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)
-- 0.00368967 seconds elapsed
-- 0.0290802 seconds elapsed
-- 0.0456356 seconds elapsed
-- 0.0193189 seconds elapsed
-- 0.00624501 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
-- 0.476416 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); -- 0.00801062 seconds elapsed -- 0.0332714 seconds elapsed -- 0.237558 seconds elapsed -- 2.08759 seconds elapsed -- 0.821166 seconds elapsed -- 0.0743017 seconds elapsed -- 0.0116593 seconds elapsed -- 10.3031 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.