intersectionValRingIdeals -- intersection of valuation ideals

Synopsis

• Usage:

  intersectionValRingIdeals(v,r)

• Inputs:
  • a matrix, values of the indeterminates, the last column contains the lower bounds w_i
  • a ring, the basering
• Outputs:
  • a hash table, the subalgebra and the generators of the module over it

Description

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...}               }
                                                                                        2
                                                generators => {w, z, y, x*w, x*z, x*y, x z}
                                                ring => R

o3 : HashTable

See also

• intersectionValRings -- intersection of ring of valuations