We compute the equation and nonminimal resolution F of the carpet of type (a,b) where a ≥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) -- 0.00155055 seconds elapsed -- 0.00536806 seconds elapsed -- 0.0250837 seconds elapsed -- 0.0103838 seconds elapsed -- 0.0029934 seconds elapsed -- 0.318577 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 , x , x , x , x , y , y , y , y , y , y ] 3 0 1 2 3 4 5 0 1 2 3 4 5 |
i4 : elapsedTime T'=minimalBetti J -- 0.20849 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); -- 0.00325242 seconds elapsed -- 0.0172002 seconds elapsed -- 0.112535 seconds elapsed -- 1.16468 seconds elapsed -- 0.365646 seconds elapsed -- 0.0406711 seconds elapsed -- 0.0053072 seconds elapsed -- 8.39963 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 |