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# ChordalNet -- a chordal network

## Description

This type describes a chordal network representation of a polynomial ideal. The constructor of this type is chordalNet

### Examples

- chordal networks examples -- a new representation of polynomial ideals

### Overview of chordal networks methods

Methods to visualize a chordal network:

- displayNet -- displays a chordal network using Graphivz
- net -- print a chordal network
- digraph(ChordalNet) -- digraph associated to a chordal network

Methods to access properties of a chordal network:

- nodes -- list of nodes of a chordal network
- elimTree -- get the underlying elimination tree
- ring -- get the underlying ring
- isTriangular -- whether a chordal network is triangular
- structure -- either "Monomial", "Binomial" or "None"

Elimination routines using chordal structure:

- chordalElim -- performs elimination on the chordal network
- chordalTria -- makes a chordal network triangular

Methods for triangular chordal networks:

- rootCount -- counts the number of roots of a chordal network
- dim(ChordalNet) -- dimension of a chordal network
- codim(ChordalNet) -- dimension of a chordal network
- codimCount -- codimension counts of the chains of a chordal network
- topComponents(ChordalNet) -- top dimension of a chordal network
- RingElement % ChordalNet -- ideal membership test
- nextChain -- iterates over the chains of a chordal network
- components(ChordalNet) -- components of a chordal network

## Functions and methods returning an object of class ChordalNet:

- `chordalNet(Ideal)` -- see chordalNet -- constructs a chordal network from a polynomial set
- `chordalNet(Ideal,List)` -- see chordalNet -- constructs a chordal network from a polynomial set
- `chordalNet(Ideal,String)` -- see chordalNet -- constructs a chordal network from a polynomial set
- chordalNet(HashTable,HashTable,ElimTree,Digraph) -- construct chordal network from a digraph
- RingMap ChordalNet -- apply ring map to a chordal network

## Methods that use an object of class ChordalNet:

- `chordalElim(ChordalNet)` -- see chordalElim -- performs elimination on the chordal network
- `net(ChordalNet)`
- `ring(ChordalNet)`
- `triaSystem(ChordalNet,ChordalNetChain)` -- see ChordalNetChain -- a chain of a chordal network
- `chordalTria(ChordalNet)` -- see chordalTria -- makes a chordal network triangular
- `codimCount(ChordalNet)` -- see codimCount -- codimension counts of the chains of a chordal network
- `components(ChordalNet)` -- see components(ChordalNet,ZZ) -- components of a chordal network
- components(ChordalNet,ZZ) -- components of a chordal network
- digraph(ChordalNet) -- digraph associated to a chordal network
- `codim(ChordalNet)` -- see dim(ChordalNet) -- dimension of a chordal network
- dim(ChordalNet) -- dimension of a chordal network
- `displayNet(ChordalNet)` -- see displayNet -- displays a chordal network using Graphivz
- `displayNet(Function,ChordalNet)` -- see displayNet -- displays a chordal network using Graphivz
- `displayNet(String,String,Function,ChordalNet)` -- see displayNet -- displays a chordal network using Graphivz
- `elimTree(ChordalNet)` -- see ElimTree -- the elimination tree of a chordal graph
- isPrimeSimple(ChordalNet) -- simple primality test of a chordal network
- `isTriangular(ChordalNet)` -- see isTriangular -- whether a chordal network is triangular
- `nextChain(ChordalNet)` -- see nextChain -- iterates over the chains of a chordal network
- `nextChain(ChordalNetChain,ChordalNet)` -- see nextChain -- iterates over the chains of a chordal network
- `nextChain(ChordalNetChain,Sequence,ZZ,ChordalNet)` -- see nextChain -- iterates over the chains of a chordal network
- `nextChain(ZZ,ChordalNet)` -- see nextChain -- iterates over the chains of a chordal network
- `nodes(ChordalNet)` -- see nodes -- list of nodes of a chordal network
- `nodes(ChordalNet,RingElement)` -- see nodes -- list of nodes of a chordal network
- `reduceNet(ChordalNet)` -- see reduceNet -- reduces a chordal network
- `pseudoRemainder(RingElement,ChordalNet)` -- see RingElement % ChordalNet -- ideal membership test
- RingElement % ChordalNet -- ideal membership test
- `rootCount(ChordalNet)` -- see rootCount -- counts the number of roots of a chordal network
- size(ChordalNet) -- size of a chordal network
- writeDotFile(String,Function,ChordalNet) -- writes a chordal network to a dot file

## For the programmer

The object ChordalNet is a type, with ancestor classes MutableHashTable < HashTable < Thing.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Chordal/ChordalDoc.m2:221:0.