# intersectionValRings -- intersection of ring of valuations

## Synopsis

• Usage:
intersectionValRings(v,r)
• Inputs:
• , rows are the values of the indeterminates
• a ring, the basering
• Outputs:
• an object of class MonomialSubalgebra, the subalgebra consisting of the elements with valuation ≥0 for all given valuations

## Description

A discrete monomial valuation v on R=K[X1,...,Xn] is determined by the values v(Xj) of the indeterminates. This function computes the subalgebra S={f∈R: vi(f)≥0, i=1,...,r} that is the intersection of the valuation rings of the given valuations v1, ...,vr, 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=(vi(Xj)) 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```