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 : h=carpetBettiTables(a,b) -- 0.00118425 seconds elapsed -- 0.00162756 seconds elapsed -- 0.00189763 seconds elapsed -- 0.00179268 seconds elapsed -- 0.00138618 seconds elapsed 0 1 2 3 4 5 6 7 8 9 o2 = HashTable{0 => total: 1 36 209 516 786 786 516 209 36 1} 0: 1 . . . . . . . . . 1: . 36 160 342 436 350 174 49 . . 2: . . 49 174 350 436 342 160 36 . 3: . . . . . . . . . 1 o2 : HashTable |
i3 : T= carpetBettiTable(h,3) 0 1 2 3 4 5 6 7 8 9 o3 = total: 1 36 209 516 786 786 516 209 36 1 0: 1 . . . . . . . . . 1: . 36 160 342 436 350 174 49 . . 2: . . 49 174 350 436 342 160 36 . 3: . . . . . . . . . 1 o3 : BettiTally |
i4 : J=canonicalCarpet(a+b+1,b,Characteristic=>3); ZZ o4 : 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 |
i5 : elapsedTime T'=minimalBetti J -- 0.227086 seconds elapsed 0 1 2 3 4 5 6 7 8 9 o5 = 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 o5 : BettiTally |
i6 : T-T' 0 1 2 3 4 5 6 7 8 9 o6 = total: . . 49 201 484 484 201 49 . . 1: . . . 27 148 336 174 49 . . 2: . . 49 174 336 148 27 . . . 3: . . . . . . . . . . o6 : BettiTally |
i7 : elapsedTime h=carpetBettiTables(6,6); -- 0.00239531 seconds elapsed -- 0.00439149 seconds elapsed -- 0.00656188 seconds elapsed -- 0.00737626 seconds elapsed -- 0.00685538 seconds elapsed -- 0.00522131 seconds elapsed -- 0.00275713 seconds elapsed -- 68.6065 seconds elapsed |
i8 : keys h o8 = {0} o8 : List |
i9 : carpetBettiTable(h,7) 0 1 2 3 4 5 6 7 8 9 10 11 o9 = total: 1 55 401 1298 2675 3788 3788 2675 1298 401 55 1 0: 1 . . . . . . . . . . . 1: . 55 320 930 1688 2060 1728 987 368 81 . . 2: . . 81 368 987 1728 2060 1688 930 320 55 . 3: . . . . . . . . . . . 1 o9 : BettiTally |
i10 : carpetBettiTable(h,5) 0 1 2 3 4 5 6 7 8 9 10 11 o10 = total: 1 55 401 1298 2675 3788 3788 2675 1298 401 55 1 0: 1 . . . . . . . . . . . 1: . 55 320 930 1688 2060 1728 987 368 81 . . 2: . . 81 368 987 1728 2060 1688 930 320 55 . 3: . . . . . . . . . . . 1 o10 : BettiTally |