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) -- 0.00195699 seconds elapsed -- 0.00590398 seconds elapsed -- 0.0253268 seconds elapsed -- 0.0101786 seconds elapsed -- 0.00317164 seconds elapsed -- 0.2305 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 -- 0.251921 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); -- 0.00393734 seconds elapsed -- 0.01807 seconds elapsed -- 0.140338 seconds elapsed -- 1.48087 seconds elapsed -- 0.49559 seconds elapsed -- 0.0398007 seconds elapsed -- 0.00553317 seconds elapsed -- 5.23962 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.