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) -- 0.00188105 seconds elapsed -- 0.0218881 seconds elapsed -- 0.0259542 seconds elapsed -- 0.0108807 seconds elapsed -- 0.00337075 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 -- 0.220667 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); -- 0.00423404 seconds elapsed -- 0.0196693 seconds elapsed -- 0.141937 seconds elapsed -- 1.44799 seconds elapsed -- 0.501593 seconds elapsed -- 0.0456089 seconds elapsed -- 0.00647089 seconds elapsed -- 5.23453 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.