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 : elapsedTime T=carpetBettiTable(a,b,3) -- 0.00179291 seconds elapsed -- 0.0054125 seconds elapsed -- 0.0227534 seconds elapsed -- 0.00994822 seconds elapsed -- 0.0031337 seconds elapsed -- 0.215268 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 , y ..y ] 3 0 5 0 5 |
i4 : elapsedTime T'=minimalBetti J -- 0.192222 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); -- 0.00379012 seconds elapsed -- 0.0172964 seconds elapsed -- 0.143034 seconds elapsed -- 1.67195 seconds elapsed -- 0.476337 seconds elapsed -- 0.0375937 seconds elapsed -- 0.00562731 seconds elapsed -- 4.89777 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 |
The object carpetBettiTable is a method function.