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



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 :

For the programmer

The object gfanInitialForms is a method function with options.