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intersectionValRingIdeals -- intersection of valuation ideals



A discrete monomial valuation v on R=K[X_1,\ldots,X_n] is determined by the values v(X_j) of the indeterminates. This function takes as input the matrix V=(v_i(X_j)), whose rows correspond to the values of the indeterminates for for r valuations v_1, \ldots,v_r, with an additional column holding lower bounds w_1,\ldots,w_r \in \ZZ. It returns the subalgebra S=\{f\in R: v_i(f)\geq 0, i=1,\ldots,n\}, the intersection of the valuation rings of the r valuations, and a system of generators of the S-submodule M=\{f\in R: v_i(f)\geq w_i, i=1,\ldots,n\} over R, which consists of the elements whose i-th valuation is greater or equal to the i-th bound w_i. If w_i>=0 for all i, then M is an ideal in S.

This method can be used with the options allComputations and grading. The additional data can be accessed via the subalgebra in the HashTable.
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)

                                        3        2     2   2    2    4     2     2      4
o3 = HashTable{"module generators" => {w , z*w, z , y*w , y w, y z, y , x*z , x*y z, x*y }    }
               "subalgebra" => MonomialSubalgebra{cache => CacheTable{...1...}               }
                                                  generators => {w, z, y, x*w, x*z, x*y, x z}
                                                  ring => R

o3 : HashTable

See also

Ways to use intersectionValRingIdeals :

For the programmer

The object intersectionValRingIdeals is a method function with options.