submatrixByDegrees -- submatrix consisting of rows and columns in an interval or box of degrees

Synopsis

Usage:

submatrixByDegrees(m, targetBox, sourceBox)
Inputs:
- m, a matrix, A matrix between free modules
- targetBox, a sequence, A sequence of 2 degree vectors, (lodegree, hidegree). All rows whose multi-degree is greater than or equal (in each component) to lodegree, and less than or equal to hidegree, are selected
- sourceBox, a sequence, Same as targetBox, except this governs which columns are selected
Outputs:
- a matrix, The submatrix of m consisting of those rows (respectively columns) whose (multi-)degree lie in the box targetBox, (respectively columnBox)

Description

If only one degree (as integer, or list of integers) is given for targetBox or sourceBox, then only rows or columns that match that exact degree are used.

i1 : R = QQ[a..d];

i2 : I = ideal"a2b-c3,abc-d3,ac2-bd2-cd2,abcd-c4"

             2     3           3     2      2      2     4
o2 = ideal (a b - c , a*b*c - d , a*c  - b*d  - c*d , - c  + a*b*c*d)

o2 : Ideal of R

i3 : C = res I;

i4 : m = C.dd_2

o4 = {3} | ac2-bd2-cd2 0           -abc+d3 0        -a3c+a2cd          
     {3} | 0           ac2-bd2-cd2 a2b-c3  -acd+cd2 a4-a3d-acd2+cd3    
     {3} | -a2b+c3     -abc+d3     0       abd-bd2  -a2c2+ac2d-acd2+cd3
     {4} | 0           0           0       b2+bc-cd a3-bcd-c2d-cd2     
     ------------------------------------------------------------------------
     -a3c+a2cd                         
     a4-a3d+a2cd-3acd2-ad3+2cd3+d4     
     -a2c2-a2bd+ac2d+2abd2-acd2-bd3+cd3
     a3-ab2+b2d+acd-c2d-2cd2-d3        
     ------------------------------------------------------------------------
     -a3c+a2cd                                
     a4-a3d+a2cd-3acd2-ad3+2cd3+d4            
     -a2c2-a2bd+ac2d+2abd2-acd2-ad3-bd3+cd3+d4
     a3-ab2+ac2+b2d+acd-c2d-bd2-3cd2-d3       
     ------------------------------------------------------------------------
     -a2bc+a2cd-acd2+d4                        a2cd-acd2            |
     a3b-a3d+abcd+ac2d+a2d2-2bcd2-c2d2-bd3-cd3 -a3d+ac2d+a2d2-c2d2  |
     -abc2-ab2d-abcd+ac2d+b2d2-c2d2+cd3+d4     -abcd+ac2d+bcd2-c2d2 |
     a2b-b3-b2c+bc2-a2d+bcd                    -b2c+c3-a2d+c2d      |

             4      9
o4 : Matrix R  <-- R

i5 : submatrixByDegrees(m, 3, 6)

o5 = {3} | ac2-bd2-cd2 0           -abc+d3 0        |
     {3} | 0           ac2-bd2-cd2 a2b-c3  -acd+cd2 |
     {3} | -a2b+c3     -abc+d3     0       abd-bd2  |

             3      4
o5 : Matrix R  <-- R

i6 : submatrixByDegrees(m, (3,3), (6,7))

o6 = {3} | ac2-bd2-cd2 0           -abc+d3 0        -a3c+a2cd          
     {3} | 0           ac2-bd2-cd2 a2b-c3  -acd+cd2 a4-a3d-acd2+cd3    
     {3} | -a2b+c3     -abc+d3     0       abd-bd2  -a2c2+ac2d-acd2+cd3
     ------------------------------------------------------------------------
     -a3c+a2cd                         
     a4-a3d+a2cd-3acd2-ad3+2cd3+d4     
     -a2c2-a2bd+ac2d+2abd2-acd2-bd3+cd3
     ------------------------------------------------------------------------
     -a3c+a2cd                                
     a4-a3d+a2cd-3acd2-ad3+2cd3+d4            
     -a2c2-a2bd+ac2d+2abd2-acd2-ad3-bd3+cd3+d4
     ------------------------------------------------------------------------
     -a2bc+a2cd-acd2+d4                        a2cd-acd2            |
     a3b-a3d+abcd+ac2d+a2d2-2bcd2-c2d2-bd3-cd3 -a3d+ac2d+a2d2-c2d2  |
     -abc2-ab2d-abcd+ac2d+b2d2-c2d2+cd3+d4     -abcd+ac2d+bcd2-c2d2 |

             3      9
o6 : Matrix R  <-- R

i7 : submatrixByDegrees(m, (4,4), (7,7))

o7 = {4} | a3-bcd-c2d-cd2 a3-ab2+b2d+acd-c2d-2cd2-d3
     ------------------------------------------------------------------------
     a3-ab2+ac2+b2d+acd-c2d-bd2-3cd2-d3 a2b-b3-b2c+bc2-a2d+bcd
     ------------------------------------------------------------------------
     -b2c+c3-a2d+c2d |

             1      5
o7 : Matrix R  <-- R

For multidegrees, the interval is a box.

i8 : S = QQ[a..d, Degrees=>{2:{1,0},2:{0,1}}];

i9 : I = ideal(a*d^4, b^3, a^2*d^2, a*b*c*d^3)

               4   3   2 2         3
o9 = ideal (a*d , b , a d , a*b*c*d )

o9 : Ideal of S

i10 : C = res I

       1      4      6      4      1
o10 = S  <-- S  <-- S  <-- S  <-- S  <-- 0
                                          
      0      1      2      3      4      5

o10 : ChainComplex

i11 : m = C.dd_2

o11 = {3, 0} | 0   -a2d2 0    0   -acd3 -ad4 |
      {2, 2} | -d2 b3    -bcd 0   0     0    |
      {1, 4} | a   0     0    -bc 0     b3   |
      {2, 4} | 0   0     a    d   b2    0    |

              4      6
o11 : Matrix S  <-- S

i12 : degrees target m

o12 = {{3, 0}, {2, 2}, {1, 4}, {2, 4}}

o12 : List

i13 : degrees source m

o13 = {{2, 4}, {5, 2}, {3, 4}, {2, 5}, {4, 4}, {4, 4}}

o13 : List

i14 : submatrixByDegrees(C.dd_2, ({2,2},{2,4}), ({2,2},{5,4}))

o14 = {2, 2} | -d2 b3 -bcd 0  0 |
      {2, 4} | 0   0  a    b2 0 |

              2      5
o14 : Matrix S  <-- S

Caveat

The degrees are taken from the target and source free modules, not from the matrix entries themselves.

Ways to use submatrixByDegrees :

submatrixByDegrees(Matrix,List,List)
submatrixByDegrees(Matrix,Sequence,Sequence)
submatrixByDegrees(Matrix,ZZ,ZZ)

For the programmer