intersectionValRings -- intersection of ring of valuations

Synopsis

• Usage:

  intersectionValRings(v,r)

• Inputs:
  • a matrix, rows are the values of the indeterminates
  • a ring, the basering
• Outputs:
  • an instance of the type MonomialSubalgebra, the subalgebra consisting of the elements with valuation \geq 0 for all given valuations

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 computes the subalgebra S=\{f\in R: v_i(f)\geq 0, i=1,\ldots,r\} that is the intersection of the valuation rings of the given valuations v_1, \ldots,v_r, i.e. it consists of all elements of R that have a nonnegative value for all r valuations. It takes as input the matrix V=(v_i(X_j)) whose rows correspond to the values of the indeterminates.

This method can be used with the options allComputations and grading.

i1 : R=QQ[x,y,z,w];

i2 : V0=matrix({{0,1,2,3},{-1,1,2,1}});

              2        4
o2 : Matrix ZZ  <--- ZZ

i3 : intersectionValRings(V0,R)

                                 2
o3 = QQ[w, z, y, x*w, x*z, x*y, x z]

o3 : monomial subalgebra of R

See also

• intersectionValRingIdeals -- intersection of valuation ideals