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Normaliz :: intersectionValRingIdeals

intersectionValRingIdeals -- intersection of valuation ideals

Synopsis

Description

A discrete monomial valuation v on R=K[X1,...,Xn] is determined by the values v(Xj) of the indeterminates. This function takes as input the matrix V=(vi(Xj)), whose rows correspond to the values of the indeterminates for for r valuations v1, ...,vr, with an additional column holding lower bounds w1,...,wr ∈ℤ. It returns the subalgebra S={f∈R: vi(f)≥0, i=1,...,n}, the intersection of the valuation rings of the r valuations, and a system of generators of the S-submodule M={f∈R: vi(f)≥wi, i=1,...,n} over R, which consists of the elements whose i-th valuation is greater or equal to the i-th bound wi. If wi>=0 for all i, then M is an ideal in S.

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 : intersectionValRingIdeals(V,R)

                                           2     2   2    2      2    4     4     2   3
o3 = HashTable{module generators => {z*w, z , x*z , y w, y z, x*y z, y , x*y , y*w , w }    }
               subalgebra => MonomialSubalgebra{cache => CacheTable{}                      }
                                                                                        2
                                                generators => {y, x*y, w, x*w, z, x*z, x z}
                                                ring => R

o3 : HashTable

See also

Ways to use intersectionValRingIdeals :