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)
-- 0.00311475 seconds elapsed
-- 0.00858654 seconds elapsed
-- 0.0339136 seconds elapsed
-- 0.0411616 seconds elapsed
-- 0.00404213 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
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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
-- 0.194453 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
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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);
-- 0.00452172 seconds elapsed
-- 0.0187834 seconds elapsed
-- 0.153709 seconds elapsed
-- 1.8903 seconds elapsed
-- 0.525033 seconds elapsed
-- 0.0393357 seconds elapsed
-- 0.00643079 seconds elapsed
-- 5.20718 seconds 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
|