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)
-- .00371911s elapsed
-- .00878603s elapsed
-- .031343s elapsed
-- .0146609s elapsed
-- .00478567s 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
-- .271393s 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);
-- .00578965s elapsed
-- .0211494s elapsed
-- .128763s elapsed
-- 1.18327s elapsed
-- .512136s elapsed
-- .0475802s elapsed
-- .00870115s elapsed
-- 7.73468s 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
|