quotient' -- factor a map through another with the same source

Usage:

h = quotient'(f, g)

h = g \\ f
Inputs:
- f, a matrix,
- g, a matrix, with the same source as f.;
Optional inputs:
- BasisElementLimit (missing documentation) => ..., default value infinity,
- DegreeLimit (missing documentation) => ..., default value {},
- MinimalGenerators (missing documentation) => ..., default value true,
- PairLimit (missing documentation) => ..., default value infinity,
- Strategy (missing documentation) => ..., default value null,
Outputs:
- h, a matrix, the quotient of f upon (opposite) division by g, as a map from target of g to target of f.

Description

This function produces a matrix h such that f - h*g is the reduction of f modulo a Gröbner basis for the image of dual g. See quotient for the dual notion.

If the remainder f - h*g is zero, then the quotient g\f satisfies the equation f === (g\f) * g. Otherwise, the equation h * g + r === f will hold, where r is the map provided by remainder'.

i1 : S = QQ[x,y,z]

o1 = S

o1 : PolynomialRing

i2 : K = koszul vars S

      1      3      3      1
o2 = S  <-- S  <-- S  <-- S
                           
     0      1      2      3

o2 : ChainComplex

i3 : f = K.dd_2

o3 = {1} | -y -z 0  |
     {1} | x  0  -z |
     {1} | 0  x  y  |

             3      3
o3 : Matrix S  <-- S

i4 : g = inducedMap(coker K.dd_3, K_2)

o4 = {2} | 1 0 0 |
     {2} | 0 1 0 |
     {2} | 0 0 1 |

o4 : Matrix

i5 : h = quotient'(f, g)

o5 = {1} | -y -z 0  |
     {1} | x  0  -z |
     {1} | 0  x  y  |

o5 : Matrix

i6 : h * g == f

o6 = true

i7 : (g \\ f) * g == f

o7 = true

Note, however, that currently the function remainder' only works if the source and target modules are free.

i8 : R = ZZ[x,y]

o8 = R

o8 : PolynomialRing

i9 : f = random(R^{2:1}, R^2)

o9 = {-1} | 8x+y  8x+3y |
     {-1} | 3x+7y 3x+7y |

             2      2
o9 : Matrix R  <-- R

i10 : f = f + id_(R^2)

o10 = {-1} | 8x+y+1 8x+3y   |
      {-1} | 3x+7y  3x+7y+1 |

              2      2
o10 : Matrix R  <-- R

i11 : g = transpose(vars R ++ vars R)

o11 = {-1} | x 0 |
      {-1} | y 0 |
      {-1} | 0 x |
      {-1} | 0 y |

              4      2
o11 : Matrix R  <-- R

i12 : h = quotient'(f, g)

o12 = {-1} | 8 1 8 3 |
      {-1} | 3 7 3 7 |

              2      4
o12 : Matrix R  <-- R

i13 : r = remainder'(f, g)

o13 = {-1} | 1 0 |
      {-1} | 0 1 |

              2      2
o13 : Matrix R  <-- R

i14 : h * g + r == f

o14 = true

Ways to use quotient':

quotient'(Matrix,Matrix)

For the programmer

The object quotient' is a method function with options.

The source of this document is in Macaulay2Doc/functions/quotient-remainder-doc.m2:183:0.

quotient' -- factor a map through another with the same source

Description

See also

Ways to use quotient':

For the programmer