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CompleteIntersectionResolutions :: makeHomotopies

makeHomotopies -- returns a system of higher homotopies

Synopsis

Description

Given a 1×n matrix f, and a chain complex F, the script attempts to make a family of higher homotopies on F for the elements of f, in the sense described, for example, in Eisenbud "Enriched Free Resolutions and Change of Rings".

The output is a hash table with entries of the form {J,i}=>s, where J is a list of non-negative integers, of length n and H#{J,i}: Fi->Fi+2|J|-1 are maps satisfying the conditions

H#{e0,i}= d;

and

H#{e0,i+1}*H#{e,i}+H#{e,i-1}H#{e0,i}= fi,

where e0 = {0,…,0} and e is the index of degree 1 with a 1 in the i-th place; and, for each index list I with |I|<=d,

sumJ<I H#{I\J, }H#{J, }= 0.

i1 : kk=ZZ/101

o1 = kk

o1 : QuotientRing
i2 : S = kk[a,b,c,d]

o2 = S

o2 : PolynomialRing
i3 : F = res ideal vars S

      1      4      6      4      1
o3 = S  <-- S  <-- S  <-- S  <-- S  <-- 0
                                         
     0      1      2      3      4      5

o3 : ChainComplex
i4 : f = matrix{{a,b,c}}

o4 = | a b c |

             1       3
o4 : Matrix S  <--- S
i5 : homot = makeHomotopies(f,F,2)

o5 = HashTable{{{0, 0, 0}, 0} => 0                        }
               {{0, 0, 0}, 1} => | a b c d |
               {{0, 0, 0}, 2} => {1} | -b -c 0  -d 0  0  |
                                 {1} | a  0  -c 0  -d 0  |
                                 {1} | 0  a  b  0  0  -d |
                                 {1} | 0  0  0  a  b  c  |
               {{0, 0, 0}, 3} => {2} | c  d  0  0  |
                                 {2} | -b 0  d  0  |
                                 {2} | a  0  0  d  |
                                 {2} | 0  -b -c 0  |
                                 {2} | 0  a  0  -c |
                                 {2} | 0  0  a  b  |
               {{0, 0, 1}, -1} => 0
               {{0, 0, 1}, 0} => {1} | 0 |
                                 {1} | 0 |
                                 {1} | 1 |
                                 {1} | 0 |
               {{0, 0, 1}, 1} => {2} | 0  0  0 0 |
                                 {2} | -1 0  0 0 |
                                 {2} | 0  -1 0 0 |
                                 {2} | 0  0  0 0 |
                                 {2} | 0  0  0 0 |
                                 {2} | 0  0  0 1 |
               {{0, 0, 1}, 2} => {3} | 1 0 0 0  0  0 |
                                 {3} | 0 0 0 0  0  0 |
                                 {3} | 0 0 0 -1 0  0 |
                                 {3} | 0 0 0 0  -1 0 |
               {{0, 0, 2}, -1} => 0
               {{0, 1, 0}, -1} => 0
               {{0, 1, 0}, 0} => {1} | 0 |
                                 {1} | 1 |
                                 {1} | 0 |
                                 {1} | 0 |
               {{0, 1, 0}, 1} => {2} | -1 0 0 0 |
                                 {2} | 0  0 0 0 |
                                 {2} | 0  0 1 0 |
                                 {2} | 0  0 0 0 |
                                 {2} | 0  0 0 1 |
                                 {2} | 0  0 0 0 |
               {{0, 1, 0}, 2} => {3} | 0 -1 0 0  0 0 |
                                 {3} | 0 0  0 -1 0 0 |
                                 {3} | 0 0  0 0  0 0 |
                                 {3} | 0 0  0 0  0 1 |
               {{0, 1, 1}, -1} => 0
               {{0, 2, 0}, -1} => 0
               {{1, 0, 0}, -1} => 0
               {{1, 0, 0}, 0} => {1} | 1 |
                                 {1} | 0 |
                                 {1} | 0 |
                                 {1} | 0 |
               {{1, 0, 0}, 1} => {2} | 0 1 0 0 |
                                 {2} | 0 0 1 0 |
                                 {2} | 0 0 0 0 |
                                 {2} | 0 0 0 1 |
                                 {2} | 0 0 0 0 |
                                 {2} | 0 0 0 0 |
               {{1, 0, 0}, 2} => {3} | 0 0 1 0 0 0 |
                                 {3} | 0 0 0 0 1 0 |
                                 {3} | 0 0 0 0 0 1 |
                                 {3} | 0 0 0 0 0 0 |
               {{1, 0, 1}, -1} => 0
               {{1, 1, 0}, -1} => 0
               {{2, 0, 0}, -1} => 0

o5 : HashTable
In this case the higher homotopies are 0:
i6 : L = sort select(keys homot, k->(homot#k!=0 and sum(k_0)>1))

o6 = {}

o6 : List
On the other hand, if we take a complete intersection and something contained in it in a more complicated situation, the program gives nonzero higher homotopies:
i7 : kk= ZZ/32003;
i8 : S = kk[a,b,c,d];
i9 : M = S^1/(ideal"a2,b2,c2,d2");
i10 : F = res M

       1      4      6      4      1
o10 = S  <-- S  <-- S  <-- S  <-- S  <-- 0
                                          
      0      1      2      3      4      5

o10 : ChainComplex
i11 : setRandomSeed 0

o11 = 0
i12 : f = random(S^1,S^{2:-5});

              1       2
o12 : Matrix S  <--- S
i13 : homot = makeHomotopies(f,F,5)

o13 = HashTable{{{0, 0}, 0} => 0                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          }
                {{0, 0}, 1} => | a2 b2 c2 d2 |
                {{0, 0}, 2} => {2} | -b2 -c2 0   -d2 0   0   |
                               {2} | a2  0   -c2 0   -d2 0   |
                               {2} | 0   a2  b2  0   0   -d2 |
                               {2} | 0   0   0   a2  b2  c2  |
                {{0, 0}, 3} => {4} | c2  d2  0   0   |
                               {4} | -b2 0   d2  0   |
                               {4} | a2  0   0   d2  |
                               {4} | 0   -b2 -c2 0   |
                               {4} | 0   a2  0   -c2 |
                               {4} | 0   0   a2  b2  |
                {{0, 0}, 4} => {6} | -d2 |
                               {6} | c2  |
                               {6} | -b2 |
                               {6} | a2  |
                {{0, 0}, 5} => 0
                {{0, 0}, 6} => 0
                {{0, 1}, -1} => 0
                {{0, 1}, 0} => {2} | 2960a3-15112a2b+2471ab2+12669b3-916a2c+7483abc-9205b2c+3008a2d+9234abd+6807b2d-10690acd+7488bcd                                                                   |
                               {2} | -7000ab2-15389b3+12240abc+14672b2c+1137abd-15456b2d+13885acd-11834bcd                                                                                             |
                               {2} | -2398a3-14934a2b-5595ab2-4436b3-9923a2c-3320abc+12237b2c+4215ac2-13156bc2+3376c3+10399a2d-8638abd+13700b2d-1674acd+1001bcd+7381c2d                                |
                               {2} | 2756a3+9794a2b+13485ab2-15948b3+5338a2c+10525abc+9304b2c-2904ac2-5320bc2-10749c3+5799a2d-11359abd+11510b2d-14115acd+2682bcd+5611c2d-15547ad2-171bd2-977cd2-9562d3 |
                {{0, 1}, 1} => {4} | 7000ab2+15389b3-12240abc-14672b2c+5595ac2+4436bc2-12237c3-1137abd+15456b2d-13885acd+11834bcd-13700c2d-13485ad2+15948bd2-9304cd2-11510d3 2960a3-15112a2b+2471ab2+12669b3-916a2c+7483abc-9205b2c+3008a2d+9234abd+6807b2d-10690acd+7488bcd                                                                   0                                                                                                                                                                  0                                                                                                                                  |
                               {4} | 2398a3+14934a2b+9923a2c+3320abc-4215ac2+13156bc2-3376c3-10399a2d+8638abd+1674acd-1001bcd-7381c2d+2904ad2+5320bd2+10749cd2-5611d3        0                                                                                                                                                                 2960a3-15112a2b+2471ab2+12669b3-916a2c+7483abc-9205b2c+3008a2d+9234abd+6807b2d-10690acd+7488bcd                                                                    0                                                                                                                                  |
                               {4} | 5595a3+4436a2b-12237a2c-13700a2d                                                                                                        2398a3+14934a2b+5595ab2+4436b3+9923a2c+3320abc-12237b2c-4215ac2+13156bc2-3376c3-10399a2d+8638abd-13700b2d+1674acd-1001bcd-7381c2d+2904ad2+5320bd2+10749cd2-5611d3 -7000ab2-15389b3+12240abc+14672b2c+1137abd-15456b2d+13885acd-11834bcd                                                                                              0                                                                                                                                  |
                               {4} | -2756a3-9794a2b-5338a2c-10525abc-5799a2d+11359abd+14115acd-2682bcd+15547ad2+171bd2+977cd2+9562d3                                        0                                                                                                                                                                 0                                                                                                                                                                  2960a3-15112a2b+2471ab2+12669b3-916a2c+7483abc-9205b2c+3008a2d+9234abd+6807b2d-10690acd+7488bcd                                    |
                               {4} | -13485a3+15948a2b-9304a2c-11510a2d                                                                                                      -2756a3-9794a2b-13485ab2+15948b3-5338a2c-10525abc-9304b2c-5799a2d+11359abd-11510b2d+14115acd-2682bcd+15547ad2+171bd2+977cd2+9562d3                                0                                                                                                                                                                  -7000ab2-15389b3+12240abc+14672b2c+1137abd-15456b2d+13885acd-11834bcd                                                              |
                               {4} | 2904a3+5320a2b+10749a2c-5611a2d                                                                                                         2904ab2+5320b3+10749b2c-5611b2d                                                                                                                                   -2756a3-9794a2b-13485ab2+15948b3-5338a2c-10525abc-9304b2c+2904ac2+5320bc2+10749c3-5799a2d+11359abd-11510b2d+14115acd-2682bcd-5611c2d+15547ad2+171bd2+977cd2+9562d3 -2398a3-14934a2b-5595ab2-4436b3-9923a2c-3320abc+12237b2c+4215ac2-13156bc2+3376c3+10399a2d-8638abd+13700b2d-1674acd+1001bcd+7381c2d |
                {{0, 1}, 2} => {6} | -2398a3-14934a2b-9923a2c-3320abc+4215ac2-13156bc2+3376c3+10399a2d-8638abd-1674acd+1001bcd+7381c2d-2904ad2-5320bd2-10749cd2+5611d3 7000ab2+15389b3-12240abc-14672b2c+5595ac2+4436bc2-12237c3-1137abd+15456b2d-13885acd+11834bcd-13700c2d+15948bd2-9304cd2-11510d3 2960a3-15112a2b+2471ab2+12669b3-916a2c+7483abc-9205b2c+3008a2d+9234abd+6807b2d-10690acd+7488bcd                                   5595ad2                                                                                                                                  0                                                                                                                                                                 0                                                                                               |
                               {6} | 2756a3+9794a2b+5338a2c+10525abc+5799a2d-11359abd-14115acd+2682bcd-15547ad2-171bd2-977cd2-9562d3                                   -13485ac2                                                                                                                      0                                                                                                                                 7000ab2+15389b3-12240abc-14672b2c+4436bc2-12237c3-1137abd+15456b2d-13885acd+11834bcd-13700c2d-13485ad2+15948bd2-9304cd2-11510d3          2960a3-15112a2b+2471ab2+12669b3-916a2c+7483abc-9205b2c+3008a2d+9234abd+6807b2d-10690acd+7488bcd                                                                   0                                                                                               |
                               {6} | 0                                                                                                                                 2756a3+9794a2b+13485ab2+5338a2c+10525abc+5799a2d-11359abd-14115acd+2682bcd-15547ad2-171bd2-977cd2-9562d3                       0                                                                                                                                 2398a3+14934a2b+5595ab2+9923a2c+3320abc-4215ac2+13156bc2-3376c3-10399a2d+8638abd+1674acd-1001bcd-7381c2d+2904ad2+5320bd2+10749cd2-5611d3 0                                                                                                                                                                 2960a3-15112a2b+2471ab2+12669b3-916a2c+7483abc-9205b2c+3008a2d+9234abd+6807b2d-10690acd+7488bcd |
                               {6} | 0                                                                                                                                 -15948a2b+9304a2c+11510a2d                                                                                                     2756a3+9794a2b+13485ab2-15948b3+5338a2c+10525abc+9304b2c+5799a2d-11359abd+11510b2d-14115acd+2682bcd-15547ad2-171bd2-977cd2-9562d3 4436a2b-12237a2c-13700a2d                                                                                                                2398a3+14934a2b+5595ab2+4436b3+9923a2c+3320abc-12237b2c-4215ac2+13156bc2-3376c3-10399a2d+8638abd-13700b2d+1674acd-1001bcd-7381c2d+2904ad2+5320bd2+10749cd2-5611d3 -7000ab2-15389b3+12240abc+14672b2c+1137abd-15456b2d+13885acd-11834bcd                           |
                {{0, 1}, 3} => {8} | -2756a3-9794a2b-13485ab2-5338a2c-10525abc-5799a2d+11359abd+14115acd-2682bcd+15547ad2+171bd2+977cd2+9562d3 -2398a3-14934a2b-5595ab2-9923a2c-3320abc+4215ac2-13156bc2+3376c3+10399a2d-8638abd-1674acd+1001bcd+7381c2d-2904ad2-5320bd2-10749cd2+5611d3 7000ab2+15389b3-12240abc-14672b2c+4436bc2-12237c3-1137abd+15456b2d-13885acd+11834bcd-13700c2d+15948bd2-9304cd2-11510d3 2960a3-15112a2b+2471ab2+12669b3-916a2c+7483abc-9205b2c+3008a2d+9234abd+6807b2d-10690acd+7488bcd |
                {{0, 1}, 4} => 0
                {{0, 1}, 5} => 0
                {{0, 2}, -1} => 0
                {{0, 2}, 0} => {6} | 15649a4+9616a3b+9402a2b2+12740ab3+2416b4+1336a3c-11327a2bc+3537ab2c-5573b3c+8042a2c2-8888abc2-8975b2c2-8017a3d-15385a2bd+6253ab2d+3992b3d+1635a2cd+5683abcd-11119b2cd-14734ac2d-6067bc2d+7456a2d2-9757abd2+80b2d2-12033acd2+15712bcd2-8731c2d2-14470ad3-10751bd3+1577cd3+14498d4 |
                               {6} | -7859a4-8129a3b+2105a2b2+2364ab3+10273b4+13845a3c-4927a2bc+9458ab2c-8906b3c+12474a2c2-3491abc2+4054b2c2-12698ac3+306bc3+8731c4-1484a3d+5085a2bd-12308ab2d-10662b3d+11501a2cd-4692abcd+4657b2cd+8906ac2d+12930bc2d-1577c3d+5166a2d2-1377abd2-5226b2d2-9635acd2-2801bcd2-14498c2d2 |
                               {6} | -12967a4-7411a3b+2243a3c+271a2bc+10840a2c2+11228abc2-590a3d+15567a2bd-3695a2cd-2571abcd+15963ac2d+967bc2d-12307a2d2+883abd2+7968acd2+4771bcd2                                                                                                                                    |
                               {6} | 2808a4+11816a3b-5333a2b2-1976ab3-5806b4-12698a3c-10067a2bc-1757ab2c+6869b3c+8731a2c2+3427abc2-1456b2c2+14470a3d-15713a2bd-6426ab2d-6628b3d-3317a2cd+6827abcd+3177b2cd-12127ac2d+8259bc2d-14498a2d2-11110abd2-4514b2d2-13433acd2-5651bcd2                                         |
                {{0, 2}, 1} => {8} | 15185a4-2004a3b-11428a2b2-3952ab3-11612b4-12499a3c+15415a2bc-3514ab2c+13738b3c+10678a2c2-9177abc2-2912b2c2-14909ac3+2202a3d+1669a2bd-12852ab2d-13256b3d-6853a2cd+3256abcd+6354b2cd+11204ac2d-15485bc2d-4577a2d2-8349abd2-9028b2d2+2657acd2-11302bcd2-13134ad3 2808a2b2+11816ab3-11428b4-12698ab2c+11563b3c+8731b2c2+14470ab2d-10174b3d-5057b2cd-14498b2d2 -1655a3b-8417a2b2-11815ab3-10273b4-14743a3c+2131a2bc+12333ab2c+8906b3c-6858a2c2-4880abc2-14720b2c2-12698ac3+11563bc3+8731c4-13595a3d-1902a2bd+4517ab2d+10662b3d+1637a2cd-1523abcd-4657b2cd-11969ac2d-12353bc2d-5057c3d-5166a2d2+1377abd2+5226b2d2+9635acd2+2801bcd2-14498c2d2 9330a3b+9453a2b2-15673ab3+2416b4+5876a3c+13215a2bc+12685ab2c-5573b3c+8042a2c2-8888abc2-8975b2c2-4199a3d+5230a2bd+4658ab2d+3992b3d-1422a2cd+2250abcd-11119b2cd-14734ac2d-6067bc2d+13072a2d2+13875abd2-10586b2d2-5426acd2-4422bcd2+8731c2d2+14470ad3-10174bd3-5057cd3-14498d4 |
                {{0, 2}, 2} => 0
                {{0, 3}, -1} => 0
                {{0, 3}, 0} => 0
                {{0, 4}, -1} => 0
                {{0, 5}, -1} => 0
                {{1, 0}, -1} => 0
                {{1, 0}, 0} => {2} | -2136a3+9349a2b-9489ab2-11250b3+8735a2c+13529abc-14212b2c-5609a2d-15802abd-1270b2d-545acd+626bcd                                                                  |
                               {2} | 13163ab2-9750b3+15956abc-12767b2c-10830abd-13006b2d+2547acd-653bcd                                                                                                |
                               {2} | -371a3-1415a2b-12600ab2-6717b3-3327a2c+11236abc+12829b2c+12268ac2+13129bc2-5914c3-4035a2d+9276abd+12697b2d+8860acd+9526bcd-9500c2d                                |
                               {2} | -2519a3+1414a2b+12981ab2+1573b3+11874a2c+3209abc+200b2c-14107ac2+8794bc2+14443c3-13874a2d+2110abd+8468b2d-4550acd+6286bcd-7217c2d-6827ad2+9265bd2+11558cd2-6490d3 |
                {{1, 0}, 1} => {4} | -13163ab2+9750b3-15956abc+12767b2c+12600ac2+6717bc2-12829c3+10830abd+13006b2d-2547acd+653bcd-12697c2d-12981ad2-1573bd2-200cd2-8468d3 -2136a3+9349a2b-9489ab2-11250b3+8735a2c+13529abc-14212b2c-5609a2d-15802abd-1270b2d-545acd+626bcd                                                                   0                                                                                                                                                                0                                                                                                                                  |
                               {4} | 371a3+1415a2b+3327a2c-11236abc-12268ac2-13129bc2+5914c3+4035a2d-9276abd-8860acd-9526bcd+9500c2d+14107ad2-8794bd2-14443cd2+7217d3     0                                                                                                                                                                  -2136a3+9349a2b-9489ab2-11250b3+8735a2c+13529abc-14212b2c-5609a2d-15802abd-1270b2d-545acd+626bcd                                                                 0                                                                                                                                  |
                               {4} | 12600a3+6717a2b-12829a2c-12697a2d                                                                                                    371a3+1415a2b+12600ab2+6717b3+3327a2c-11236abc-12829b2c-12268ac2-13129bc2+5914c3+4035a2d-9276abd-12697b2d-8860acd-9526bcd+9500c2d+14107ad2-8794bd2-14443cd2+7217d3 13163ab2-9750b3+15956abc-12767b2c-10830abd-13006b2d+2547acd-653bcd                                                                                               0                                                                                                                                  |
                               {4} | 2519a3-1414a2b-11874a2c-3209abc+13874a2d-2110abd+4550acd-6286bcd+6827ad2-9265bd2-11558cd2+6490d3                                     0                                                                                                                                                                  0                                                                                                                                                                -2136a3+9349a2b-9489ab2-11250b3+8735a2c+13529abc-14212b2c-5609a2d-15802abd-1270b2d-545acd+626bcd                                   |
                               {4} | -12981a3-1573a2b-200a2c-8468a2d                                                                                                      2519a3-1414a2b-12981ab2-1573b3-11874a2c-3209abc-200b2c+13874a2d-2110abd-8468b2d+4550acd-6286bcd+6827ad2-9265bd2-11558cd2+6490d3                                    0                                                                                                                                                                13163ab2-9750b3+15956abc-12767b2c-10830abd-13006b2d+2547acd-653bcd                                                                 |
                               {4} | 14107a3-8794a2b-14443a2c+7217a2d                                                                                                     14107ab2-8794b3-14443b2c+7217b2d                                                                                                                                   2519a3-1414a2b-12981ab2-1573b3-11874a2c-3209abc-200b2c+14107ac2-8794bc2-14443c3+13874a2d-2110abd-8468b2d+4550acd-6286bcd+7217c2d+6827ad2-9265bd2-11558cd2+6490d3 -371a3-1415a2b-12600ab2-6717b3-3327a2c+11236abc+12829b2c+12268ac2+13129bc2-5914c3-4035a2d+9276abd+12697b2d+8860acd+9526bcd-9500c2d |
                {{1, 0}, 2} => {6} | -371a3-1415a2b-3327a2c+11236abc+12268ac2+13129bc2-5914c3-4035a2d+9276abd+8860acd+9526bcd-9500c2d-14107ad2+8794bd2+14443cd2-7217d3 -13163ab2+9750b3-15956abc+12767b2c+12600ac2+6717bc2-12829c3+10830abd+13006b2d-2547acd+653bcd-12697c2d-1573bd2-200cd2-8468d3 -2136a3+9349a2b-9489ab2-11250b3+8735a2c+13529abc-14212b2c-5609a2d-15802abd-1270b2d-545acd+626bcd                                 12600ad2                                                                                                                                  0                                                                                                                                                                  0                                                                                                |
                               {6} | -2519a3+1414a2b+11874a2c+3209abc-13874a2d+2110abd-4550acd+6286bcd-6827ad2+9265bd2+11558cd2-6490d3                                 -12981ac2                                                                                                                   0                                                                                                                                -13163ab2+9750b3-15956abc+12767b2c+6717bc2-12829c3+10830abd+13006b2d-2547acd+653bcd-12697c2d-12981ad2-1573bd2-200cd2-8468d3               -2136a3+9349a2b-9489ab2-11250b3+8735a2c+13529abc-14212b2c-5609a2d-15802abd-1270b2d-545acd+626bcd                                                                   0                                                                                                |
                               {6} | 0                                                                                                                                 -2519a3+1414a2b+12981ab2+11874a2c+3209abc-13874a2d+2110abd-4550acd+6286bcd-6827ad2+9265bd2+11558cd2-6490d3                  0                                                                                                                                371a3+1415a2b+12600ab2+3327a2c-11236abc-12268ac2-13129bc2+5914c3+4035a2d-9276abd-8860acd-9526bcd+9500c2d+14107ad2-8794bd2-14443cd2+7217d3 0                                                                                                                                                                  -2136a3+9349a2b-9489ab2-11250b3+8735a2c+13529abc-14212b2c-5609a2d-15802abd-1270b2d-545acd+626bcd |
                               {6} | 0                                                                                                                                 1573a2b+200a2c+8468a2d                                                                                                      -2519a3+1414a2b+12981ab2+1573b3+11874a2c+3209abc+200b2c-13874a2d+2110abd+8468b2d-4550acd+6286bcd-6827ad2+9265bd2+11558cd2-6490d3 6717a2b-12829a2c-12697a2d                                                                                                                 371a3+1415a2b+12600ab2+6717b3+3327a2c-11236abc-12829b2c-12268ac2-13129bc2+5914c3+4035a2d-9276abd-12697b2d-8860acd-9526bcd+9500c2d+14107ad2-8794bd2-14443cd2+7217d3 13163ab2-9750b3+15956abc-12767b2c-10830abd-13006b2d+2547acd-653bcd                               |
                {{1, 0}, 3} => {8} | 2519a3-1414a2b-12981ab2-11874a2c-3209abc+13874a2d-2110abd+4550acd-6286bcd+6827ad2-9265bd2-11558cd2+6490d3 -371a3-1415a2b-12600ab2-3327a2c+11236abc+12268ac2+13129bc2-5914c3-4035a2d+9276abd+8860acd+9526bcd-9500c2d-14107ad2+8794bd2+14443cd2-7217d3 -13163ab2+9750b3-15956abc+12767b2c+6717bc2-12829c3+10830abd+13006b2d-2547acd+653bcd-12697c2d-1573bd2-200cd2-8468d3 -2136a3+9349a2b-9489ab2-11250b3+8735a2c+13529abc-14212b2c-5609a2d-15802abd-1270b2d-545acd+626bcd |
                {{1, 0}, 4} => 0
                {{1, 0}, 5} => 0
                {{1, 1}, -1} => 0
                {{1, 1}, 0} => {6} | -1296a4-4551a3b+191a2b2+15481ab3-10627b4-11274a3c+1939a2bc-4676ab2c+3269b3c+6088a2c2+6739abc2+7817b2c2-9307a3d+7026a2bd+14808ab2d+8480b3d-2399a2cd-14900abcd+8651b2cd-10307ac2d-2191bc2d-5044a2d2-7209abd2+14125b2d2+7356acd2+14081bcd2+7116c2d2+7ad3+5479bd3+5728cd3+9185d4              |
                               {6} | 13103a4+13659a3b-9052a2b2-15750ab3+3350b4+12586a3c+15450a2bc+15597ab2c-10997b3c+4506a2c2+10732abc2-4252b2c2+8652ac3-7700bc3-7116c4+11213a3d+7108a2bd-13748ab2d+13859b3d+11755a2cd+10617abcd-11890b2cd+7998ac2d+1304bc2d-5728c3d+12183a2d2-2212abd2-11944b2d2-13205acd2-15026bcd2-9185c2d2 |
                               {6} | -1557a4+15698a3b-13724a3c+2848a2bc+8262a2c2-1249abc2-841a3d+14765a2bd-3829a2cd+2677abcd+11542ac2d-2803bc2d-8151a2d2-3759abd2-4790acd2-4327bcd2                                                                                                                                            |
                               {6} | -3856a4+9208a3b+12895a2b2+13208ab3-2958b4+8652a3c+1675a2bc+7449ab2c+10021b3c-7116a2c2+9219abc2+12551b2c2-7a3d+9306a2bd-12211ab2d+4009b3d+15402a2cd+10971abcd+9320b2cd+4881ac2d+12002bc2d-9185a2d2+6394abd2-5671b2d2-11227acd2-6333bcd2                                                    |
                {{1, 1}, 1} => {8} | -13982a4-6941a3b+6920a2b2-5587ab3-5916b4-829a3c+666a2bc+14898ab2c-11961b3c-6858a2c2+12379abc2-6901b2c2+12935ac3+12682a3d+10217a2bd+7581ab2d+8018b3d-8921a2cd-14325abcd-13363b2cd+6050ac2d-7999bc2d-858a2d2+14733abd2-11342b2d2+3359acd2-12666bcd2-12612ad3 -3856a2b2+9208ab3+6920b4+8652ab2c+11050b3c-7116b2c2-7ab2d-7912b3d+4529b2cd-9185b2d2 -2622a3b+11178a2b2+2905ab3-3350b4-15538a3c-12630a2bc-9838ab2c+10997b3c-12218a2c2+7684abc2-1961b2c2+8652ac3+11050bc3-7116c4-125a3d-8431a2bd-15294ab2d-13859b3d+10765a2cd-12052abcd+11890b2cd-8012ac2d-14695bc2d+4529c3d-12183a2d2+2212abd2+11944b2d2+13205acd2+15026bcd2-9185c2d2 6049a3b+2488a2b2+10894ab3-10627b4+5502a3c-10683a2bc-11863ab2c+3269b3c+6088a2c2+6739abc2+7817b2c2+860a3d+9435a2bd+15632ab2d+8480b3d-36a2cd-996abcd+8651b2cd-10307ac2d-2191bc2d-12756a2d2+11207abd2+7912b2d2-7343acd2-14572bcd2-7116c2d2-7ad3-7912bd3+4529cd3-9185d4 |
                {{1, 1}, 2} => 0
                {{1, 2}, -1} => 0
                {{1, 2}, 0} => 0
                {{1, 3}, -1} => 0
                {{1, 4}, -1} => 0
                {{2, 0}, -1} => 0
                {{2, 0}, 0} => {6} | 923a4-15811a3b+9155a2b2+3650ab3-7167b4+10859a3c+5838a2bc-1670ab2c-4335b3c+13191a2c2-11272abc2+4657b2c2+3475a3d+4218a2bd+1955ab2d-5931b3d+10862a2cd+1683abcd+599b2cd+15151ac2d+1799bc2d+2971a2d2-2385abd2-15549b2d2+3200acd2+7293bcd2+3326c2d2-15859ad3+306bd3-5974cd3+12732d4     |
                               {6} | 12818a4-4380a3b+12565a2b2-12643ab3-1409b4+8755a3c-11979a2bc-822ab2c-4831b3c-11011a2c2-3976abc2+5360b2c2+4017ac3+13132bc3-3326c4+9557a3d-15249a2bd-12141ab2d+5593b3d-5370a2cd-4869abcd-10696b2cd-3153ac2d+2506bc2d+5974c3d+4560a2d2+6793abd2+1352b2d2+6628acd2+11530bcd2-12732c2d2 |
                               {6} | 14274a4+203a3b+12451a3c+4374a2bc-3779a2c2+10971abc2-4813a3d+413a2bd-1816a2cd+2967abcd-1303ac2d+15531bc2d+3642a2d2+15658abd2+3104acd2+5419bcd2                                                                                                                                     |
                               {6} | 7827a4+13572a3b-11741a2b2+5073ab3+5463b4+4017a3c-4678a2bc+9517ab2c+12524b3c-3326a2c2+1095abc2-7505b2c2+15859a3d+3206a2bd+8116ab2d+4889b3d-2866a2cd+3459abcd-809b2cd-14874ac2d-9606bc2d-12732a2d2-8784abd2+497b2d2+11977acd2-8260bcd2                                              |
                {{2, 0}, 1} => {8} | -98a4+5908a3b-2706a2b2+10146ab3+10926b4-13348a3c+7968a2bc-12969ab2c-6955b3c+654a2c2+13764abc2-15010b2c2+5563ac3+14421a3d+14695a2bd-15771ab2d+9778b3d+548a2cd+507abcd-1618b2cd-9686ac2d+12791bc2d-10669a2d2+6704abd2+994b2d2-11893acd2+15483bcd2+14572ad3 7827a2b2+13572ab3-2706b4+4017ab2c+9515b3c-3326b2c2+15859ab2d+6718b3d-11706b2cd-12732b2d2 387a3b-15403a2b2-12799ab3+1409b4-11161a3c-7509a2bc-10501ab2c+4831b3c-5338a2c2-883abc2+3161b2c2+4017ac3+9515bc3-3326c4-5953a3d+1781a2bd-15537ab2d-5593b3d+7352a2cd+7525abcd+10696b2cd+2868ac2d+3906bc2d-11706c3d-4560a2d2-6793abd2-1352b2d2-6628acd2-11530bcd2-12732c2d2 -10168a3b+7347a2b2+12363ab3-7167b4+8176a3c-11584a2bc+12745ab2c-4335b3c+13191a2c2-11272abc2+4657b2c2+14251a3d-13248a2bd+2455ab2d-5931b3d-2783a2cd-13179abcd+599b2cd+15151ac2d+1799bc2d-13378a2d2-7244abd2-7028b2d2+11234acd2-2063bcd2-3326c2d2+15859ad3+6718bd3-11706cd3-12732d4 |
                {{2, 0}, 2} => 0
                {{2, 1}, -1} => 0
                {{2, 1}, 0} => 0
                {{2, 2}, -1} => 0
                {{2, 3}, -1} => 0
                {{3, 0}, -1} => 0
                {{3, 0}, 0} => 0
                {{3, 1}, -1} => 0
                {{3, 2}, -1} => 0
                {{4, 0}, -1} => 0
                {{4, 1}, -1} => 0
                {{5, 0}, -1} => 0

o13 : HashTable
We can see that all 6 potential higher homotopies are nontrivial:
i14 : L = sort select(keys homot, k->(homot#k!=0 and sum(k_0)>1))

o14 = {{{0, 2}, 0}, {{0, 2}, 1}, {{1, 1}, 0}, {{1, 1}, 1}, {{2, 0}, 0}, {{2,
      -----------------------------------------------------------------------
      0}, 1}}

o14 : List
i15 : #L

o15 = 6
i16 : netList L

      +------+-+
o16 = |{0, 2}|0|
      +------+-+
      |{0, 2}|1|
      +------+-+
      |{1, 1}|0|
      +------+-+
      |{1, 1}|1|
      +------+-+
      |{2, 0}|0|
      +------+-+
      |{2, 0}|1|
      +------+-+
For example we have:
i17 : homot#(L_0)

o17 = {6} | 15649a4+9616a3b+9402a2b2+12740ab3+2416b4+1336a3c-11327a2bc+
      {6} | -7859a4-8129a3b+2105a2b2+2364ab3+10273b4+13845a3c-4927a2bc+
      {6} | -12967a4-7411a3b+2243a3c+271a2bc+10840a2c2+11228abc2-590a3d
      {6} | 2808a4+11816a3b-5333a2b2-1976ab3-5806b4-12698a3c-10067a2bc-
      -----------------------------------------------------------------------
      3537ab2c-5573b3c+8042a2c2-8888abc2-8975b2c2-8017a3d-15385a2bd+6253ab2d+
      9458ab2c-8906b3c+12474a2c2-3491abc2+4054b2c2-12698ac3+306bc3+8731c4-148
      +15567a2bd-3695a2cd-2571abcd+15963ac2d+967bc2d-12307a2d2+883abd2+7968ac
      1757ab2c+6869b3c+8731a2c2+3427abc2-1456b2c2+14470a3d-15713a2bd-6426ab2d
      -----------------------------------------------------------------------
      3992b3d+1635a2cd+5683abcd-11119b2cd-14734ac2d-6067bc2d+7456a2d2-9757abd
      4a3d+5085a2bd-12308ab2d-10662b3d+11501a2cd-4692abcd+4657b2cd+8906ac2d+1
      d2+4771bcd2                                                            
      -6628b3d-3317a2cd+6827abcd+3177b2cd-12127ac2d+8259bc2d-14498a2d2-11110a
      -----------------------------------------------------------------------
      2+80b2d2-12033acd2+15712bcd2-8731c2d2-14470ad3-10751bd3+1577cd3+14498d4
      2930bc2d-1577c3d+5166a2d2-1377abd2-5226b2d2-9635acd2-2801bcd2-14498c2d2
                                                                             
      bd2-4514b2d2-13433acd2-5651bcd2                                        
      -----------------------------------------------------------------------
      |
      |
      |
      |

              4       1
o17 : Matrix S  <--- S
But all the homotopies are minimal in this case:
i18 : k1 = S^1/(ideal vars S);
i19 : select(keys homot,k->(k1**homot#k)!=0)

o19 = {}

o19 : List

See also

Ways to use makeHomotopies :