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) -- 0.0021099 seconds elapsed -- 0.00673504 seconds elapsed -- 0.0281943 seconds elapsed -- 0.0115969 seconds elapsed -- 0.00349515 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 -- 0.229968 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); -- 0.00462002 seconds elapsed -- 0.0222237 seconds elapsed -- 0.155129 seconds elapsed -- 1.67583 seconds elapsed -- 0.539048 seconds elapsed -- 0.0507745 seconds elapsed -- 0.00784779 seconds elapsed -- 5.8532 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.