We compute the equation and nonminimal resolution F of the carpet of type (a,b) where $a \ge b$ 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) -- 0.00353076 seconds elapsed -- 0.028139 seconds elapsed -- 0.0440783 seconds elapsed -- 0.01781 seconds elapsed -- 0.00593351 seconds elapsed 0 1 2 3 4 5 6 7 8 9 o2 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1} 0: 1 . . . . . . . . . 1: . 36 160 315 288 . . . . . 2: . . . . . 288 315 160 36 . 3: . . . . . . . . . 1 0 1 2 3 4 5 6 7 8 9 2 => total: 1 36 167 370 476 476 370 167 36 1 0: 1 . . . . . . . . . 1: . 36 160 322 336 140 48 7 . . 2: . . 7 48 140 336 322 160 36 . 3: . . . . . . . . . 1 0 1 2 3 4 5 6 7 8 9 3 => 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 : HashTable |
i3 : T= carpetBettiTable(h,3) 0 1 2 3 4 5 6 7 8 9 o3 = 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 o3 : BettiTally |
i4 : J=canonicalCarpet(a+b+1,b,Characteristic=>3); ZZ o4 : Ideal of --[x ..x , y ..y ] 3 0 5 0 5 |
i5 : elapsedTime T'=minimalBetti J -- 0.402018 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: . . . . . . . . . . 1: . . . . . . . . . . 2: . . . . . . . . . . 3: . . . . . . . . . . o6 : BettiTally |
i7 : elapsedTime h=carpetBettiTables(6,6); -- 0.00714377 seconds elapsed -- 0.0321474 seconds elapsed -- 0.237285 seconds elapsed -- 2.56999 seconds elapsed -- 0.950979 seconds elapsed -- 0.075164 seconds elapsed -- 0.0110542 seconds elapsed -- 11.2692 seconds elapsed |
i8 : keys h o8 = {0, 2, 3, 5} o8 : List |
i9 : carpetBettiTable(h,7) 0 1 2 3 4 5 6 7 8 9 10 11 o9 = 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 o9 : BettiTally |
i10 : carpetBettiTable(h,5) 0 1 2 3 4 5 6 7 8 9 10 11 o10 = 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 o10 : BettiTally |
The object carpetBettiTables is a method function.