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 : elapsedTime T=carpetBettiTable(a,b,3)
-- 0.00103129 seconds elapsed
-- 0.00309179 seconds elapsed
-- 0.0590814 seconds elapsed
-- 0.00547621 seconds elapsed
-- 0.00165633 seconds elapsed
-- 0.212756 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
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i3 : J=canonicalCarpet(a+b+1,b,Characteristic=>3);
ZZ
o3 : Ideal of --[x ..x , y ..y ]
3 0 5 0 5
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i4 : elapsedTime T'=minimalBetti J
-- 0.0966655 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
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i5 : T-T'
0 1 2 3 4 5 6 7 8 9
o5 = total: . . . . . . . . . .
1: . . . . . . . . . .
2: . . . . . . . . . .
3: . . . . . . . . . .
o5 : BettiTally
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i6 : elapsedTime h=carpetBettiTables(6,6);
-- 0.00234554 seconds elapsed
-- 0.00953667 seconds elapsed
-- 0.135429 seconds elapsed
-- 1.28342 seconds elapsed
-- 0.271165 seconds elapsed
-- 0.0267541 seconds elapsed
-- 0.00317246 seconds elapsed
-- 2.69221 seconds elapsed
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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
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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
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