gfanHomogenize -- homogenize a list of polynomials with respect to a weight vector

Synopsis

Usage:

G = gfanHomogenize(L,X)

H = gfanHomogenize(M,X)

H = gfanHomogenize(I,X)
Inputs:
- L, a list, of polynomials
- M, a marked polynomial list,
- I, an ideal,
- X, a symbol, the homogenizing variable
Optional inputs:
- i
- w
Outputs:
- G, a list, polynomials from L homogenized with variable X
- H, a marked polynomial list, polynomials from M homogenized with variable X

Description

The method homogenizes the polynomials in L or M or I with respect to a given weight vector provided by the optional argument w. If the w option is not specified, the polynomials are homogenized with respect to total degree. This functionality is also provided by the homogenize method which does not use gfan.

i1 : QQ[x,y];

i2 : L = {x+y, x^2*y + x};

i3 : gfanHomogenize(L, symbol z, "w" => {2,3})

                  5    2
o3 = {x*z + y, x*z  + x y}

o3 : List

i4 : QQ[x,y,z];

i5 : L = {x+y, x^2*y + x};

i6 : homogenize(matrix{L}, z, {2,3,1})

o6 = | xz+y xz5+x2y |

                      1               2
o6 : Matrix (QQ[x..z])  <-- (QQ[x..z])

Using the variant that accepts a MarkedPolynomialList as input produces a MarkedPolynomialList as output.

i7 : QQ[x,y];

i8 : L = markedPolynomialList {{y}, {x+y}};

i9 : gfanHomogenize(L, symbol z,  "w" => {2,3})

o9 = {(y) + x*z}

o9 : MarkedPolynomialList

gfan Documentation

This program homogenises a list of polynomials by introducing an extra variable. The name of the variable to be introduced is read from the input after the list of polynomials. Without the -w option the homogenisation is done with respect to total degree.
Example:
Input:
Q[x,y]{y-1}
z
Output:
Q[x,y,z]{y-z}
Options:
-i:
 Treat input as an ideal. This will make the program compute the homogenisation of the input ideal. This is done by computing a degree Groebner basis and homogenising it.
-w:
 Specify a homogenisation vector. The length of the vector must be the same as the number of variables in the ring. The vector is read from the input after the list of polynomials.

-H:
 Let the name of the new variable be H rather than reading in a name from the input.

Ways to use gfanHomogenize :

gfanHomogenize(Ideal,Symbol)
gfanHomogenize(List,Symbol)
gfanHomogenize(MarkedPolynomialList,Symbol)

For the programmer

The object gfanHomogenize is a method function with options.