/97yv3+57/25zv3-109/81uv3-121/87v4,-4971x2+14389xy+1607y2+59/119xz+12020yz+103/122z2+8894xu+7091yu+54/19zu-50/77u2+28/25xv-113/56yv+68/29zv-14620uv+79/107v2;TestGRRes(Name, I); kill R, Name, @p;  "";
==> ==============================================
==> 
==> === Example: [ ell.d7.g6 ]
==>  = Ring:  (ZZ/31991),(x,y,z,u,v),(dp(5),C,L(1048575))
==>  ! Resolution via 'grres': 
==>  = Non-minimal betti numbers: 
==>            0     1     2
==> ------------------------
==>     0:     1     -     -
==>     1:     -     1     -
==>     2:     -     -     -
==>     3:     -     2     2
==> ------------------------
==> total:     1     3     2
==> 
==>  = Degrees of (ordered) maps: 
==> Graded resolution: 
==> R <-- d_1 --
==> R(-2) + R(-4)^2 <-- d_2 --
==> R(-5)^2, given by maps: 
==> d_1 :
==> Graded homomorphism: R <- R(-2) + R(-4)^2, given by a matrix, with degree\
   s: 
==>     .1 .2 .3 ...
==>     -- -- -- +..
==>  0 : 2  4  4 |.1
==>     == == ==    
==>      2  4  4    
==> d_2 :
==> Graded homomorphism: R(-2) + R(-4)^2 <- R(-5)^2, given by a matrix, with \
   degrees: 
==>     .1 .2 ...
==>     -- -- +..
==>  2 : 3  3 |.1
==>  4 : 1  1 |.2
==>  4 : 1  1 |.3
==>     == ==    
==>      5  5    
==>  = TRANSPOSE'd complex: %%%%%%%%%%%%%%
==>            0     1     2
==> ------------------------
==>    -5:     2     2     -
==>    -4:     -     -     -
==>    -3:     -     1     -
==>    -2:     -     -     1
==> ------------------------
==> total:     2     3     1
==> 
==> Graded resolution: 
==> R(5)^2 <-- d_1 --
==> R(4)^2 + R(2) <-- d_2 --
==> R, given by maps: 
==> d_1 :
==> Graded homomorphism: R(5)^2 <- R(4)^2 + R(2), given by a matrix, with deg\
   rees: 
==>      ..1 ..2 ..3 ....
==>      --- --- --- +...
==>  -5 :  1   1   3 |..1
==>  -5 :  1   1   3 |..2
==>      === === ===     
==>       -4  -4  -2     
==> d_2 :
==> Graded homomorphism: R(4)^2 + R(2) <- R, given by a matrix, with degrees: 
==>      ..1 ....
==>      --- +...
==>  -4 :  4 |..1
==>  -4 :  4 |..2
==>  -2 :  2 |..3
==>      ===     
==>        0     
==> 
==> 
string Name = "k3.d7.g5"; int @p=31991; ring R = (@p),(x,y,z,u,v), dp;ideal I = -97/108x2y-31/118xy2-73/61y3-79/14x2z-15930xyz-2324y2z+1842xz2+656yz2-8852z3-89/38x2u-102/43xyu+14719y2u+70/67xzu+7335yzu+27/56z2u-10744xu2-55/83yu2+120/73zu2+120/61u3-126/125x2v+691xyv-15385y2v+117/16xzv-17/97yzv+80/121z2v-48/119xuv+21/34yuv-103/65zuv-49/32u2v-41/42xv2+11/75yv2-502zv2-7583uv2+26/69v3,97/108x3+77/114x2y+71/21xy2+13679y3-1645x2z-1/33xyz-79/7y2z-52/53xz2+11940yz2-5800z3+109/13x2u-115/64xyu-125/56y2u-2365xzu+2103yzu+56/87z2u-84/79xu2+107/106yu2-79/70zu2-419u3+5354x2v+92/53xyv-32/19y2v+11/74xzv+4193yzv+45/79z2v-113/72xuv+17/71yuv+11164zuv-17/33u2v+103/66xv2+55/79yv2+118/15zv2-2646uv2+57/106v3,x3-61/113x2y-64/21xy2-107/8y3-13/60x2z+43/35xyz+41/114y2z-13683xz2-5829yz2+71/38z3+90/17x2u-39/29xyu+42/5y2u-61/55xzu+111/77yzu-87/100z2u+10735xu2-83/91yu2-4884zu2-7965u3-65/12x2v+109/86xyv+10606y2v-14164xzv-6678yzv+83/18z2v-93/10xuv+120/49yuv-1592zuv-8710u2v-73/57xv2+10762yv2-2956zv2-89/63uv2-12/7v3;TestGRRes(Name, I); kill R, Name, @p;  "";
==> ==============================================
==> 
==> === Example: [ k3.d7.g5 ]
==>  = Ring:  (ZZ/31991),(x,y,z,u,v),(dp(5),C,L(1048575))
==>  ! Resolution via 'grres': 
==>  = Non-minimal betti numbers: 
==>            0     1     2
==> ------------------------
==>     0:     1     -     -
==>     1:     -     -     -
==>     2:     -     3     1
==>     3:     -     -     1
==> ------------------------
==> total:     1     3     2
==> 
==>  = Degrees of (ordered) maps: 
==> Graded resolution: 
==> R <-- d_1 --
==> R(-3)^3 <-- d_2 --
==> R(-4) + R(-5), given by maps: 
==> d_1 :
==> Graded homomorphism: R <- R(-3)^3, given by a matrix, with degrees: 
==>     .1 .2 .3 ...
==>     -- -- -- +..
==>  0 : 3  3  3 |.1
==>     == == ==    
==>      3  3  3    
==> d_2 :
==> Graded homomorphism: R(-3)^3 <- R(-4) + R(-5), given by a matrix, with de\
   grees: 
==>     .1 .2 ...
==>     -- -- +..
==>  3 : 1  2 |.1
==>  3 : 1  2 |.2
==>  3 : 1  2 |.3
==>     == ==    
==>      4  5    
==>  = TRANSPOSE'd complex: %%%%%%%%%%%%%%
==>            0     1     2
==> ------------------------
==>    -5:     1     -     -
==>    -4:     1     3     -
==>    -3:     -     -     -
==>    -2:     -     -     1
==> ------------------------
==> total:     2     3     1
==> 
==> Graded resolution: 
==> R(5) + R(4) <-- d_1 --
==> R(3)^3 <-- d_2 --
==> R, given by maps: 
==> d_1 :
==> Graded homomorphism: R(5) + R(4) <- R(3)^3, given by a matrix, with degre\
   es: 
==>      ..1 ..2 ..3 ....
==>      --- --- --- +...
==>  -5 :  2   2   2 |..1
==>  -4 :  1   1   1 |..2
==>      === === ===     
==>       -3  -3  -3     
==> d_2 :
==> Graded homomorphism: R(3)^3 <- R, given by a matrix, with degrees: 
==>      ..1 ....
==>      --- +...
==>  -3 :  3 |..1
==>  -3 :  3 |..2
==>  -3 :  3 |..3
==>      ===     
==>        0     
==> 
==> 
string Name = "rat.d8.g6"; int @p=31991; ring R = (@p),(x,y,z,u,v), dp;ideal I = -19/125x2y2-87/119xy3-97/21y4+36/53x2yz+2069xy2z-59/50y3z-65/33x2z2-14322xyz2+79/60y2z2-9035xz3-14890yz3+87/47z4-23/48x2yu+45/44xy2u+1972y3u+79/118x2zu-5173xyzu+115/121y2zu+1239xz2u-115/17yz2u-15900z3u-78/95x2u2+67/101xyu2-12757y2u2+12752xzu2+68/21yzu2+103/90z2u2-12917xu3+97/92yu3-24/49zu3-13/79u4-51/61x2yv-3103xy2v+77/117y3v+73/115x2zv-79/33xyzv+123/110y2zv+11969xz2v-31/95yz2v-123/95z3v-105/124x2uv+12624xyuv+2/63y2uv+6579xzuv+13/62yzuv+4388z2uv-12747xu2v-26/105yu2v-78/61zu2v-125/53u3v-5/71xyv2+62/77y2v2+21/44xzv2-9806yzv2+3/91z2v2+361xuv2+568yuv2+2926zuv2+53/38u2v2-14523yv3+2082zv3+113/115uv3,108/73x2y2+4028xy3+38/43y4-1944x2yz+39/80xy2z+8/109y3z+52/27x2z2+103/45xyz2+5834y2z2+63/101xz3+107/80yz3+1178z4-1/6x2yu+78/25xy2u-21/43y3u+50/71x2zu-14693xyzu+15074y2zu+9/103xz2u-7396yz2u-14493z3u+93/25x2u2+61/4xyu2-11306y2u2-79/81xzu2+59/82yzu2-5/106z2u2+89/71xu3-34/11yu3+15/103zu3-115/52u4-54/65x2yv+67/16xy2v-7/68y3v-10/13x2zv+32/85xyzv+1/91y2zv+107/118xz2v+7594yz2v-98/103z3v+9919x2uv-965xyuv+53/34y2uv+119/11xzuv-3400yzuv-8329z2uv+75/98xu2v-24yu2v+55/87zu2v-82/71u3v-73/115x2v2+85/19xyv2-213y2v2-7704xzv2-15347yzv2+14960z2v2+15065xuv2-125/17yuv2+32/83zuv2-14/73u2v2-21/44xv3+79/2yv3-61/32zv3+46/119uv3-2082v4,9/20x2y2+113/71xy3-88/65y4+9983x2yz-6722xy2z+87/68y3z+1893x2z2+65/32xyz2+51/55y2z2-102/53xz3+58/5yz3-7187z4-96/7x2yu-14/87xy2u-3532y3u+95/54x2zu+19/65xyzu-6728y2zu+31/121xz2u+73/106yz2u-91/5z3u-12928x2u2+707xyu2-55/48y2u2-96/25xzu2+15869yzu2-20/107z2u2-10030xu3-13786yu3-122/9zu3+19/59u4-7/52x2yv+101/74xy2v+83/6y3v-91/55x2zv-5266xyzv+85/61y2zv+126/95xz2v+56/51yz2v+13073z3v-50/21x2uv-13553xyuv-116/53y2uv+68/71xzuv-111/98yzuv-11037z2uv+68/121xu2v-124/53yu2v+54/55zu2v+5862u3v+12318x2v2-119/29xyv2+101/17y2v2-51/40xzv2-82/33yzv2-30/41z2v2-29/52xuv2+7817yuv2+8121zuv2-28/99u2v2+1125xv3-73/55yv3-14141zv3+8742uv3-1203v4,x2y2+11357xy3+295y4+144x2yz-31/54xy2z+89/119y3z+1/46x2z2+29/26xyz2+1384y2z2+1461xz3+113/91yz3+9494z4-7/32x2yu+12850xy2u-3626y3u-33/106x2zu-7/60xyzu-5935y2zu-8597xz2u+5527yz2u+1708z3u+6182x2u2-15780xyu2+4669y2u2-38/69xzu2+8412yzu2+9265z2u2-5679xu3-67/18yu3-34/67zu3-7178u4+113/56x2yv-3669xy2v+17/113y3v-87/35x2zv-4871xyzv-111/11y2zv-1131xz2v-72/13yz2v+838z3v-115/4x2uv+3395xyuv-43/68y2uv-82/13xzuv+7042yzuv-88/119z2uv+100/19xu2v+24/11yu2v+89/3zu2v+7395u3v-119/109x2v2+1/104xyv2+18/25y2v2+700xzv2-59/9yzv2-92/87z2v2+2486xuv2-67/103yuv2+1469zuv2-101/91u2v2-79/33xv3+10838yv3+81/4zv3-11843uv3+7204v4,19/125x3-15698x2y-22/117xy2-95/107y3+2027x2z-7750xyz+85/104y2z-15326xz2+31/101yz2+67/81z3-7879x2u-112/115xyu+124/81y2u+99/61xzu-7458yzu+40/33z2u-1502xu2+6591yu2-7/73zu2-42/95u3+93/83x2v-15/112xyv-84/95y2v+35/36xzv+5/24yzv-12768z2v+13232xuv-76/103yuv-79/52zuv-7217u2v+75/92xv2-49/64yv2+17/14zv2-6109uv2+1695v3;TestGRRes(Name, I); kill R, Name, @p; "";
==> ==============================================
==> 
==> === Example: [ rat.d8.g6 ]
==>  = Ring:  (ZZ/31991),(x,y,z,u,v),(dp(5),C,L(1048575))
==>  ! Resolution via 'grres': 
==>  = Non-minimal betti numbers: 
==>            0     1     2     3
==> ------------------------------
==>     0:     1     -     -     -
==>     1:     -     -     -     -
==>     2:     -     1     -     -
==>     3:     -     4     5     1
==> ------------------------------
==> total:     1     5     5     1
==> 
==>  = Degrees of (ordered) maps: 
==> Graded resolution: 
==> R <-- d_1 --
==> R(-3) + R(-4)^4 <-- d_2 --
==> R(-5)^5 <-- d_3 --
==> R(-6), given by maps: 
==> d_1 :
==> Graded homomorphism: R <- R(-3) + R(-4)^4, given by a matrix, with degree\
   s: 
==>     .1 .2 .3 .4 .5 ...
==>     -- -- -- -- -- +..
==>  0 : 3  4  4  4  4 |.1
==>     == == == == ==    
==>      3  4  4  4  4    
==> d_2 :
==> Graded homomorphism: R(-3) + R(-4)^4 <- R(-5)^5, given by a square matrix\
   , with degrees: 
==>      ..1 ..2 ..3 ..4 ..5 ....
==>      --- --- --- --- --- +...
==>   3 :  -   2   2   2   2 |..1
==>   4 :  1   1   1   1   1 |..2
==>   4 :  1   1   1   1   1 |..3
==>   4 :  1   1   1   1   1 |..4
==>   4 :  1   1   1   1   1 |..5
==>      === === === === ===     
==>        5   5   5   5   5     
==> d_3 :
==> Graded homomorphism: R(-5)^5 <- R(-6), given by a matrix, with degrees: 
==>     .1 ...
==>     -- +..
==>  5 : 1 |.1
==>  5 : 1 |.2
==>  5 : 1 |.3
==>  5 : 1 |.4
==>  5 : 1 |.5
==>     ==    
==>      6    
==>  = TRANSPOSE'd complex: %%%%%%%%%%%%%%
==>            0     1     2     3
==> ------------------------------
==>    -6:     1     5     4     -
==>    -5:     -     -     1     -
==>    -4:     -     -     -     -
==>    -3:     -     -     -     1
==> ------------------------------
==> total:     1     5     5     1
==> 
==> Graded resolution: 
==> R(6) <-- d_1 --
==> R(5)^5 <-- d_2 --
==> R(4)^4 + R(3) <-- d_3 --
==> R, given by maps: 
==> d_1 :
==> Graded homomorphism: R(6) <- R(5)^5, given by a matrix, with degrees: 
==>      ..1 ..2 ..3 ..4 ..5 ....
==>      --- --- --- --- --- +...
==>  -6 :  1   1   1   1   1 |..1
==>      === === === === ===     
==>       -5  -5  -5  -5  -5     
==> d_2 :
==> Graded homomorphism: R(5)^5 <- R(4)^4 + R(3), given by a square matrix, w\
   ith degrees: 
==>      ..1 ..2 ..3 ..4 ..5 ....
==>      --- --- --- --- --- +...
==>  -5 :  1   1   1   1   - |..1
==>  -5 :  1   1   1   1   2 |..2
==>  -5 :  1   1   1   1   2 |..3
==>  -5 :  1   1   1   1   2 |..4
==>  -5 :  1   1   1   1   2 |..5
==>      === === === === ===     
==>       -4  -4  -4  -4  -3     
==> d_3 :
==> Graded homomorphism: R(4)^4 + R(3) <- R, given by a matrix, with degrees: 
==>      ..1 ....
==>      --- +...
==>  -4 :  4 |..1
==>  -4 :  4 |..2
==>  -4 :  4 |..3
==>  -4 :  4 |..4
==>  -3 :  3 |..5
==>      ===     
==>        0     
==> 
==> 
string Name = "k3.d14.g19"; int @p=31991; ring R = (@p),(x,y,z,u,v), dp;ideal I = 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I); kill R, Name, @p;  "";
==> ==============================================
==> 
==> === Example: [ k3.d14.g19 ]
==>  = Ring:  (ZZ/31991),(x,y,z,u,v),(dp(5),C,L(1048575))
==>  ! Resolution via 'grres': 
==>  = Non-minimal betti numbers: 
==>            0     1     2     3
==> ------------------------------
==>     0:     1     -     -     -
==>     1:     -     -     -     -
==>     2:     -     -     -     -
==>     3:     -     -     -     -
==>     4:     -     4     2     -
==>     5:     -     4     8     3
==> ------------------------------
==> total:     1     8    10     3
==> 
==>  = Degrees of (ordered) maps: 
==> Graded resolution: 
==> R <-- d_1 --
==> R(-5)^4 + R(-6)^4 <-- d_2 --
==> R(-6)^2 + R(-7)^8 <-- d_3 --
==> R(-8)^3, given by maps: 
==> d_1 :
==> Graded homomorphism: R <- R(-5)^4 + R(-6)^4, given by a matrix, with degr\
   ees: 
==>     .1 .2 .3 .4 .5 .6 .7 .8 ...
==>     -- -- -- -- -- -- -- -- +..
==>  0 : 5  5  5  5  6  6  6  6 |.1
==>     == == == == == == == ==    
==>      5  5  5  5  6  6  6  6    
==> d_2 :
==> Graded homomorphism: R(-5)^4 + R(-6)^4 <- R(-6)^2 + R(-7)^8, given by a m\
   atrix, with degrees: 
==>      ..1 ..2 ..3 ..4 ..5 ..6 ..7 ..8 ..9 .10 ....
==>      --- --- --- --- --- --- --- --- --- --- +...
==>   5 :  1   1   2   2   2   2   2   2   2   2 |..1
==>   5 :  1   1   2   2   2   2   2   2   2   2 |..2
==>   5 :  1   1   2   2   2   2   2   2   2   2 |..3
==>   5 :  1   1   2   2   2   2   2   2   2   2 |..4
==>   6 :  -   -   1   1   1   1   1   1   1   1 |..5
==>   6 :  -   -   1   1   1   1   1   1   1   1 |..6
==>   6 :  -   -   1   1   1   1   1   1   1   1 |..7
==>   6 :  -   -   1   1   1   1   1   1   1   1 |..8
==>      === === === === === === === === === ===     
==>        6   6   7   7   7   7   7   7   7   7     
==> d_3 :
==> Graded homomorphism: R(-6)^2 + R(-7)^8 <- R(-8)^3, given by a matrix, wit\
   h degrees: 
==>      ..1 ..2 ..3 ....
==>      --- --- --- +...
==>   6 :  2   2   2 |..1
==>   6 :  2   2   2 |..2
==>   7 :  1   1   1 |..3
==>   7 :  1   1   1 |..4
==>   7 :  1   1   1 |..5
==>   7 :  1   1   1 |..6
==>   7 :  1   1   1 |..7
==>   7 :  1   1   1 |..8
==>   7 :  1   1   1 |..9
==>   7 :  1   1   1 |.10
==>      === === ===     
==>        8   8   8     
==>  = TRANSPOSE'd complex: %%%%%%%%%%%%%%
==>            0     1     2     3
==> ------------------------------
==>    -8:     3     8     4     -
==>    -7:     -     2     4     -
==>    -6:     -     -     -     -
==>    -5:     -     -     -     -
==>    -4:     -     -     -     -
==>    -3:     -     -     -     1
==> ------------------------------
==> total:     3    10     8     1
==> 
==> Graded resolution: 
==> R(8)^3 <-- d_1 --
==> R(7)^8 + R(6)^2 <-- d_2 --
==> R(6)^4 + R(5)^4 <-- d_3 --
==> R, given by maps: 
==> d_1 :
==> Graded homomorphism: R(8)^3 <- R(7)^8 + R(6)^2, given by a matrix, with d\
   egrees: 
==>      ..1 ..2 ..3 ..4 ..5 ..6 ..7 ..8 ..9 .10 ....
==>      --- --- --- --- --- --- --- --- --- --- +...
==>  -8 :  1   1   1   1   1   1   1   1   2   2 |..1
==>  -8 :  1   1   1   1   1   1   1   1   2   2 |..2
==>  -8 :  1   1   1   1   1   1   1   1   2   2 |..3
==>      === === === === === === === === === ===     
==>       -7  -7  -7  -7  -7  -7  -7  -7  -6  -6     
==> d_2 :
==> Graded homomorphism: R(7)^8 + R(6)^2 <- R(6)^4 + R(5)^4, given by a matri\
   x, with degrees: 
==>      ..1 ..2 ..3 ..4 ..5 ..6 ..7 ..8 ....
==>      --- --- --- --- --- --- --- --- +...
==>  -7 :  1   1   1   1   2   2   2   2 |..1
==>  -7 :  1   1   1   1   2   2   2   2 |..2
==>  -7 :  1   1   1   1   2   2   2   2 |..3
==>  -7 :  1   1   1   1   2   2   2   2 |..4
==>  -7 :  1   1   1   1   2   2   2   2 |..5
==>  -7 :  1   1   1   1   2   2   2   2 |..6
==>  -7 :  1   1   1   1   2   2   2   2 |..7
==>  -7 :  1   1   1   1   2   2   2   2 |..8
==>  -6 :  -   -   -   -   1   1   1   1 |..9
==>  -6 :  -   -   -   -   1   1   1   1 |.10
==>      === === === === === === === ===     
==>       -6  -6  -6  -6  -5  -5  -5  -5     
==> d_3 :
==> Graded homomorphism: R(6)^4 + R(5)^4 <- R, given by a matrix, with degree\
   s: 
==>      ..1 ....
==>      --- +...
==>  -6 :  6 |..1
==>  -6 :  6 |..2
==>  -6 :  6 |..3
==>  -6 :  6 |..4
==>  -5 :  5 |..5
==>  -5 :  5 |..6
==>  -5 :  5 |..7
==>  -5 :  5 |..8
==>      ===     
==>        0     
==> 
==> 
string Name = "k3.d11.g11.ss0"; int @p=31991; ring R = (@p),(x,y,z,u,v), dp;ideal I = 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y2z-5297xy3z+53/14y4z-17/122x3z2+5470x2yz2+14889xy2z2-123/49y3z2+131x2z3+13/43xyz3-82/37y2z3-106/91xz4+11066yz4-72/13z5+15904x4u+44/47x3yu+10450x2y2u+86/41xy3u-40/41y4u+17/92x3zu-81/29x2yzu-40/101xy2zu+103/59y3zu-29/72x2z2u+17/123xyz2u+38/3y2z2u+3358xz3u-69/97yz3u+7627z4u+15451x3u2+5371x2yu2-9009xy2u2-62/35y3u2-44/75x2zu2+13188xyzu2+50/51y2zu2+53/93xz2u2-6796yz2u2-13/6z3u2+39/62x2u3-22/71xyu3-32/97y2u3+43/14xzu3-9/37yzu3-48/125z2u3+100/103xu4+13286yu4+6246zu4-26/107u5-15634x4v+112/9x3yv+54/109x2y2v+23/93xy3v+101/13y4v+79/84x3zv+5290x2yzv-52/29xy2zv-109/111y3zv+107/88x2z2v+10186xyz2v+122/97y2z2v-8370xz3v-108/47yz3v+6311z4v-1126x3uv+51/10x2yuv+9201xy2uv-111/119y3uv+105/58x2zuv-74/23xyzuv-130y2zuv-99/28xz2uv+24/65yz2uv-111/49z3uv-4961x2u2v-23/42xyu2v-32/101y2u2v+48/83xzu2v-13001yzu2v-31/86z2u2v+85/91xu3v+4019yu3v-10/43zu3v+11/105u4v+9790x3v2+73/78x2yv2+63/50xy2v2+35/2y3v2-52/119x2zv2+13680xyzv2-46/55y2zv2+89/12xz2v2-23/82yz2v2-107/85z3v2-20/79x2uv2-85/111xyuv2+31/95y2uv2-13xzuv2+5661yzuv2+9399z2uv2+73/75xu2v2+4782yu2v2-5440zu2v2+45/64u3v2-1484x2v3+1/54xyv3+43/115y2v3+139xzv3-47/85yzv3-103/43z2v3+48/97xuv3+67/15yuv3-69/95zuv3-67/90u2v3+6540xv4+6276yv4-9756zv4-10/121uv4+118/63v5,x5+13/23x4y-10277x3y2-81/107x2y3+18/97xy4+193x4z-206x3yz+1473x2y2z-3907xy3z+9620y4z-11603x3z2+7./examples/TestGRRes.sing  11> 430x2yz2-3745xy2z2-23/55y3z2-31/13x2z3-115/51xyz3+54/5y2z3+107/6xz4+8432yz4-23/71z5-21/79x4u-7130x3yu-55/46x2y2u+61/113xy3u+11/95y4u-31/125x3zu+47/80x2yzu+12/79xy2zu-51/112y3zu-28/75x2z2u+1962xyz2u-12942y2z2u-93/37xz3u-7/9yz3u+81/7z4u-79/62x3u2+98/29x2yu2-113/36xy2u2-59/51y3u2-5931x2zu2+8/29xyzu2-22/117y2zu2-12146xz2u2+12607yz2u2-8748z3u2-11878x2u3-6xyu3-11798y2u3+97/111xzu3+122/3yzu3+10228z2u3-99/67xu4-8058yu4+116/43zu4-6801u5-7565x4v-21/58x3yv-23/90x2y2v-111/49xy3v+62/3y4v-118/109x3zv-91/27x2yzv-2256xy2zv-6909y3zv+47/126x2z2v+397xyz2v-65/103y2z2v+106/43xz3v-43/9yz3v+4502z4v+5383x3uv-19/48x2yuv+91/92xy2uv-15545y3uv+11204x2zuv+7609xyzuv-125/44y2zuv+15/7xz2uv-4157yz2uv+27/10z3uv+70/67x2u2v+5/121xyu2v-12337y2u2v-12417xzu2v+46/113yzu2v+27/89z2u2v-9419xu3v-59/47yu3v+52/97zu3v+71/56u4v+56/123x3v2+122/73x2yv2+100/103xy2v2+43/8y3v2+99/98x2zv2-14242xyzv2-93/38y2zv2+61/109xz2v2-21/125yz2v2+80/23z3v2+11603x2uv2-1734xyuv2-55/73y2uv2+5/89xzuv2+77/100yzuv2-71/38z2uv2+41/47xu2v2+981yu2v2+11/41zu2v2-8983u3v2+98/59x2v3+105/97xyv3-6372y2v3+12829xzv3+79/96yzv3+110/57z2v3+8495xuv3-11228yuv3-2396zuv3-101/9u2v3-71/39xv4-11746yv4-43/96zv4+64/55uv4-769v5,122/49x4y+3959x3y2+7277x2y3-41/93xy4-39/59y5+32/97x4z+97/96x3yz-43/116x2y2z+19/30xy3z+18/17y4z+39/122x3z2-98/99x2yz2+23/65xy2z2-50/103y3z2-11445x2z3+111/119xyz3-82/67y2z3+49/15xz4+59/74yz4+87/124z5+97/113x3yu+59/113x2y2u-17/16xy3u-1892y4u-15886x3zu+75/124x2yzu-13/108xy2zu+14203y3zu+13871x2z2u+24xyz2u+1483y2z2u-34/91xz3u+51/29yz3u+88/107z4u+6268x2yu2+13776xy2u2+1935y3u2-9062x2zu2-4534xyzu2+9553y2zu2-8429xz2u2-11141yz2u2-89/88z3u2-79/50xyu3-121/15y2u3-7891xzu3-27/94yzu3+96/71z2u3+27/53yu4+15307zu4-775x4v-2648x3yv-2095x2y2v+7594xy3v+12989y4v-17/29x3zv+11801x2yzv+32/45xy2zv-55/73y3zv+9756x2z2v+6886xyz2v+5939y2z2v+39/23xz3v+2632yz3v+77/6z4v+36/121x3uv-76/55x2yuv+5296xy2uv-32/47y3uv-125/84x2zuv-30/67xyzuv-14914y2zuv+73/23xz2uv+123/101yz2uv+49/43z3uv-76/71x2u2v+79/50xyu2v+38/7y2u2v-5262xzu2v-5211yzu2v+2/95z2u2v-3570xu3v-9280yu3v+32/91zu3v+1089u4v-73/33x3v2-3931x2yv2-21/113xy2v2+191y3v2-93/2x2zv2-75/47xyzv2+16/115y2zv2+120/23xz2v2+23/114yz2v2-115/108z3v2-103/12x2uv2+20/107xyuv2+28/11y2uv2+23/42xzuv2-15290yzuv2+3651z2uv2-9566xu2v2-113/68yu2v2+7705zu2v2-12261u3v2+9882x2v3-8520xyv3+14858y2v3+200xzv3+14/73yzv3+45/53z2v3-4/113xuv3-11327yuv3+59/51zuv3-13869u2v3-33/29xv4-13026yv4+4236zv4+10782uv4+2817v5,-123/61x4y+4/87x3y2-15558x2y3+78/5xy4-31/83y5+10267x4z-29/21x3yz+2197x2y2z-57/98xy3z+121/31y4z-1310x3z2+9386x2yz2-37/50xy2z2-1767y3z2-489x2z3-117/31xyz3+13576y2z3-50/9xz4-47/77yz4+53/89z5-57/11x4u-8374x3yu+104/103x2y2u+29/12xy3u+7368y4u+102/37x3zu+10508x2yzu+28xy2zu-14484y3zu-240x2z2u-115/114xyz2u+97/74y2z2u+105/11xz3u+125/46yz3u-6746z4u+3454x3u2+14656x2yu2-82/111xy2u2+13130y3u2+121/58x2zu2-42/11xyzu2-89/81y2zu2+68/49xz2u2+8243yz2u2-11615z3u2-40/87x2u3+80/39xyu3+14735y2u3+10706xzu3-2448yzu3+64/51z2u3+4514xu4+658yu4-39/49zu4+21/23u5+11/105x4v-108x3yv-31/98x2y2v-73/64xy3v+205y4v-15348x3zv-13734x2yzv-52/11xy2zv+5897y3zv+106/27x2z2v+9613xyz2v+81/59y2z2v+1172xz3v+3716yz3v-3581z4v+50x3uv+81/38x2yuv+84/109xy2uv-34/69y3uv-58/35x2zuv-23/105xyzuv+43/78y2zuv-57/31xz2uv+98/37yz2uv+11/96z3uv+147x2u2v+11414xyu2v+5/96y2u2v+19/77xzu2v+1274yzu2v+4/63z2u2v-4396xu3v+7060yu3v+5044zu3v-102/121u4v+67/25x3v2-13601x2yv2+68/65xy2v2+82/77y3v2-8394x2zv2-13318xyzv2+71/104y2zv2-75/107xz2v2+39/100yz2v2+11229z3v2+113/84x2uv2-2988xyuv2-9616y2uv2-41/50xzuv2+3/34yzuv2-12322z2uv2-120/11xu2v2+4/75yu2v2+12562zu2v2-92/39u3v2+1497x2v3+80/103xyv3+15/64y2v3-16/109xzv3-85/81yzv3-52/81z2v3-23/49xuv3+1295yuv3+13460zuv3+52/73u2v3-24/5xv4+11079yv4-25/57zv4-119/47uv4-7/120v5,123/61x4z-4/87x3yz+15558x2y2z-78/5xy3z+31/83y4z-2966x3z2+73/89x2yz2+75/112xy2z2+51/29y3z2-46/47x2z3+3812xyz3-8592y2z3+7582xz4+113/92yz4-11815z5+34/111x3zu-25/63x2yzu+115/79xy2zu+36/115y3zu-17/38x2z2u+7906xyz2u-108/71y2z2u+7541xz3u-12683yz3u-111/110z4u-107/24x2zu2-70/9xyzu2+42/113y2zu2-75xz2u2+64/101yz2u2+65/106z3u2-118/97xzu3+12960yzu3-918z2u3-72/113zu4+17/73x4v+15/17x3yv-22/91x2y2v+78/121xy3v+9/77y4v-91/102x3zv+15814x2yzv-12480xy2zv+1177y3zv-7/40x2z2v+47/69xyz2v-2847y2z2v+987xz3v-1954yz3v+43/97z4v-7692x3uv-109/44x2yuv-94/111xy2uv+84/17y3uv+13682x2zuv+64/7xyzuv-13069y2zuv+92/67xz2uv-109/84yz2uv-93/56z3uv-5368x2u2v-93/101xyu2v-118/43y2u2v-40/57xzu2v-58/55yzu2v+84/19z2u2v+4/9xu3v-540yu3v+11588zu3v-93/25u4v-85/89x3v2-1800x2yv2-40/51xy2v2-7293y3v2+15927x2zv2-33/106xyzv2+25/98y2zv2-13657xz2v2+43/37yz2v2+79/115z3v2-74/11x2uv2+29/114xyuv2-50/69y2uv2+252xzuv2-65/69yzuv2-88/5z2uv2+4418xu2v2-6634yu2v2-9854zu2v2-20/73u3v2+94/11x2v3+1430xyv3+1/41y2v3+21/101xzv3-68/59yzv3-9515z2v3+55/8xuv3-12014yuv3+57/73zuv3-97/58u2v3-124/91xv4+5575yv4-5470zv4+15340uv4+123/7v5,-14327x4y-14863x3y2+3913x2y3+4595xy4+75/77y5-5371x3yz+86/123x2y2z-84/115xy3z-117/100y4z-23/37x2yz2-82/125xy2z2+2426y3z2+17/50xyz3+11188y2z3-110/39yz4-126/97x4u-85/41x3yu-22/25x2y2u-10619xy3u+3473y4u-75/52x3zu-121/37x2yzu+48/125xy2zu+7246y3zu-78/49x2z2u+106/3xyz2u-50/59y2z2u-48/23xz3u+79/111yz3u-14913z4u+31/87x3u2+101/84x2yu2-1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I); kill R, Name, @p; "";
==> ==============================================
==> 
==> === Example: [ k3.d11.g11.ss0 ]
==>  = Ring:  (ZZ/31991),(x,y,z,u,v),(dp(5),C,L(1048575))
==>  ! Resolution via 'grres': 
==>  = Non-minimal betti numbers: 
==>            0     1     2     3     4
==> ------------------------------------
==>     0:     1     -     -     -     -
==>     1:     -     -     -     -     -
==>     2:     -     -     -     -     -
==>     3:     -     -     -     -     -
==>     4:     -     9     8     -     -
==>     5:     -     -     5     7     2
==> ------------------------------------
==> total:     1     9    13     7     2
==> 
==>  = Degrees of (ordered) maps: 
==> Graded resolution: 
==> R <-- d_1 --
==> R(-5)^9 <-- d_2 --
==> R(-6)^8 + R(-7)^5 <-- d_3 --
==> R(-8)^7 <-- d_4 --
==> R(-9)^2, given by maps: 
==> d_1 :
==> Graded homomorphism: R <- R(-5)^9, given by a matrix, with degrees: 
==>     .1 .2 .3 .4 .5 .6 .7 .8 .9 ...
==>     -- -- -- -- -- -- -- -- -- +..
==>  0 : 5  5  5  5  5  5  5  5  5 |.1
==>     == == == == == == == == ==    
==>      5  5  5  5  5  5  5  5  5    
==> d_2 :
==> Graded homomorphism: R(-5)^9 <- R(-6)^8 + R(-7)^5, given by a matrix, wit\
   h degrees: 
==>      ..1 ..2 ..3 ..4 ..5 ..6 ..7 ..8 ..9 .10 .11 .12 .13 ....
==>      --- --- --- --- --- --- --- --- --- --- --- --- --- +...
==>   5 :  1   1   1   1   1   1   1   1   2   2   2   2   2 |..1
==>   5 :  1   1   1   1   1   1   1   1   2   2   2   2   2 |..2
==>   5 :  1   1   1   1   1   1   1   1   2   2   2   2   2 |..3
==>   5 :  1   1   1   1   1   1   1   1   2   2   2   2   2 |..4
==>   5 :  1   1   1   1   1   1   1   1   2   2   2   2   2 |..5
==>   5 :  1   1   1   1   1   1   1   1   2   2   2   2   2 |..6
==>   5 :  1   1   1   1   1   1   1   1   2   2   2   2   2 |..7
==>   5 :  1   1   1   1   1   1   1   1   2   2   2   2   2 |..8
==>   5 :  1   1   1   1   1   1   1   1   -   -   -   -   - |..9
==>      === === === === === === === === === === === === ===     
==>        6   6   6   6   6   6   6   6   7   7   7   7   7     
==> d_3 :
==> Graded homomorphism: R(-6)^8 + R(-7)^5 <- R(-8)^7, given by a matrix, wit\
   h degrees: 
==>      ..1 ..2 ..3 ..4 ..5 ..6 ..7 ....
==>      --- --- --- --- --- --- --- +...
==>   6 :  -   -   2   2   2   2   2 |..1
==>   6 :  -   -   2   2   2   2   2 |..2
==>   6 :  -   -   2   2   2   2   2 |..3
==>   6 :  -   -   2   2   2   2   2 |..4
==>   6 :  -   -   2   2   2   2   2 |..5
==>   6 :  -   -   2   2   2   2   2 |..6
==>   6 :  -   -   2   2   2   2   2 |..7
==>   6 :  -   -   2   2   2   2   2 |..8
==>   7 :  1   1   1   1   1   1   1 |..9
==>   7 :  1   1   1   1   1   1   1 |.10
==>   7 :  1   1   1   1   1   1   1 |.11
==>   7 :  1   1   1   1   1   1   1 |.12
==>   7 :  1   1   1   1   1   1   1 |.13
==>      === === === === === === ===     
==>        8   8   8   8   8   8   8     
==> d_4 :
==> Graded homomorphism: R(-8)^7 <- R(-9)^2, given by a matrix, with degrees: 
==>     .1 .2 ...
==>     -- -- +..
==>  8 : 1  1 |.1
==>  8 : 1  1 |.2
==>  8 : 1  1 |.3
==>  8 : 1  1 |.4
==>  8 : 1  1 |.5
==>  8 : 1  1 |.6
==>  8 : 1  1 |.7
==>     == ==    
==>      9  9    
==>  = TRANSPOSE'd complex: %%%%%%%%%%%%%%
==>            0     1     2     3     4
==> ------------------------------------
==>    -9:     2     7     5     -     -
==>    -8:     -     -     8     9     -
==>    -7:     -     -     -     -     -
==>    -6:     -     -     -     -     -
==>    -5:     -     -     -     -     -
==>    -4:     -     -     -     -     1
==> ------------------------------------
==> total:     2     7    13     9     1
==> 
==> Graded resolution: 
==> R(9)^2 <-- d_1 --
==> R(8)^7 <-- d_2 --
==> R(7)^5 + R(6)^8 <-- d_3 --
==> R(5)^9 <-- d_4 --
==> R, given by maps: 
==> d_1 :
==> Graded homomorphism: R(9)^2 <- R(8)^7, given by a matrix, with degrees: 
==>      ..1 ..2 ..3 ..4 ..5 ..6 ..7 ....
==>      --- --- --- --- --- --- --- +...
==>  -9 :  1   1   1   1   1   1   1 |..1
==>  -9 :  1   1   1   1   1   1   1 |..2
==>      === === === === === === ===     
==>       -8  -8  -8  -8  -8  -8  -8     
==> d_2 :
==> Graded homomorphism: R(8)^7 <- R(7)^5 + R(6)^8, given by a matrix, with d\
   egrees: 
==>      ..1 ..2 ..3 ..4 ..5 ..6 ..7 ..8 ..9 .10 .11 .12 .13 ....
==>      --- --- --- --- --- --- --- --- --- --- --- --- --- +...
==>  -8 :  1   1   1   1   1   -   -   -   -   -   -   -   - |..1
==>  -8 :  1   1   1   1   1   -   -   -   -   -   -   -   - |..2
==>  -8 :  1   1   1   1   1   2   2   2   2   2   2   2   2 |..3
==>  -8 :  1   1   1   1   1   2   2   2   2   2   2   2   2 |..4
==>  -8 :  1   1   1   1   1   2   2   2   2   2   2   2   2 |..5
==>  -8 :  1   1   1   1   1   2   2   2   2   2   2   2   2 |..6
==>  -8 :  1   1   1   1   1   2   2   2   2   2   2   2   2 |..7
==>      === === === === === === === === === === === === ===     
==>       -7  -7  -7  -7  -7  -6  -6  -6  -6  -6  -6  -6  -6     
==> d_3 :
==> Graded homomorphism: R(7)^5 + R(6)^8 <- R(5)^9, given by a matrix, with d\
   egrees: 
==>      ..1 ..2 ..3 ..4 ..5 ..6 ..7 ..8 ..9 ....
==>      --- --- --- --- --- --- --- --- --- +...
==>  -7 :  2   2   2   2   2   2   2   2   - |..1
==>  -7 :  2   2   2   2   2   2   2   2   - |..2
==>  -7 :  2   2   2   2   2   2   2   2   - |..3
==>  -7 :  2   2   2   2   2   2   2   2   - |..4
==>  -7 :  2   2   2   2   2   2   2   2   - |..5
==>  -6 :  1   1   1   1   1   1   1   1   1 |..6
==>  -6 :  1   1   1   1   1   1   1   1   1 |..7
==>  -6 :  1   1   1   1   1   1   1   1   1 |..8
==>  -6 :  1   1   1   1   1   1   1   1   1 |..9
==>  -6 :  1   1   1   1   1   1   1   1   1 |.10
==>  -6 :  1   1   1   1   1   1   1   1   1 |.11
==>  -6 :  1   1   1   1   1   1   1   1   1 |.12
==>  -6 :  1   1   1   1   1   1   1   1   1 |.13
==>      === === === === === === === === ===     
==>       -5  -5  -5  -5  -5  -5  -5  -5  -5     
==> d_4 :
==> Graded homomorphism: R(5)^9 <- R, given by a matrix, with degrees: 
==>      ..1 ....
==>      --- +...
==>  -5 :  5 |..1
==>  -5 :  5 |..2
==>  -5 :  5 |..3
==>  -5 :  5 |..4
==>  -5 :  5 |..5
==>  -5 :  5 |..6
==>  -5 :  5 |..7
==>  -5 :  5 |..8
==>  -5 :  5 |..9
==>      ===     
==>        0     
==> 
==> 
string Name = "ell.d10.g9"; int @p=31991; ring R = (@p),(x,y,z,u,v), dp;ideal I = 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106y4v-14412x3zv+6361x2yzv+10187xy2zv-76/101y3zv+4991x2z2v+14252xyz2v-107/18y2z2v+4857xz3v-13162yz3v+1088z4v-3/64x3uv-14/113x2yuv+96/41xy2uv-76/103y3uv+8924x2zuv-11/30xyzuv-4493y2zuv+3070xz2uv+10/41yz2uv-35/51z3uv+4318x2u2v+15309xyu2v+107/101y2u2v-1748xzu2v+11082yzu2v+29/90z2u2v-19/8xu3v+107/64yu3v+3039zu3v-21/59u4v+14/87x3v2+16/59x2yv2+77/9xy2v2+3679y3v2+53/113x2zv2+97/80xyzv2-899y2zv2+10016xz2v2-10049yz2v2+29/10z3v2+43/7x2uv2-5919xyuv2-7/34y2uv2+42/85xzuv2+102/23yzuv2-8300z2uv2+125/2xu2v2-1700yu2v2-107/21zu2v2+12942u3v2-3054x2v3-87/53xyv3+4745y2v3+52/83xzv3-11747yzv3-119/64z2v3-5987xuv3+2016yuv3-2889zuv3+4535u2v3-1588xv4-27/5yv4+8541zv4-15581uv4-3/119v5,-x3y2-7492x2y3+118/35xy4-43/5y5-2479x3yz-8522x2y2z-105/26xy3z-103/18y4z+5868x3z2-10040x2yz2-50/81xy2z2+3378y3z2-71/44x2z3-12323xyz3-25/44y2z3+12979xz4-15146yz4+5962z5+8852x3yu-39/77x2y2u-9554xy3u-14590y4u+61/84x3zu-2295x2yzu-5969xy2zu+14241y3zu-29/74x2z2u+107/17xyz2u+107y2z2u-14/99xz3u+109/93yz3u-73/41z4u-9170x3u2+8877x2yu2-109/118xy2u2+65/36y3u2+122/93x2zu2-15020xyzu2+33/50y2zu2-11/46xz2u2-91/116yz2u2-47/10z3u2-34/57x2u3-90/79xyu3+12760y2u3+81/71xzu3+1436yzu3+14839z2u3+6598xu4-12418yu4-47/59zu4-1357u5+6/91x3yv-46/21x2y2v+100/53xy3v-69/125y4v+98x3zv-63/23x2yzv+96/121xy2zv+12/11y3zv-37/120x2z2v-37/106xyz2v-25/122y2z2v-136./examples/TestGRRes.sing  13> 57xz3v+79/47yz3v+15519z4v+45/46x3uv+73/100x2yuv-5361xy2uv-39/101y3uv+1161x2zuv-105/22xyzuv+79/34y2zuv+96/31xz2uv-20/51yz2uv-2683z3uv-122/125x2u2v-66/109xyu2v-6357y2u2v-3021xzu2v+86/33yzu2v-105/8z2u2v+95/22xu3v-9046yu3v+69/83zu3v-3166u4v+91/115x3v2+9661x2yv2-8757xy2v2+79/23y3v2+125/107x2zv2+6133xyzv2+8975y2zv2-38/103xz2v2-120/43yz2v2+59/124z3v2-9252x2uv2+27/64xyuv2+99/80y2uv2+125/28xzuv2-60/109yzuv2-73/107z2uv2-15162xu2v2+91/71yu2v2-89/122zu2v2-1/9u3v2+40/117x2v3-71/41xyv3+10502y2v3-5618xzv3-995yzv3-117/112z2v3-15185xuv3-4470yuv3-13/4zuv3-14631u2v3-5804xv4-15527yv4-47/87zv4+5084uv4-64/11v5,x4y+46/47x3y2+13/47x2y3-13553xy4-5/59y5-15/76x4z+370x3yz+15/56x2y2z-8199xy3z+17/37y4z+32/47x3z2-26/29x2yz2-1/109xy2z2+68/77y3z2-88/123x2z3-1397xyz3-90y2z3+117/80xz4+14919yz4+11248z5+1742x4u-6060x3yu+97/38x2y2u+54/53xy3u-26/45y4u-315x3zu+9/76x2yzu+113/14xy2zu-28/79y3zu-109/14x2z2u+53/54xyz2u-213y2z2u-18/11xz3u-12129yz3u+6321z4u-13361x3u2+109/81x2yu2+37/84xy2u2-19/69y3u2+82/31x2zu2-20/41xyzu2-13135y2zu2-52/99xz2u2-12942yz2u2-8237z3u2-12377x2u3-41/43xyu3+13143y2u3-6701xzu3-98/53yzu3+97/43z2u3+1570xu4-41/5yu4-7/86zu4+306u5-7920x4v+76/53x3yv-91/73x2y2v-49/64xy3v+12976y4v-1787x3zv-95/123x2yzv-10033xy2zv-83/100y3zv-17/112x2z2v+1060xyz2v-121/54y2z2v+25/77xz3v+23/73yz3v-3576z4v+41/98x3uv+15265x2yuv+9014xy2uv-86/33y3uv-83/51x2zuv-15747xyzuv+11/16y2zuv-12110xz2uv-15554yz2uv-7654z3uv+68/53x2u2v+1673xyu2v-123/64y2u2v-3/19xzu2v+29/34yzu2v-5457z2u2v+51/58xu3v-4242yu3v-70/67zu3v+7159u4v+8620x3v2-9381x2yv2+52/79xy2v2-73/113y3v2+83/22x2zv2-1445xyzv2+6524y2zv2-122xz2v2-107/85yz2v2-1240z3v2+1710x2uv2-101/59xyuv2-2950y2uv2-12343xzuv2-79/27yzuv2-5/26z2uv2+89/68xu2v2-14985yu2v2+8382zu2v2-5109u3v2-12322x2v3-97/106xyv3-86/23y2v3+115/11xzv3-45/2yzv3-79/110z2v3+7472xuv3-17/71yuv3+1659zuv3-4468u2v3-85/47xv4+11/73yv4-1700zv4-32/51uv4+7855v5,62/49x4-125/36x3y+2x2y2+56/39xy3+7307y4-42/43x3z+83/42x2yz-11648xy2z-43/106y3z-49/109x2z2-95/28xyz2-194y2z2+71/95xz3-83/59yz3-66/95z4-1/51x3u-82/21x2yu+121/103xy2u-11106y3u-55/91x2zu+55/119xyzu-123/79y2zu-20/11xz2u+6943yz2u+69z3u+5908x2u2-56/89xyu2-13119y2u2+32/115xzu2+11412yzu2+63/50z2u2+73/85xu3-12527yu3+1991zu3+26/33u4+79/91x3v-10/27x2yv+20/117xy2v-49/102y3v+98/33x2zv+115/76xyzv+12412y2zv-47/48xz2v+121/113yz2v+105/26z3v-10954x2uv+10730xyuv+15721y2uv-101/63xzuv-32/19yzuv+67/71z2uv+77/45xu2v+27/98yu2v-670zu2v-13396u3v-55/119x2v2-7890xyv2-101/7y2v2-9563xzv2+3/95yzv2+4857z2v2+9397xuv2+3/116yuv2-12257zuv2-79/49u2v2+11697xv3+26/19yv3+4885zv3-53/22uv3-8355v4;TestGRRes(Name, I); kill R, Name, @p; "";
==> ==============================================
==> 
==> === Example: [ ell.d10.g9 ]
==>  = Ring:  (ZZ/31991),(x,y,z,u,v),(dp(5),C,L(1048575))
==>  ! Resolution via 'grres': 
==>  = Non-minimal betti numbers: 
==>            0     1     2     3     4
==> ------------------------------------
==>     0:     1     -     -     -     -
==>     1:     -     -     -     -     -
==>     2:     -     -     -     -     -
==>     3:     -     1     -     -     -
==>     4:     -     9    14     5     -
==>     5:     -     -     1     2     1
==> ------------------------------------
==> total:     1    10    15     7     1
==> 
==>  = Degrees of (ordered) maps: 
==> Graded resolution: 
==> R <-- d_1 --
==> R(-4) + R(-5)^9 <-- d_2 --
==> R(-6)^14 + R(-7) <-- d_3 --
==> R(-7)^5 + R(-8)^2 <-- d_4 --
==> R(-9), given by maps: 
==> d_1 :
==> Graded homomorphism: R <- R(-4) + R(-5)^9, given by a matrix, with degree\
   s: 
==>      ..1 ..2 ..3 ..4 ..5 ..6 ..7 ..8 ..9 .10 ....
==>      --- --- --- --- --- --- --- --- --- --- +...
==>   0 :  4   5   5   5   5   5   5   5   5   5 |..1
==>      === === === === === === === === === ===     
==>        4   5   5   5   5   5   5   5   5   5     
==> d_2 :
==> Graded homomorphism: R(-4) + R(-5)^9 <- R(-6)^14 + R(-7), given by a matr\
   ix, with degrees: 
==>      ..1 ..2 ..3 ..4 ..5 ..6 ..7 ..8 ..9 .10 .11 .12 .13 .14 .15 ....
==>      --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- +...
==>   4 :  -   -   -   -   -   2   2   2   2   2   2   2   2   2   - |..1
==>   5 :  1   1   1   1   1   1   1   1   1   1   1   1   1   1   2 |..2
==>   5 :  1   1   1   1   1   1   1   1   1   1   1   1   1   1   2 |..3
==>   5 :  1   1   1   1   1   1   1   1   1   1   1   1   1   1   2 |..4
==>   5 :  1   1   1   1   1   1   1   1   1   1   1   1   1   1   2 |..5
==>   5 :  1   1   1   1   1   1   1   1   1   1   1   1   1   1   2 |..6
==>   5 :  1   1   1   1   1   1   1   1   1   1   1   1   1   1   - |..7
==>   5 :  1   1   1   1   1   1   1   1   1   1   1   1   1   1   - |..8
==>   5 :  1   1   1   1   1   1   1   1   1   1   1   1   1   1   - |..9
==>   5 :  1   1   1   1   1   1   1   1   1   1   1   1   1   1   - |.10
==>      === === === === === === === === === === === === === === ===     
==>        6   6   6   6   6   6   6   6   6   6   6   6   6   6   7     
==> d_3 :
==> Graded homomorphism: R(-6)^14 + R(-7) <- R(-7)^5 + R(-8)^2, given by a ma\
   trix, with degrees: 
==>      ..1 ..2 ..3 ..4 ..5 ..6 ..7 ....
==>      --- --- --- --- --- --- --- +...
==>   6 :  1   1   1   1   1   2   - |..1
==>   6 :  1   1   1   1   1   2   - |..2
==>   6 :  1   1   1   1   1   2   - |..3
==>   6 :  1   1   1   1   1   2   - |..4
==>   6 :  1   1   1   1   1   2   - |..5
==>   6 :  1   1   1   1   1   -   2 |..6
==>   6 :  1   1   1   1   1   -   2 |..7
==>   6 :  1   1   1   1   1   -   2 |..8
==>   6 :  1   1   1   1   1   -   2 |..9
==>   6 :  1   1   1   1   1   -   2 |.10
==>   6 :  1   1   1   1   1   -   2 |.11
==>   6 :  1   1   1   1   1   -   2 |.12
==>   6 :  1   1   1   1   1   -   2 |.13
==>   6 :  1   1   1   1   1   -   - |.14
==>   7 :  -   -   -   -   -   1   1 |.15
==>      === === === === === === ===     
==>        7   7   7   7   7   8   8     
==> d_4 :
==> Graded homomorphism: R(-7)^5 + R(-8)^2 <- R(-9), given by a matrix, with \
   degrees: 
==>     .1 ...
==>     -- +..
==>  7 : 2 |.1
==>  7 : 2 |.2
==>  7 : 2 |.3
==>  7 : 2 |.4
==>  7 : 2 |.5
==>  8 : 1 |.6
==>  8 : 1 |.7
==>     ==    
==>      9    
==>  = TRANSPOSE'd complex: %%%%%%%%%%%%%%
==>            0     1     2     3     4
==> ------------------------------------
==>    -9:     1     2     1     -     -
==>    -8:     -     5    14     9     -
==>    -7:     -     -     -     1     -
==>    -6:     -     -     -     -     -
==>    -5:     -     -     -     -     -
==>    -4:     -     -     -     -     1
==> ------------------------------------
==> total:     1     7    15    10     1
==> 
==> Graded resolution: 
==> R(9) <-- d_1 --
==> R(8)^2 + R(7)^5 <-- d_2 --
==> R(7) + R(6)^14 <-- d_3 --
==> R(5)^9 + R(4) <-- d_4 --
==> R, given by maps: 
==> d_1 :
==> Graded homomorphism: R(9) <- R(8)^2 + R(7)^5, given by a matrix, with deg\
   rees: 
==>      ..1 ..2 ..3 ..4 ..5 ..6 ..7 ....
==>      --- --- --- --- --- --- --- +...
==>  -9 :  1   1   2   2   2   2   2 |..1
==>      === === === === === === ===     
==>       -8  -8  -7  -7  -7  -7  -7     
==> d_2 :
==> Graded homomorphism: R(8)^2 + R(7)^5 <- R(7) + R(6)^14, given by a matrix\
   , with degrees: 
==>      ..1 ..2 ..3 ..4 ..5 ..6 ..7 ..8 ..9 .10 .11 .12 .13 .14 .15 ....
==>      --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- +...
==>  -8 :  1   2   2   2   2   2   -   -   -   -   -   -   -   -   - |..1
==>  -8 :  1   -   -   -   -   -   2   2   2   2   2   2   2   2   - |..2
==>  -7 :  -   1   1   1   1   1   1   1   1   1   1   1   1   1   1 |..3
==>  -7 :  -   1   1   1   1   1   1   1   1   1   1   1   1   1   1 |..4
==>  -7 :  -   1   1   1   1   1   1   1   1   1   1   1   1   1   1 |..5
==>  -7 :  -   1   1   1   1   1   1   1   1   1   1   1   1   1   1 |..6
==>  -7 :  -   1   1   1   1   1   1   1   1   1   1   1   1   1   1 |..7
==>      === === === === === === === === === === === === === === ===     
==>       -7  -6  -6  -6  -6  -6  -6  -6  -6  -6  -6  -6  -6  -6  -6     
==> d_3 :
==> Graded homomorphism: R(7) + R(6)^14 <- R(5)^9 + R(4), given by a matrix, \
   with degrees: 
==>      ..1 ..2 ..3 ..4 ..5 ..6 ..7 ..8 ..9 .10 ....
==>      --- --- --- --- --- --- --- --- --- --- +...
==>  -7 :  2   2   2   2   2   -   -   -   -   - |..1
==>  -6 :  1   1   1   1   1   1   1   1   1   - |..2
==>  -6 :  1   1   1   1   1   1   1   1   1   - |..3
==>  -6 :  1   1   1   1   1   1   1   1   1   - |..4
==>  -6 :  1   1   1   1   1   1   1   1   1   - |..5
==>  -6 :  1   1   1   1   1   1   1   1   1   - |..6
==>  -6 :  1   1   1   1   1   1   1   1   1   2 |..7
==>  -6 :  1   1   1   1   1   1   1   1   1   2 |..8
==>  -6 :  1   1   1   1   1   1   1   1   1   2 |..9
==>  -6 :  1   1   1   1   1   1   1   1   1   2 |.10
==>  -6 :  1   1   1   1   1   1   1   1   1   2 |.11
==>  -6 :  1   1   1   1   1   1   1   1   1   2 |.12
==>  -6 :  1   1   1   1   1   1   1   1   1   2 |.13
==>  -6 :  1   1   1   1   1   1   1   1   1   2 |.14
==>  -6 :  1   1   1   1   1   1   1   1   1   2 |.15
==>      === === === === === === === === === ===     
==>       -5  -5  -5  -5  -5  -5  -5  -5  -5  -4     
==> d_4 :
==> Graded homomorphism: R(5)^9 + R(4) <- R, given by a matrix, with degrees: 
==>      ..1 ....
==>      --- +...
==>  -5 :  5 |..1
==>  -5 :  5 |..2
==>  -5 :  5 |..3
==>  -5 :  5 |..4
==>  -5 :  5 |..5
==>  -5 :  5 |..6
==>  -5 :  5 |..7
==>  -5 :  5 |..8
==>  -5 :  5 |..9
==>  -4 :  4 |.10
==>      ===     
==>        0     
==> 
==> 
string Name = "k3.d10.g9.quart2"; int @p=31991; ring R = (@p),(x,y,z,u,v), dp;ideal I = 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-487xyuv2-12143y2uv2+13825xzuv2+55/6yzuv2-4250z2uv2+4237xu2v2-109/9yu2v2+67/53zu2v2+82/33u3v2+8660x2v3+15046xyv3-79/84y2v3-10310xzv3+110yzv3-7636z2v3+57/92xuv3-22/119yuv3-95/103zuv3+5138u2v3+123/49xv4-7587yv4+30/41zv4-124/121uv4+54/71v5,-29/60x4y-108/77x3y2-109/37x2y3-3619xy4+109/6x4z-37/67x3yz+53/45x2y2z+5291xy3z-2927y4z+34/5x3z2+87/17x2yz2+100/89xy2z2-114/29y3z2-4057x2z3-1/42xyz3-14/61y2z3-398xz4-122/73yz4+66/37z5+99/37x4u-5691x3yu-8778x2y2u+17/115xy3u+51/113y4u-71/101x3zu+85/91x2yzu-92/9xy2zu-3442y3zu+109/26x2z2u+50/37xyz2u+77/94y2z2u+16/35xz3u+9985yz3u+5/102z4u-5932x3u2+89/125x2yu2-895xy2u2-12455y3u2-630x2zu2-64/47xyzu2+25/9y2zu2+7906xz2u2+6827yz2u2+9808z3u2-113/118x2u3+79/8xyu3+9484y2u3+62/39xzu3+6/85yzu3-23/49z2u3-93/115xu4-11/93yu4-15177zu4-13/2u5-7623x4v-103/73x3yv-96/115x2y2v+39/76xy3v+80/79y4v+43/68x3zv+45/97x2yzv+101/87xy2zv+4632y3zv-918x2z2v+8248xyz2v-4276y2z2v+8853xz3v-39/61yz3v-121/87z4v+9968x3uv+473x2yuv+117/56xy2uv-19/21y3uv+121/119x2zuv+3/98xyzuv-65/42y2zuv-3723xz2uv+7/34yz2uv-112/87z3uv+103x2u2v+25/41xyu2v-14459y2u2v-56/41xzu2v-59/81yzu2v-109/102z2u2v-87/16xu3v-13011yu3v+49/123zu3v+106/89u4v-61/51x3v2+14107x2yv2+8035xy2v2-8853y3v2+5723x2zv2+123/53xyzv2-9727y2zv2-102/83xz2v2+1111yz2v2-15745z3v2+83/118x2uv2-57/35xyuv2-48/73y2uv2-28/37xzuv2-27/97yzuv2./examples/TestGRRes.sing  15> -27/58z2uv2+71/93xu2v2+117/8yu2v2+12344zu2v2-2497u3v2-118/71x2v3-11/19xyv3+21/104y2v3+32/113xzv3+15544yzv3+31/18z2v3+5909xuv3-67/58yuv3+27/35zuv3+115/9u2v3+79/13xv4+6722yv4-37/114zv4-71/124uv4+4657v5,-77/61x4y-88/101x3y2+93/88x2y3-11/70xy4+9806y5+7896x4z-4699x3yz+55/122x2y2z-63/122xy3z-125/74y4z+47/45x3z2+101/17x2yz2+92/47xy2z2+69/82y3z2+12402x2z3+113/98xyz3-101/33y2z3-15376xz4+47/71yz4-73/10z5+65/74x4u-14409x3yu-14478x2y2u+13593xy3u+102/97y4u+39/62x3zu-34/125x2yzu-83/9xy2zu+45/113y3zu+14484x2z2u-15293xyz2u-26/55y2z2u-958xz3u+67/35yz3u-93/19z4u+25/16x3u2+107/52x2yu2-4599xy2u2-86/51y3u2-9885x2zu2-77/47xyzu2+33/65y2zu2+90/109xz2u2-61/26yz2u2+6198z3u2-38/37x2u3-13935xyu3-142y2u3-64/5xzu3-7228yzu3+1251z2u3+1556xu4+117/121yu4-92/35zu4+99/92u5+13493x4v+12654x3yv+32/101x2y2v-11118xy3v+43/51y4v-575x3zv+103/21x2yzv+85/24xy2zv+1788y3zv+85/3x2z2v-64/25xyz2v+57/35y2z2v+37/120xz3v-69/110yz3v+48/49z4v+55/114x3uv-6439x2yuv+31/51xy2uv-90/49y3uv-45/104x2zuv-12018xyzuv+6/119y2zuv+40/63xz2uv+20/91yz2uv+50/43z3uv+1/26x2u2v-109/47xyu2v+99/7y2u2v+72/83xzu2v+61/118yzu2v+3530z2u2v+6146xu3v+117yu3v-9921zu3v-8708u4v-10/47x3v2-15294x2yv2-7336xy2v2+1/66y3v2-3057x2zv2+74/123xyzv2+146y2zv2-103/34xz2v2-117/76yz2v2+8472z3v2-7/92x2uv2+10033xyuv2+43/53y2uv2+4694xzuv2-49/2yzuv2-71/73z2uv2-125/17xu2v2-9817yu2v2+7218zu2v2+6897u3v2-19/90x2v3+11899xyv3-11779y2v3-5456xzv3+17/42yzv3+15340z2v3+12/7xuv3+9580yuv3-502zuv3-14069u2v3-4371xv4+14452yv4-9423zv4-117/122uv4+1126v5,49/108x4-39/4x3y-67/21x2y2-8/69xy3-9779y4+57/14x3z-11145x2yz+6928xy2z-7824y3z+1/79x2z2+5173xyz2-62/15y2z2-123/112xz3+88/79yz3+1/125z4+57/23x3u-11856x2yu-7444xy2u+115/8y3u-11133x2zu+71/73xyzu-7941y2zu+69/65xz2u+22/75yz2u+65/121z3u+9471x2u2+9167xyu2+51/59y2u2+12835xzu2+15047yzu2+11102z2u2-10059xu3+19/28yu3+65/21zu3-39/28u4-3/73x3v+94/61x2yv+8778xy2v-12922y3v-8711x2zv-37/97xyzv+14270y2zv+4487xz2v-59/112yz2v-14183z3v+15553x2uv+3579xyuv+114/91y2uv-4/97xzuv+13/85yzuv-89/15z2uv+58/75xu2v-34/7yu2v-90/61zu2v+90/101u3v-14673x2v2+90/19xyv2-45/37y2v2+23/49xzv2-71/11yzv2+119/8z2v2+89/10xuv2+109/91yuv2+36/49zuv2-7/31u2v2-40/113xv3-21/121yv3+9910zv3+33/14uv3-23/79v4,93/70x4-43/125x3y+9582x2y2+7565xy3-11511y4-3/79x3z-36/107x2yz-2038xy2z+879y3z-4700x2z2+103/14xyz2+102/79y2z2-67/68xz3-44/25yz3+105/79z4-29/24x3u-74/83x2yu+67/43xy2u+49/12y3u-115/11x2zu+23/67xyzu-61/27y2zu+12257xz2u+14068yz2u+23/15z3u+607x2u2+73/8xyu2+14237y2u2-13/33xzu2+110/71yzu2+41/101z2u2+5708xu3+88/67yu3+1460zu3-2472u4-1629x3v-51/70x2yv-88/73xy2v-36/97y3v+38/11x2zv+15899xyzv+54/19y2zv+9460xz2v-5150yz2v+3462z3v+5522x2uv-19/123xyuv+14871y2uv+53/5xzuv-7535yzuv-13430z2uv+107/47xu2v-8307yu2v-55/79zu2v-11945u3v-16/83x2v2+115/48xyv2+12389y2v2+11545xzv2-25/26yzv2-3755z2v2+4724xuv2-31/21yuv2+7872zuv2+89/45u2v2+87/47xv3+7625yv3+13494zv3-15376uv3-25/126v4;TestGRRes(Name, I); kill R, Name, @p; "";
==> ==============================================
==> 
==> === Example: [ k3.d10.g9.quart2 ]
==>  = Ring:  (ZZ/31991),(x,y,z,u,v),(dp(5),C,L(1048575))
==>  ! Resolution via 'grres': 
==>  = Non-minimal betti numbers: 
==>            0     1     2     3     4
==> ------------------------------------
==>     0:     1     -     -     -     -
==>     1:     -     -     -     -     -
==>     2:     -     -     -     -     -
==>     3:     -     2     -     -     -
==>     4:     -     4     7     2     -
==>     5:     -     3     8     7     2
==> ------------------------------------
==> total:     1     9    15     9     2
==> 
==>  = Degrees of (ordered) maps: 
==> Graded resolution: 
==> R <-- d_1 --
==> R(-4)^2 + R(-5)^4 + R(-6)^3 <-- d_2 --
==> R(-6)^7 + R(-7)^8 <-- d_3 --
==> R(-7)^2 + R(-8)^7 <-- d_4 --
==> R(-9)^2, given by maps: 
==> d_1 :
==> Graded homomorphism: R <- R(-4)^2 + R(-5)^4 + R(-6)^3, given by a matrix,\
    with degrees: 
==>     .1 .2 .3 .4 .5 .6 .7 .8 .9 ...
==>     -- -- -- -- -- -- -- -- -- +..
==>  0 : 4  4  5  5  5  5  6  6  6 |.1
==>     == == == == == == == == ==    
==>      4  4  5  5  5  5  6  6  6    
==> d_2 :
==> Graded homomorphism: R(-4)^2 + R(-5)^4 + R(-6)^3 <- R(-6)^7 + R(-7)^8, gi\
   ven by a matrix, with degrees: 
==>      ..1 ..2 ..3 ..4 ..5 ..6 ..7 ..8 ..9 .10 .11 .12 .13 .14 .15 ....
==>      --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- +...
==>   4 :  2   2   2   2   2   2   2   -   -   3   3   3   3   3   3 |..1
==>   4 :  2   2   2   2   2   2   2   -   -   3   3   3   3   3   3 |..2
==>   5 :  1   1   1   1   1   1   1   2   2   2   2   2   2   2   2 |..3
==>   5 :  1   1   1   1   1   1   1   2   2   2   2   2   2   2   2 |..4
==>   5 :  1   1   1   1   1   1   1   -   -   2   2   2   2   2   2 |..5
==>   5 :  1   1   1   1   1   1   1   -   -   2   2   2   2   2   2 |..6
==>   6 :  -   -   -   -   -   -   -   1   1   1   1   1   1   1   1 |..7
==>   6 :  -   -   -   -   -   -   -   1   1   1   1   1   1   1   1 |..8
==>   6 :  -   -   -   -   -   -   -   1   1   1   1   1   1   1   1 |..9
==>      === === === === === === === === === === === === === === ===     
==>        6   6   6   6   6   6   6   7   7   7   7   7   7   7   7     
==> d_3 :
==> Graded homomorphism: R(-6)^7 + R(-7)^8 <- R(-7)^2 + R(-8)^7, given by a m\
   atrix, with degrees: 
==>      ..1 ..2 ..3 ..4 ..5 ..6 ..7 ..8 ..9 ....
==>      --- --- --- --- --- --- --- --- --- +...
==>   6 :  1   1   2   2   2   2   2   2   2 |..1
==>   6 :  1   1   2   2   2   2   2   2   2 |..2
==>   6 :  1   1   2   2   2   2   2   2   2 |..3
==>   6 :  1   1   2   2   2   2   2   2   2 |..4
==>   6 :  1   1   2   2   2   2   2   2   2 |..5
==>   6 :  1   1   2   2   2   2   2   2   2 |..6
==>   6 :  1   1   2   2   2   2   2   2   2 |..7
==>   7 :  -   -   1   1   1   1   1   1   1 |..8
==>   7 :  -   -   1   1   1   1   1   1   1 |..9
==>   7 :  -   -   1   1   1   1   1   1   1 |.10
==>   7 :  -   -   1   1   1   1   1   1   1 |.11
==>   7 :  -   -   1   1   1   1   1   1   1 |.12
==>   7 :  -   -   1   1   1   1   1   1   1 |.13
==>   7 :  -   -   1   1   1   1   1   1   1 |.14
==>   7 :  -   -   1   1   1   1   1   1   1 |.15
==>      === === === === === === === === ===     
==>        7   7   8   8   8   8   8   8   8     
==> d_4 :
==> Graded homomorphism: R(-7)^2 + R(-8)^7 <- R(-9)^2, given by a matrix, wit\
   h degrees: 
==>     .1 .2 ...
==>     -- -- +..
==>  7 : 2  2 |.1
==>  7 : 2  2 |.2
==>  8 : 1  1 |.3
==>  8 : 1  1 |.4
==>  8 : 1  1 |.5
==>  8 : 1  1 |.6
==>  8 : 1  1 |.7
==>  8 : 1  1 |.8
==>  8 : 1  1 |.9
==>     == ==    
==>      9  9    
==>  = TRANSPOSE'd complex: %%%%%%%%%%%%%%
==>            0     1     2     3     4
==> ------------------------------------
==>    -9:     2     7     8     3     -
==>    -8:     -     2     7     4     -
==>    -7:     -     -     -     2     -
==>    -6:     -     -     -     -     -
==>    -5:     -     -     -     -     -
==>    -4:     -     -     -     -     1
==> ------------------------------------
==> total:     2     9    15     9     1
==> 
==> Graded resolution: 
==> R(9)^2 <-- d_1 --
==> R(8)^7 + R(7)^2 <-- d_2 --
==> R(7)^8 + R(6)^7 <-- d_3 --
==> R(6)^3 + R(5)^4 + R(4)^2 <-- d_4 --
==> R, given by maps: 
==> d_1 :
==> Graded homomorphism: R(9)^2 <- R(8)^7 + R(7)^2, given by a matrix, with d\
   egrees: 
==>      ..1 ..2 ..3 ..4 ..5 ..6 ..7 ..8 ..9 ....
==>      --- --- --- --- --- --- --- --- --- +...
==>  -9 :  1   1   1   1   1   1   1   2   2 |..1
==>  -9 :  1   1   1   1   1   1   1   2   2 |..2
==>      === === === === === === === === ===     
==>       -8  -8  -8  -8  -8  -8  -8  -7  -7     
==> d_2 :
==> Graded homomorphism: R(8)^7 + R(7)^2 <- R(7)^8 + R(6)^7, given by a matri\
   x, with degrees: 
==>      ..1 ..2 ..3 ..4 ..5 ..6 ..7 ..8 ..9 .10 .11 .12 .13 .14 .15 ....
==>      --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- +...
==>  -8 :  1   1   1   1   1   1   1   1   2   2   2   2   2   2   2 |..1
==>  -8 :  1   1   1   1   1   1   1   1   2   2   2   2   2   2   2 |..2
==>  -8 :  1   1   1   1   1   1   1   1   2   2   2   2   2   2   2 |..3
==>  -8 :  1   1   1   1   1   1   1   1   2   2   2   2   2   2   2 |..4
==>  -8 :  1   1   1   1   1   1   1   1   2   2   2   2   2   2   2 |..5
==>  -8 :  1   1   1   1   1   1   1   1   2   2   2   2   2   2   2 |..6
==>  -8 :  1   1   1   1   1   1   1   1   2   2   2   2   2   2   2 |..7
==>  -7 :  -   -   -   -   -   -   -   -   1   1   1   1   1   1   1 |..8
==>  -7 :  -   -   -   -   -   -   -   -   1   1   1   1   1   1   1 |..9
==>      === === === === === === === === === === === === === === ===     
==>       -7  -7  -7  -7  -7  -7  -7  -7  -6  -6  -6  -6  -6  -6  -6     
==> d_3 :
==> Graded homomorphism: R(7)^8 + R(6)^7 <- R(6)^3 + R(5)^4 + R(4)^2, given b\
   y a matrix, with degrees: 
==>      ..1 ..2 ..3 ..4 ..5 ..6 ..7 ..8 ..9 ....
==>      --- --- --- --- --- --- --- --- --- +...
==>  -7 :  1   1   1   2   2   -   -   -   - |..1
==>  -7 :  1   1   1   2   2   -   -   -   - |..2
==>  -7 :  1   1   1   2   2   2   2   3   3 |..3
==>  -7 :  1   1   1   2   2   2   2   3   3 |..4
==>  -7 :  1   1   1   2   2   2   2   3   3 |..5
==>  -7 :  1   1   1   2   2   2   2   3   3 |..6
==>  -7 :  1   1   1   2   2   2   2   3   3 |..7
==>  -7 :  1   1   1   2   2   2   2   3   3 |..8
==>  -6 :  -   -   -   1   1   1   1   2   2 |..9
==>  -6 :  -   -   -   1   1   1   1   2   2 |.10
==>  -6 :  -   -   -   1   1   1   1   2   2 |.11
==>  -6 :  -   -   -   1   1   1   1   2   2 |.12
==>  -6 :  -   -   -   1   1   1   1   2   2 |.13
==>  -6 :  -   -   -   1   1   1   1   2   2 |.14
==>  -6 :  -   -   -   1   1   1   1   2   2 |.15
==>      === === === === === === === === ===     
==>       -6  -6  -6  -5  -5  -5  -5  -4  -4     
==> d_4 :
==> Graded homomorphism: R(6)^3 + R(5)^4 + R(4)^2 <- R, given by a matrix, wi\
   th degrees: 
==>      ..1 ....
==>      --- +...
==>  -6 :  6 |..1
==>  -6 :  6 |..2
==>  -6 :  6 |..3
==>  -5 :  5 |..4
==>  -5 :  5 |..5
==>  -5 :  5 |..6
==>  -5 :  5 |..7
==>  -4 :  4 |..8
==>  -4 :  4 |..9
==>      ===     
==>        0     
==> 
==> 
string Name = "rat.d10.g9.quart2"; int @p=31991; ring R = (@p),(x,y,z,u,v), dp;ideal I = 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+76/101xy3u+123/88y4u-5932x3zu-11857x2yzu+7128xy2zu-45/79y3zu+119/18x2z2u+9/74xyz2u+7042y2z2u-1114xz3u-11/82yz3u-1466z4u-6/85x3u2+27/106x2yu2+14246xy2u2-6216y3u2+47/6x2zu2-45/59xyzu2+89/41y2zu2+41/80xz2u2-7583yz2u2-75/113z3u2-14808x2u3-10873xyu3-90/67y2u3-11081xzu3-7369yzu3-7131z2u3-1402xu4-15386yu4-108/73zu4-5039u5+120/113x4v+10617x3yv-50/87x2y2v-2395xy3v-20/69y4v-8587x3zv+12960x2yzv-41/50xy2zv-13844y3zv-65/32x2z2v-77/122xyz2v-85/66y2z2v+13/100xz3v-20/51yz3v-13676z4v+76/97x3uv+1046x2yuv-8059xy2uv-117/59y3uv-29/105x2zuv+7287xyzuv-107/119y2zuv-35/118xz2uv+79/86yz2uv-2211z3uv+5448x2u2v+62/35xyu2v-2275y2u2v+29/121xzu2v-1674yzu2v-56/43z2u2v-3377xu3v-43/110yu3v+23/10zu3v-24/61u4v+121/53x3v2-4745x2yv2-57/64xy2v2+9554y3v2-12741x2zv2+10449xyzv2+37/108y2zv2+8621xz2v2-11/57yz2v2+1566z3v2+125/49x2uv2-121/118xyuv2+109/84y2uv2-335xzuv2+10167yzuv2-59/109z2uv2-103/119xu2v2+43/13yu2v2-73/87zu2v2+2037u3v2+13002x2v3+83/48xyv3-10713y2v3+1026xzv3-105/64yzv3-37/6z2v3+14779xuv3-6448yuv3+19/69zuv3-1/110u2v3+10010xv4+79/12yv4+12/19zv4-35/61uv4-11/57v5,-1592x3y2-75/121x2y3-40/19xy4+2651y5+39/121x3yz+122/77x2y2z-114/31xy3z+1544y4z+2/3x3z2-10271x2yz2-8373xy2z2+56/61y3z2+55/48x2z3-116xyz3-25/7y2z3-108/113xz4-34/53yz4+5548z5-122x3yu-9690x2y2u+43/87xy3u-5/19y4u+97/54x3zu-17/19x2yzu+4355xy2zu+12./examples/TestGRRes.sing  17> /5y3zu-1/100x2z2u+12754xyz2u+13600y2z2u+17/45xz3u-12091yz3u+5145z4u-63/64x3u2-84/31x2yu2-97/41xy2u2+7/13y3u2-79/62x2zu2-80/103xyzu2-69/14y2zu2+119/4xz2u2-35/87yz2u2-13840z3u2+14101x2u3+7952xyu3-1857y2u3-9861xzu3+3180yzu3+75/107z2u3-250xu4-15134yu4+4717zu4-2/41u5+22/27x3yv-8983x2y2v+10520xy3v-113/2y4v+10/73x3zv-1986x2yzv-110/13xy2zv+1550y3zv+32/111x2z2v-111/35xyz2v+101/98y2z2v+8045xz3v-2/89yz3v+2924z4v-79/11x3uv-15178x2yuv+10874xy2uv+54/11y3uv-8950x2zuv+70/53xyzuv-2403y2zuv-8249xz2uv+6935yz2uv+20/89z3uv+885x2u2v-76/71xyu2v-4/17y2u2v-31/52xzu2v-4/99yzu2v+10333z2u2v-93/104xu3v+82/101yu3v-71/37zu3v+9397u4v-15/112x3v2-6614x2yv2+119/2xy2v2+88/119y3v2+306x2zv2+2790xyzv2+10992y2zv2-115/74xz2v2-14711yz2v2+11612z3v2-1788x2uv2-75/97xyuv2+79/30y2uv2+99/59xzuv2-11439yzuv2-121/113z2uv2+108/37xu2v2+37/36yu2v2-3/65zu2v2-55/42u3v2+13/100x2v3-209xyv3-1272y2v3-117/68xzv3+63/94yzv3+32/59z2v3+1013xuv3-3463yuv3+6946zuv3-37/86u2v3+67/117xv4+85/28yv4-3024zv4-82/9uv4-32/65v5,-35/52x4y-12140x3y2+23/83x2y3+69/5xy4-80/79y5+120/43x4z-11865x3yz-3487x2y2z+53/59xy3z+53/102y4z-14083x3z2-14430x2yz2-2442xy2z2-33/104y3z2-91/38x2z3+4/87xyz3-26/57y2z3+4097xz4-9/122yz4+6364z5+9634x4u-97/95x3yu-46/99x2y2u+3847xy3u+121/106y4u+12765x3zu-5292x2yzu+1607xy2zu-67/121y3zu-12/35x2z2u+4/55xyz2u-17/27y2z2u+91/122xz3u-23/31yz3u+65/49z4u+73/46x3u2-124/27x2yu2-9933xy2u2+46/75y3u2+53/114x2zu2+3503xyzu2-14147y2zu2-11283xz2u2+11889yz2u2+99/104z3u2+3117x2u3+12624xyu3-10060y2u3+2193xzu3-80/47yzu3-77/13z2u3+11/31xu4-47/90yu4+49/48zu4-2/105u5-92/61x4v+7443x3yv+35/76x2y2v+114/67xy3v-73/126y4v+97/107x3zv+9464x2yzv+10869xy2zv+15718y3zv-37/33x2z2v+124/13xyz2v-11/26y2z2v-61/40xz3v+91/100yz3v-18/103z4v+60/29x3uv+21/125x2yuv-11117xy2uv+11748y3uv-16/117x2zuv+18/103xyzuv-1711y2zuv+1872xz2uv-109/123yz2uv-18/113z3uv-26/103x2u2v+14140xyu2v+11065y2u2v+8686xzu2v-5/111yzu2v+30/101z2u2v-10501xu3v-36/113yu3v-73/74zu3v+12753u4v-43/52x3v2-76/15x2yv2-5793xy2v2+18/13y3v2+1/79x2zv2+84/23xyzv2-172y2zv2+86/77xz2v2+15/37yz2v2+11835z3v2-6482x2uv2+94/113xyuv2+10727y2uv2-102/41xzuv2+15914yzuv2-12973z2uv2-9038xu2v2-13107yu2v2+1533zu2v2+12549u3v2-13528x2v3+903xyv3+23/114y2v3-123/64xzv3-81/5yzv3+111/103z2v3+4734xuv3-33/20yuv3-7954zuv3-2478u2v3+15518xv4-6723yv4-14/31zv4-3482uv4+10919v5,-3/94x4y-12936x3y2+2/11x2y3+32/23xy4-15921y5+61/93x4z+82/111x3yz-93/2x2y2z-6659xy3z-97/90y4z+402x3z2-14586x2yz2-121/39xy2z2+68/7y3z2+1212x2z3-2980xyz3+49/52y2z3-72/89xz4+92/47yz4+8478z5+2733x4u-103/89x3yu+1166x2y2u-7/53xy3u-106/23y4u+677x3zu+907x2yzu+7891xy2zu-9014y3zu+76/47x2z2u+49/116xyz2u-49/78y2z2u+12261xz3u+118/105yz3u-126/13z4u-8812x3u2-97/120x2yu2-9534xy2u2+92/5y3u2-54/71x2zu2+94/103xyzu2+2256y2zu2+4182xz2u2-5798yz2u2-31/115z3u2-73/98x2u3+15822xyu3+1004y2u3-578xzu3+9494yzu3-6779z2u3+14506xu4+10/121yu4+58/27zu4-2817u5-19/119x4v+7128x3yv+75/64x2y2v-65/109xy3v+5129y4v-53/55x3zv+54/125x2yzv-3009xy2zv+6144y3zv+15601x2z2v+123/55xyz2v-58/77y2z2v-56/61xz3v+121/10yz3v-103/86z4v-93/25x3uv+94/123x2yuv-25/107xy2uv+14807y3uv+65/7x2zuv+87/44xyzuv+6605y2zuv+23/99xz2uv-413yz2uv-17/15z3uv-79/46x2u2v+15240xyu2v-42/67y2u2v+8932xzu2v-5888yzu2v-4204z2u2v+7002xu3v-36/97yu3v-1634zu3v+61/102u4v-14/33x3v2-6520x2yv2+9004xy2v2-67/36y3v2-7/8x2zv2-24/11xyzv2-9373y2zv2+1556xz2v2-79/74yz2v2-6691z3v2+108x2uv2-76/61xyuv2+220y2uv2-1191xzuv2-4/9yzuv2+4546z2uv2+12205xu2v2+9/22yu2v2+64/93zu2v2-44/125u3v2+292x2v3+41/74xyv3+16/79y2v3-15892xzv3+5733yzv3+6796z2v3-42/55xuv3+71/79yuv3-19/104zuv3-38/15u2v3+6436xv4+28/15yv4+87/55zv4+2270uv4-30/41v5,-117/4x3y+97/122x2y2-3618xy3+6566y4+97/113x3z-12634x2yz+9865xy2z-1764y3z+114/31x2z2+5006xyz2+7/44y2z2-15040xz3+8/125yz3+11134z4-12980x3u-79/41x2yu-79/98xy2u+89/65y3u-1217x2zu+89/87xyzu+83/66y2zu+115/11xz2u+123/107yz2u+10920z3u-86/73x2u2-11/94xyu2-14054y2u2+6752xzu2-123/124yzu2+12129z2u2-13310xu3-52/63yu3+12847zu3-1545u4-11064x3v+11499x2yv-37/64xy2v+50/103y3v+123/94x2zv-126xyzv-111/44y2zv+95/14xz2v+113/83yz2v-77/103z3v+41/64x2uv+91/90xyuv-4932y2uv+103/31xzuv+62/63yzuv+1161z2uv-99/106xu2v-3181yu2v-11741zu2v-33/8u3v-3/118x2v2-9369xyv2+527y2v2-113/39xzv2-88/49yzv2-113/101z2v2+95/68xuv2-5930yuv2-20/43zuv2+7/41u2v2+109/93xv3-107/61yv3-8352zv3-5255uv3+12021v4,-2159x4-94/3x3y-4602x2y2+1609xy3+10721y4+28/9x3z-99/35x2yz+1/110xy2z+113/114y3z-118/75x2z2-103/93xyz2-68/67y2z2+13687xz3-1531yz3+61/107z4+6076x3u+9004x2yu+2211xy2u+110/53y3u+47/102x2zu+8495xyzu-9238y2zu+57/121xz2u-8543yz2u+8/19z3u-13527x2u2-13293xyu2+1138y2u2+26/115xzu2+78/53yzu2-12556z2u2+7299xu3+70/19yu3-14687zu3+13559u4+113/9x3v-85/126x2yv-83/3xy2v-3/46y3v+1814x2zv+28/79xyzv+103/51y2zv+78/31xz2v-14387yz2v+1/88z3v+116/75x2uv-101/59xyuv-70/3y2uv+109/71xzuv+13/88yzuv-147z2uv-113/76xu2v-9661yu2v+13855zu2v-6162u3v-1857x2v2-8208xyv2-4634y2v2-6178xzv2-7352yzv2-8247z2v2-113/15xuv2+99/40yuv2+21/97zuv2+11/37u2v2-6605xv3+8964yv3+35/121zv3+8543uv3-6008v4;TestGRRes(Name, I); kill R, Name, @p; "";
==> ==============================================
==> 
==> === Example: [ rat.d10.g9.quart2 ]
==>  = Ring:  (ZZ/31991),(x,y,z,u,v),(dp(5),C,L(1048575))
==>  ! Resolution via 'grres': 
==>  = Non-minimal betti numbers: 
==>            0     1     2     3     4
==> ------------------------------------
==>     0:     1     -     -     -     -
==>     1:     -     -     -     -     -
==>     2:     -     -     -     -     -
==>     3:     -     2     -     -     -
==>     4:     -     5     9     3     -
==>     5:     -     1     3     3     1
==> ------------------------------------
==> total:     1     8    12     6     1
==> 
==>  = Degrees of (ordered) maps: 
==> Graded resolution: 
==> R <-- d_1 --
==> R(-4)^2 + R(-5)^5 + R(-6) <-- d_2 --
==> R(-6)^9 + R(-7)^3 <-- d_3 --
==> R(-7)^3 + R(-8)^3 <-- d_4 --
==> R(-9), given by maps: 
==> d_1 :
==> Graded homomorphism: R <- R(-4)^2 + R(-5)^5 + R(-6), given by a matrix, w\
   ith degrees: 
==>     .1 .2 .3 .4 .5 .6 .7 .8 ...
==>     -- -- -- -- -- -- -- -- +..
==>  0 : 4  4  5  5  5  5  5  6 |.1
==>     == == == == == == == ==    
==>      4  4  5  5  5  5  5  6    
==> d_2 :
==> Graded homomorphism: R(-4)^2 + R(-5)^5 + R(-6) <- R(-6)^9 + R(-7)^3, give\
   n by a matrix, with degrees: 
==>      ..1 ..2 ..3 ..4 ..5 ..6 ..7 ..8 ..9 .10 .11 .12 ....
==>      --- --- --- --- --- --- --- --- --- --- --- --- +...
==>   4 :  2   2   2   2   2   2   2   2   2   -   3   3 |..1
==>   4 :  2   2   2   2   2   2   2   2   2   -   3   3 |..2
==>   5 :  1   1   1   1   1   1   1   1   1   2   2   2 |..3
==>   5 :  1   1   1   1   1   1   1   1   1   2   2   2 |..4
==>   5 :  1   1   1   1   1   1   1   1   1   2   2   2 |..5
==>   5 :  1   1   1   1   1   1   1   1   1   -   2   2 |..6
==>   5 :  1   1   1   1   1   1   1   1   1   -   2   2 |..7
==>   6 :  -   -   -   -   -   -   -   -   -   1   1   1 |..8
==>      === === === === === === === === === === === ===     
==>        6   6   6   6   6   6   6   6   6   7   7   7     
==> d_3 :
==> Graded homomorphism: R(-6)^9 + R(-7)^3 <- R(-7)^3 + R(-8)^3, given by a m\
   atrix, with degrees: 
==>      ..1 ..2 ..3 ..4 ..5 ..6 ....
==>      --- --- --- --- --- --- +...
==>   6 :  1   1   1   2   2   2 |..1
==>   6 :  1   1   1   2   2   2 |..2
==>   6 :  1   1   1   2   2   2 |..3
==>   6 :  1   1   1   2   2   2 |..4
==>   6 :  1   1   1   2   2   2 |..5
==>   6 :  1   1   1   2   2   2 |..6
==>   6 :  1   1   1   2   2   2 |..7
==>   6 :  1   1   1   2   2   2 |..8
==>   6 :  1   1   1   -   -   2 |..9
==>   7 :  -   -   -   1   1   1 |.10
==>   7 :  -   -   -   1   1   1 |.11
==>   7 :  -   -   -   1   1   1 |.12
==>      === === === === === ===     
==>        7   7   7   8   8   8     
==> d_4 :
==> Graded homomorphism: R(-7)^3 + R(-8)^3 <- R(-9), given by a matrix, with \
   degrees: 
==>     .1 ...
==>     -- +..
==>  7 : 2 |.1
==>  7 : 2 |.2
==>  7 : 2 |.3
==>  8 : 1 |.4
==>  8 : 1 |.5
==>  8 : 1 |.6
==>     ==    
==>      9    
==>  = TRANSPOSE'd complex: %%%%%%%%%%%%%%
==>            0     1     2     3     4
==> ------------------------------------
==>    -9:     1     3     3     1     -
==>    -8:     -     3     9     5     -
==>    -7:     -     -     -     2     -
==>    -6:     -     -     -     -     -
==>    -5:     -     -     -     -     -
==>    -4:     -     -     -     -     1
==> ------------------------------------
==> total:     1     6    12     8     1
==> 
==> Graded resolution: 
==> R(9) <-- d_1 --
==> R(8)^3 + R(7)^3 <-- d_2 --
==> R(7)^3 + R(6)^9 <-- d_3 --
==> R(6) + R(5)^5 + R(4)^2 <-- d_4 --
==> R, given by maps: 
==> d_1 :
==> Graded homomorphism: R(9) <- R(8)^3 + R(7)^3, given by a matrix, with deg\
   rees: 
==>      ..1 ..2 ..3 ..4 ..5 ..6 ....
==>      --- --- --- --- --- --- +...
==>  -9 :  1   1   1   2   2   2 |..1
==>      === === === === === ===     
==>       -8  -8  -8  -7  -7  -7     
==> d_2 :
==> Graded homomorphism: R(8)^3 + R(7)^3 <- R(7)^3 + R(6)^9, given by a matri\
   x, with degrees: 
==>      ..1 ..2 ..3 ..4 ..5 ..6 ..7 ..8 ..9 .10 .11 .12 ....
==>      --- --- --- --- --- --- --- --- --- --- --- --- +...
==>  -8 :  1   1   1   2   2   2   2   2   2   2   2   - |..1
==>  -8 :  1   1   1   2   2   2   2   2   2   2   2   - |..2
==>  -8 :  1   1   1   2   2   2   2   2   2   2   2   2 |..3
==>  -7 :  -   -   -   1   1   1   1   1   1   1   1   1 |..4
==>  -7 :  -   -   -   1   1   1   1   1   1   1   1   1 |..5
==>  -7 :  -   -   -   1   1   1   1   1   1   1   1   1 |..6
==>      === === === === === === === === === === === ===     
==>       -7  -7  -7  -6  -6  -6  -6  -6  -6  -6  -6  -6     
==> d_3 :
==> Graded homomorphism: R(7)^3 + R(6)^9 <- R(6) + R(5)^5 + R(4)^2, given by \
   a matrix, with degrees: 
==>      ..1 ..2 ..3 ..4 ..5 ..6 ..7 ..8 ....
==>      --- --- --- --- --- --- --- --- +...
==>  -7 :  1   2   2   2   -   -   -   - |..1
==>  -7 :  1   2   2   2   2   2   3   3 |..2
==>  -7 :  1   2   2   2   2   2   3   3 |..3
==>  -6 :  -   1   1   1   1   1   2   2 |..4
==>  -6 :  -   1   1   1   1   1   2   2 |..5
==>  -6 :  -   1   1   1   1   1   2   2 |..6
==>  -6 :  -   1   1   1   1   1   2   2 |..7
==>  -6 :  -   1   1   1   1   1   2   2 |..8
==>  -6 :  -   1   1   1   1   1   2   2 |..9
==>  -6 :  -   1   1   1   1   1   2   2 |.10
==>  -6 :  -   1   1   1   1   1   2   2 |.11
==>  -6 :  -   1   1   1   1   1   2   2 |.12
==>      === === === === === === === ===     
==>       -6  -5  -5  -5  -5  -5  -4  -4     
==> d_4 :
==> Graded homomorphism: R(6) + R(5)^5 + R(4)^2 <- R, given by a matrix, with\
    degrees: 
==>      ..1 ....
==>      --- +...
==>  -6 :  6 |..1
==>  -5 :  5 |..2
==>  -5 :  5 |..3
==>  -5 :  5 |..4
==>  -5 :  5 |..5
==>  -5 :  5 |..6
==>  -4 :  4 |..7
==>  -4 :  4 |..8
==>      ===     
==>        0     
==> 
==> 
//  if( defined(assumeLevel0) ){ assumeLevel = assumeLevel0; } else { kill assumeLevel; } // restore the state of aL