constantStrands -- all constant strands of a chain complex

Synopsis

• Usage:
Cs = constantStrands(C, kk)
• Inputs:
• C, , A chain complex created using res(I, Strategy=>4.1)
• kk, a ring, if the coefficient ring of the ring of C is QQ, then this should be either: RR_{53}, RR_{1000}, ZZ/1073741891, or ZZ/1073741909.
• Outputs:
• Cs, a list, the list of chain complex over kk, which for each degree degree deg, consisting of the submatrices of C of degree deg

Description

Warning! This function is very rough currently. It works if one uses it in the intended manner, as in the example below. But it should be much more general, handling other rings with grace, and also it should handle arbitrary (graded) chain complexes.

 i1 : R = QQ[a..d] o1 = R o1 : PolynomialRing i2 : I = ideal(a^3, b^3, c^3, d^3, (a+3*b+7*c-4*d)^3) 3 3 3 3 3 2 2 3 2 o2 = ideal (a , b , c , d , a + 9a b + 27a*b + 27b + 21a c + 126a*b*c + ------------------------------------------------------------------------ 2 2 2 3 2 2 189b c + 147a*c + 441b*c + 343c - 12a d - 72a*b*d - 108b d - 168a*c*d ------------------------------------------------------------------------ 2 2 2 2 3 - 504b*c*d - 588c d + 48a*d + 144b*d + 336c*d - 64d ) o2 : Ideal of R i3 : C = res(ideal gens gb I, Strategy=>4.1) 1 9 25 31 18 4 o3 = R <-- R <-- R <-- R <-- R <-- R 0 1 2 3 4 5 o3 : ChainComplex i4 : betti C 0 1 2 3 4 5 o4 = total: 1 9 25 31 18 4 0: 1 . . . . . 1: . . . . . . 2: . 5 1 . . . 3: . 1 3 1 . . 4: . 3 17 13 4 . 5: . . 4 13 10 3 6: . . . 4 3 1 7: . . . . 1 . o4 : BettiTally i5 : Cs = constantStrands(C, RR_53) 1 o5 = HashTable{0 => RR <-- 0 <-- 0 <-- 0 <-- 0 <-- 0 } 53 1 2 3 4 5 0 5 3 => 0 <-- RR <-- 0 <-- 0 <-- 0 <-- 0 53 0 2 3 4 5 1 1 1 4 => 0 <-- RR <-- RR <-- 0 <-- 0 <-- 0 53 53 0 3 4 5 1 2 3 3 5 => 0 <-- RR <-- RR <-- 0 <-- 0 <-- 0 53 53 0 3 4 5 1 2 17 1 6 => 0 <-- 0 <-- RR <-- RR <-- 0 <-- 0 53 53 0 1 4 5 2 3 4 13 7 => 0 <-- 0 <-- RR <-- RR <-- 0 <-- 0 53 53 0 1 4 5 2 3 13 4 8 => 0 <-- 0 <-- 0 <-- RR <-- RR <-- 0 53 53 0 1 2 5 3 4 4 10 9 => 0 <-- 0 <-- 0 <-- RR <-- RR <-- 0 53 53 0 1 2 5 3 4 3 3 10 => 0 <-- 0 <-- 0 <-- 0 <-- RR <-- RR 53 53 0 1 2 3 4 5 1 1 11 => 0 <-- 0 <-- 0 <-- 0 <-- RR <-- RR 53 53 0 1 2 3 4 5 o5 : HashTable i6 : CR=Cs#8 13 4 o6 = 0 <-- 0 <-- 0 <-- RR <-- RR <-- 0 53 53 0 1 2 5 3 4 o6 : ChainComplex i7 : SVDBetti C, betti C 0 1 2 3 4 5 0 1 2 3 4 5 o7 = (total: 1 5 17 20 7 ., total: 1 9 25 31 18 4) 0: 1 . . . . . 0: 1 . . . . . 1: . . . . . . 1: . . . . . . 2: . 5 . . . . 2: . 5 1 . . . 3: . . . . . . 3: . 1 3 1 . . 4: . . 16 10 1 . 4: . 3 17 13 4 . 5: . . 1 10 6 . 5: . . 4 13 10 3 6: . . . . . . 6: . . . 4 3 1 7: . . . . . . 7: . . . . 1 . o7 : Sequence

Caveat

This function should be defined for any graded chain complex, not just ones created using res(I, Strategy=>4.1). Currently, it is used to extract information from the not yet implemented ring QQhybrid, whose elements, coming from QQ, are stored as real number approximations (as doubles, and as 1000 bit floating numbers), together with its remainders under a couple of primes, together with information about how many multiplications were performed to obtain this number.