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)
-- 0.0018999 seconds elapsed
-- 0.016429 seconds elapsed
-- 0.0263101 seconds elapsed
-- 0.0104578 seconds elapsed
-- 0.00333138 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
-- 0.304778 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); -- 0.00421851 seconds elapsed -- 0.01846 seconds elapsed -- 0.144523 seconds elapsed -- 1.41178 seconds elapsed -- 0.508975 seconds elapsed -- 0.043318 seconds elapsed -- 0.00635353 seconds elapsed -- 7.10233 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.