SVDBetti -- the Betti table computed with SVD methods

Synopsis

Usage:

SVDBetti C
Inputs:
- C, a chain complex, A chain complex created using res(I, Strategy=>4.1) 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:
- a Betti tally, the betti table of the minimal resolution using SVD of complexes and the numerical data

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 : SVDBetti C, betti C

             0 1  2  3 4 5         0 1  2  3  4 5
o4 = (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 .

o4 : Sequence

i5 : Rp=ZZ/32003[gens R]

o5 = Rp

o5 : PolynomialRing

i6 : betti res sub(I,Rp)

            0 1  2  3 4
o6 = total: 1 5 17 20 7
         0: 1 .  .  . .
         1: . .  .  . .
         2: . 5  .  . .
         3: . .  .  . .
         4: . . 16 10 1
         5: . .  1 10 6

o6 : BettiTally

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.

Ways to use `SVDBetti` :

"SVDBetti(ChainComplex)"

For the programmer

The object SVDBetti is a method function.