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# intersectionValRingIdeals -- intersection of valuation ideals

- Usage:
    `intersectionValRingIdeals(v,r)`
- Inputs:
  - a matrix, values of the indeterminates, the last column contains the lower bounds w_i
  - a ring, the basering
- Optional inputs:
  - allComputations (missing documentation) => ..., default value false,
  - grading (missing documentation) => ..., default value {},
- 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

## Ways to use intersectionValRingIdeals:

- `intersectionValRingIdeals(Matrix,Ring)`

## For the programmer

The object intersectionValRingIdeals is a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Normaliz.m2:2024:0.