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
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i2 : h=carpetBettiTables(a,b)
-- .00263931s elapsed
-- .0067657s elapsed
-- .0228892s elapsed
-- .0179541s elapsed
-- .00752605s 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
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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
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i4 : J=canonicalCarpet(a+b+1,b,Characteristic=>3);
ZZ
o4 : Ideal of --[x ..x , y ..y ]
3 0 5 0 5
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i5 : elapsedTime T'=minimalBetti J
-- .147958s 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
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i6 : T-T'
0 1 2 3 4 5 6 7 8 9
o6 = total: . . . . . . . . . .
1: . . . . . . . . . .
2: . . . . . . . . . .
3: . . . . . . . . . .
o6 : BettiTally
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i7 : elapsedTime h=carpetBettiTables(6,6);
-- .0124662s elapsed
-- .0171248s elapsed
-- .102548s elapsed
-- 1.22637s elapsed
-- .673584s elapsed
-- .0455392s elapsed
-- .00671012s elapsed
-- 7.67619s elapsed
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i8 : keys h
o8 = {0, 2, 3, 5}
o8 : List
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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
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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
|