gfanInitialForms -- initial forms of polynomials with respect to a weight vector

Synopsis

Usage:

G = gfanInitialForms(L,W)

G = gfanInitialForms(I,W)

H = gfanInitialForms(M,W)
Inputs:
- L, a list, of polynomials
- I, an ideal,
- M, a marked polynomial list,
- W, a list, a weight vector
Optional inputs:
- ideal
- mark
- pair
Outputs:
- G, a list, initial forms of the polynomials in L or I with respect to weight W
- H, a marked polynomial list, marked initial forms of the polynomials in M with respect to weight W

Description

This method gives the initial forms of a list of polynomials. If the "ideals" option is used, generators for the initial ideal are given. If the "pair" option is used, then the output is a pair of MarkedPolynomialLists.

i1 : QQ[x,y,z]

o1 = QQ[x..z]

o1 : PolynomialRing

i2 : L = {x + y, x + z}

o2 = {x + y, x + z}

o2 : List

i3 : gfanInitialForms(L, {1,2,1})

o3 = {y, x + z}

o3 : List

i4 : gfanInitialForms(L, {1,2,1}, "ideal"=>true)

o4 = {x + z, y}

o4 : List

i5 : gfanInitialForms(L, {1,2,1}, "ideal"=>true, "pair"=>true)

o5 = {{x + z, y}, {x + z, y - z}}

o5 : List

i6 : gfanInitialForms({x*y+z, x*z + y}, {1,1,1}, "ideal"=>true)

            2    2
o6 = {x*z, y  - z , x*y}

o6 : List

gfan Documentation

This program converts a list of polynomials to a list of their initial forms with respect to the vector given after the list.
Options:
--ideal:
 Treat input as an ideal. This will make the program compute the initial ideal of the ideal generated by the input polynomials. The computation is done by computing a Groebner basis with respect to the given vector. The vector must be positive or the input polynomials must be homogeneous in a positive grading. None of these conditions are checked by the program.

--pair:
 Produce a pair of polynomial lists. Used together with --ideal this option will also write a compatible reduced Groebner basis for the input ideal to the output. This is useful for finding the Groebner cone of a non-monomial initial ideal.

--mark:
 If the --pair option is and the --ideal option is not used this option will still make sure that the second output basis is marked consistently with the vector.
--list:
 Read in a list of vectors instead of a single vector and produce a list of polynomial sets as output.

Ways to use gfanInitialForms :

gfanInitialForms(Ideal,List)
gfanInitialForms(List,List)
gfanInitialForms(MarkedPolynomialList,List)

For the programmer

The object gfanInitialForms is a method function with options.