schreyerName -- get the names of generators in the (nonminimal) Schreyer resolution according to Schreyer's convention

Synopsis

Usage:

L = schreyerName(F,i,n)

h = schreyerName(F,a)

L = schreyerName(h,i,n)
Inputs:
- F, a chain complex, the nonminimal resolution from a Groebner basis
- i, an integer, the homological degree
- n, an integer, the number of the generator
- a, an integer, homological degree
- h, a hash table, of lists of names
Outputs:
- L, a list, list of monomials
- h, a hash table, of names

Description

We name the generators of the syzygies by the list of the monomial parts of the leadTerm with position m recursively:

schreyerName(F,i,n) = append(schreyerName(F,i-1,m),mon)

where mon denotes the monomial part and m the position in F_(i-1) of leadTerm F.dd_i_n.

i1 : (a,b)=(5,4)

o1 = (5, 4)

o1 : Sequence

i2 : I = carpet(a,b);

                ZZ
o2 : Ideal of -----[x ..x , y ..y ]
              32003  0   5   0   4

i3 : F = res(I, FastNonminimal =>true)

        ZZ                  1        ZZ                  28        ZZ                  127        ZZ                  286        ZZ                  385        ZZ                  323        ZZ                  166        ZZ                  48        ZZ                  6
o3 = (-----[x ..x , y ..y ])  <-- (-----[x ..x , y ..y ])   <-- (-----[x ..x , y ..y ])    <-- (-----[x ..x , y ..y ])    <-- (-----[x ..x , y ..y ])    <-- (-----[x ..x , y ..y ])    <-- (-----[x ..x , y ..y ])    <-- (-----[x ..x , y ..y ])   <-- (-----[x ..x , y ..y ])  <-- 0
      32003  0   5   0   4         32003  0   5   0   4          32003  0   5   0   4           32003  0   5   0   4           32003  0   5   0   4           32003  0   5   0   4           32003  0   5   0   4           32003  0   5   0   4          32003  0   5   0   4         
                                                                                                                                                                                                                                                                                      9
     0                            1                             2                              3                              4                              5                              6                              7                             8

o3 : ChainComplex

i4 : betti F

            0  1   2   3   4   5   6  7 8
o4 = total: 1 28 127 286 385 323 166 48 6
         0: 1  .   .   .   .   .   .  . .
         1: . 28 105 184 185 110  36  5 .
         2: .  .  22 101 195 203 120 38 5
         3: .  .   .   1   5  10  10  5 1

o4 : BettiTally

i5 : i=3,n=10

o5 = (3, 10)

o5 : Sequence

i6 : schreyerName(F,3,10)

o6 = {x x , x , x }
       3 4   3   1

o6 : List

i7 : h=schreyerName F;

i8 : h#8

              2                                   2                          
o8 = {{y y , x , y , y , x , x , x , x }, {y y , x , y , y , x , x , x , x },
        1 3   1   2   1   5   4   3   2     2 3   1   2   1   5   4   3   2  
     ------------------------------------------------------------------------
       2                                   2                                
     {y , x y , y , y , x , x , x , x }, {y , x y , y , y , x , x , x , x },
       3   2 0   2   1   4   3   2   1     3   3 0   2   1   4   3   2   1  
     ------------------------------------------------------------------------
       2                                   2         2
     {y , x y , y , y , x , x , x , x }, {y , x y , x , y , y , x , x , x }}
       3   4 0   2   1   4   3   2   1     3   5 0   1   2   1   4   3   2

o8 : List

i9 : h#7_20

       2
o9 = {y , x x , y , y , x , x , x }
       3   1 4   2   1   3   2   1

o9 : List

i10 : h#7_20 == schreyerName(F,7,20)

o10 = true

Ways to use schreyerName :

schreyerName(ChainComplex)
schreyerName(ChainComplex,ZZ)
schreyerName(ChainComplex,ZZ,ZZ)
schreyerName(HashTable,ZZ,ZZ)

For the programmer

The object schreyerName is a method function.