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# 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,
• 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, , 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 .