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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 a function, 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

Authors

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
    • leadTermValuation(PolynomialRing) -- see leadTermValuation -- The valuation defined by leading terms
    • 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 $\QQ^n$
    • padicValuation(ZZ) -- see padicValuation -- The p-adic valuation
    • 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 a package.