The minimal resolution of a module over a complete intersection has betti numbers that grow as a polynomial of degree at most equal to the codimension-1. The complexity is one more than the degree of this polynomial.
i1 : setRandomSeed 0
o1 = 0
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i2 : S = ZZ/101[a,b,c,d];
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i3 : ff1 = matrix"a3,b3,c3,d3";
1 4
o3 : Matrix S <--- S
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i4 : ff =ff1*random(source ff1, source ff1);
1 4
o4 : Matrix S <--- S
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i5 : R = S/ideal ff;
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i6 : M = highSyzygy (R^1/ideal"a2b2");
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i7 : complexity M
o7 = 2
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i8 : mf = matrixFactorization (ff, M)
o8 = {{7} | a -36b 0 0 |, {8} | -35a2 -48b -43a2 0 0 |}
{6} | -b2 a2 7b2 0 | {8} | -35b2 -35a 0 0 0 |
{7} | 0 0 a -b | {8} | 0 0 -35a2 -48b 0 |
{8} | 0 0 -48b2 -48a 33b2 |
o8 : List
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i9 : complexity mf
o9 = 2
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i10 : betti res (R^1/ideal"a2b2", LengthLimit=>10)
0 1 2 3 4 5 6 7 8 9 10
o10 = total: 1 1 2 3 4 5 6 7 8 9 10
0: 1 . . . . . . . . . .
1: . . . . . . . . . . .
2: . . . . . . . . . . .
3: . 1 2 1 . . . . . . .
4: . . . 2 4 2 . . . . .
5: . . . . . 3 6 3 . . .
6: . . . . . . . 4 8 4 .
7: . . . . . . . . . 5 10
o10 : BettiTally
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