We compute the equation and nonminimal resolution F of the carpet of type (a,b) where a ≥b over a larger finite prime field, lift the complex to the integers, which is possible since the coefficients are small. Finally we study the nonminimal strands over ZZ by computing the Smith normal form. The resulting data allow us to compute the Betti tables for arbitrary primes.
i1 : a=5,b=5 o1 = (5, 5) o1 : Sequence |
i2 : elapsedTime T=carpetBettiTable(a,b,3) -- 0.00177953 seconds elapsed -- 0.00580638 seconds elapsed -- 0.0273021 seconds elapsed -- 0.0114669 seconds elapsed -- 0.00335139 seconds elapsed -- 0.6607 seconds elapsed 0 1 2 3 4 5 6 7 8 9 o2 = total: 1 36 160 315 302 302 315 160 36 1 0: 1 . . . . . . . . . 1: . 36 160 315 288 14 . . . . 2: . . . . 14 288 315 160 36 . 3: . . . . . . . . . 1 o2 : BettiTally |
i3 : J=canonicalCarpet(a+b+1,b,Characteristic=>3); ZZ o3 : Ideal of --[x , x , x , x , x , x , y , y , y , y , y , y ] 3 0 1 2 3 4 5 0 1 2 3 4 5 |
i4 : elapsedTime T'=minimalBetti J -- 0.498834 seconds elapsed 0 1 2 3 4 5 6 7 8 9 o4 = total: 1 36 160 315 302 302 315 160 36 1 0: 1 . . . . . . . . . 1: . 36 160 315 288 14 . . . . 2: . . . . 14 288 315 160 36 . 3: . . . . . . . . . 1 o4 : BettiTally |
i5 : T-T' 0 1 2 3 4 5 6 7 8 9 o5 = total: . . . . . . . . . . 1: . . . . . . . . . . 2: . . . . . . . . . . 3: . . . . . . . . . . o5 : BettiTally |
i6 : elapsedTime h=carpetBettiTables(6,6); -- 0.00400062 seconds elapsed -- 0.0215324 seconds elapsed -- 0.131365 seconds elapsed -- 1.54211 seconds elapsed -- 0.622588 seconds elapsed -- 0.0465833 seconds elapsed -- 0.00579589 seconds elapsed -- 16.572 seconds elapsed |
i7 : carpetBettiTable(h,7) 0 1 2 3 4 5 6 7 8 9 10 11 o7 = total: 1 55 320 891 1408 1155 1155 1408 891 320 55 1 0: 1 . . . . . . . . . . . 1: . 55 320 891 1408 1155 . . . . . . 2: . . . . . . 1155 1408 891 320 55 . 3: . . . . . . . . . . . 1 o7 : BettiTally |
i8 : carpetBettiTable(h,5) 0 1 2 3 4 5 6 7 8 9 10 11 o8 = total: 1 55 320 891 1408 1275 1275 1408 891 320 55 1 0: 1 . . . . . . . . . . . 1: . 55 320 891 1408 1155 120 . . . . . 2: . . . . . 120 1155 1408 891 320 55 . 3: . . . . . . . . . . . 1 o8 : BettiTally |