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# Valuations -- A package for constructing and using valuations.

## Description

A valuation is a function $v:R\rightarrow G\cup\{\infty\}$ where $R$ is a ring and $G$ is a linearly ordered group with the following properties:

• $v(ab)=v(a)+v(b)$,
• $v(a+b)\geq\min\{v(a),v(b)\}$, and
• $v(a)=\infty$ iff $a=0$.

The Valuations package provides uniform constructions of common valuations and also offers user-defined valuations. A valuation acts like , but may contain extra information.

 i1 : pval = padicValuation 3; i2 : pval(54) o2 = 3 i3 : pval(2) o3 = 0 i4 : R = QQ[x,y]; i5 : leadval = leadTermValuation R; i6 : leadval(x^3+3*x^3*y^2+2*y^4) o6 = | -3 | | -2 | o6 : Ordered QQ^2 module i7 : lowestval = lowestTermValuation R; i8 : lowestval(x^3+3*x^3*y^2+2*y^4) o8 = | 3 | | 0 | o8 : Ordered QQ^2 module i9 : lowestval(0) o9 = infinity o9 : InfiniteNumber

## Version

This documentation describes version 1.0 of Valuations.

## Source code

The source code from which this documentation is derived is in the file Valuations.m2.

## Exports

• Types
• OrderedQQn -- The class of all ordered modules $\QQ^n$
• OrderedQQVector -- The class of all vectors of an ordered module $\QQ^n$
• Functions and commands
• Methods
• coneToValuation(Matrix,Subring) -- see coneToValuation -- Convert a prime cone of a tropical ideal to a (quasi-)valuation
• coneToValuation(Matrix,Subring,Ring) -- see coneToValuation -- Convert a prime cone of a tropical ideal to a (quasi-)valuation
• localRingValuation(LocalRing) -- see localRingValuation -- The valuation defined by a local ring.
• lowestTermValuation(PolynomialRing) -- see lowestTermValuation -- The valuation defined by lowest terms
• OrderedQQn == OrderedQQn -- see Ordered modules -- Overview of the ordered module $\QQ^n$
• orderedQQn(PolynomialRing) -- see orderedQQn -- Construct an ordered module $\QQ^n$
• orderedQQn(ZZ,List) -- see orderedQQn -- Construct an ordered module $\QQ^n$
• InfiniteNumber == OrderedQQVector -- see OrderedQQVector -- The class of all vectors of an ordered module $\QQ^n$
• OrderedQQVector == InfiniteNumber -- see OrderedQQVector -- The class of all vectors of an ordered module $\QQ^n$
• OrderedQQVector ? OrderedQQVector -- see OrderedQQVector -- The class of all vectors of an ordered module $\QQ^n$
• primeConesOfIdeal(Ideal) -- see primeConesOfSubalgebra -- Finds the prime cones of the tropicalization of a given subalgebra or ideal.
• primeConesOfSubalgebra(Subring) -- see primeConesOfSubalgebra -- Finds the prime cones of the tropicalization of a given subalgebra or ideal.
• valuation(Function) -- see valuation -- User-defined valuation object
• valuation(Function,LocalRing,LocalRing) -- see valuation -- User-defined valuation object
• valuation(Function,LocalRing,OrderedQQn) -- see valuation -- User-defined valuation object
• valuation(Function,LocalRing,Ring) -- see valuation -- User-defined valuation object
• valuation(Function,LocalRing,RingOfInvariants) -- see valuation -- User-defined valuation object
• valuation(Function,LocalRing,Subring) -- see valuation -- User-defined valuation object
• valuation(Function,Ring,LocalRing) -- see valuation -- User-defined valuation object
• valuation(Function,Ring,OrderedQQn) -- see valuation -- User-defined valuation object
• valuation(Function,Ring,Ring) -- see valuation -- User-defined valuation object
• valuation(Function,Ring,RingOfInvariants) -- see valuation -- User-defined valuation object
• valuation(Function,Ring,Subring) -- see valuation -- User-defined valuation object
• valuation(Function,RingOfInvariants,LocalRing) -- see valuation -- User-defined valuation object
• valuation(Function,RingOfInvariants,OrderedQQn) -- see valuation -- User-defined valuation object
• valuation(Function,RingOfInvariants,Ring) -- see valuation -- User-defined valuation object
• valuation(Function,RingOfInvariants,RingOfInvariants) -- see valuation -- User-defined valuation object
• valuation(Function,RingOfInvariants,Subring) -- see valuation -- User-defined valuation object
• valuation(Function,Subring,LocalRing) -- see valuation -- User-defined valuation object
• valuation(Function,Subring,OrderedQQn) -- see valuation -- User-defined valuation object
• valuation(Function,Subring,Ring) -- see valuation -- User-defined valuation object
• valuation(Function,Subring,RingOfInvariants) -- see valuation -- User-defined valuation object
• valuation(Function,Subring,Subring) -- see valuation -- User-defined valuation object
• Other things

## For the programmer

The object Valuations is .