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valRingIdeal -- valuation ideal

Synopsis

Description

A discrete monomial valuation v on R=K[X1,...,Xn] is determined by the values v(Xj) of the indeterminates. The function returns two ideals, both to be considered as lists of monomials. The first is the system of monomial generators of the subalgebra S={f∈R: vi(f)≥0, i=1,...,n} for several such valuations vi, i=1,...,r, the second the system of generators of the submodule M={f∈R: vi(f)≥wi, i=1,...,n} for integers w1,...,wr.
i1 : R=QQ[x,y,z,w]; 
i2 : V=matrix({{0,1,2,3,4},{-1,1,2,1,3}});

              2        5
o2 : Matrix ZZ  <--- ZZ
i3 : valRingIdeal(V,R)

                                      2                   2   2   2    2  
o3 = {ideal (y, x*y, w, x*w, z, x*z, x z), ideal (z*w, x*z , z , y w, y z,
     ------------------------------------------------------------------------
        2    4     4     2   3
     x*y z, y , x*y , y*w , w )}

o3 : List

Caveat

It is of course possible that S=K. At present, Normaliz cannot deal with the zero cone and will issue the (wrong) error message that the cone is not pointed.

See also

Ways to use valRingIdeal :