# JBMill

© 2011-2015 Mario Albert
GNU Free Documentation License, Version 1.2

CoCoALib Documentation Index

## User documentation for using Janet bases

The files JBDatastructure.H, JBSets.H, JBAlgorithm.H and JBMill.H introduce several classes for computing and working with Janet basis. The normal user should only use the classes Involutive::JBMill and Involutive::JBMill::Builder to interact with Janet bases.

### Computing a Janet basis

To compute a Janet basis the user should use the class Involutive::JBMill::Builder. To construct a Involutive::JBMill::Builder object the user has to use the standard constructor. For configuration of the building process there are several methods:

• setInput(v) -- v must be a vector<RingElem>. It sets the generating set of the ideal to v.
• setInput(cBegin, cEnd) -- cBegin and cEnd must be a vector<RingElem>::const_iterator and must define a range of RingElem. The method sets the generating set of the ideal to this range.
• setStrategy(strat) -- strat must be a Involutive::StrategyFlag. Possible enums are TQDegree, TQBlockHigh, TQBlockLow and GBCompletion. It defines the algorithm which should be used to compute a Janet basis. If this method is never called the Builder object uses the TQBlockLow strategy.
• setInvolutiveCriteria(crits) -- crits must be a bitset<3>. Every bit represents one of the three involutive criteria. If this method is never called the Builder object uses the first two involutive criteria.

The methods are chainable, e.g. the user can do the following: builder.setInput(input).setStrategy(Involutive::TQDegree). If the user calls a method more than one time only the input of the last method call is taken into account. To construct a JBMill out of a correctly configured builder object build the user has to use JBMill(build). If the user does not set a input the construction of a JBMill will fail.

### Using the JBMill

In the following let elem be a RingElem.

• myReturnJB() -- returns the minimal Janet basis as vector<RingElem>
• myReturnGB() -- returns the minimal Groebner basis as vector<RingElem>
• myPrintMultVar() -- prints the multiplicative variables of every element in the given Janet basis
• myPrintNonMultVar() -- prints the nonmultiplicative variables of every element in the given Janet basis
• myMultVars() -- compute the multiplicative variables of the given Janet basis. It returns a map<PPMonoidElem, vector<bool> > where the key is a LPP of an element in the Janet basis.
• myNonMultVars() -- compute the nonmultiplicative variables of the given Janet basis. It returns a map<PPMonoidElem, vector<bool> > where the key is a LPP of an element in the Janet basis.
• myNonMultVarsOf(elem) -- computes the nonmultiplicative variables of elem which must be a member of the Janet basis. If not we assume that every variable is nonmultiplicative. It returns a vector<bool>.
• IamPommaretBasis -- checks if the Janet basis is also a Pommaret basis. It returns a boolean.
• IamHomogenous -- checks if the Janet basis is also homogeneous. It returns a boolean.
• IamMonomialIdeal -- checks if the Janet basis is also a monomial ideal. It returns a boolean.
• myStandardRepresentation(elem) -- compute the involutive standard representation of elem. It returns pair<map<PPMonoidElem, RingElem>, RingElem>. The first entry of the pair is a map, where the key represents the LPP of an element in the Janet basis and the value the corresponding factor. The second entry of the pair corresponds to the rest.
• myOutputStandardRepresentation(elem) -- computes an involutive standard representation of elem.
• myHilbertPol(elem) -- elem must be a single indent. The method computes the Hilbert polynomial of the ideal in terms of elem.
• myHilbertFunc(n) -- n must be a BigInt. The method computes the dimension of P/I in degree n.
• myHilbertSeries(elem) -- elem must be a single indent of a fraction field. The method computes the Hilbert series of the ideal in terms of elem.
• mySyzygy() -- Compute the first involutive syzygy and returns a FGModule.
• myDim() -- Computes the dimension of P/I.
• myCls(ppelem) -- Computes the class of ppelem which is of type PPMonoidElem. the class starts counting at 0 up to n - 1. The cls of 1 is -1. It returns a long.
• myMinCls() -- Computes the minimal class of all LPP's of the Janet basis as long.
• myMaxCls() -- Computes the maximal class of all LPP's of the Janet basis as long.
• myElementsWithClass(InputCls) -- Computes all elements of the Janet basis where the class of the LPP is InputCls. InputCls is a long and the method returns a vector<RingElem>.
• myComplementaryDecomposition() -- Computes the complementary decomposition of I. it returns vector<pair<PPMonoidElem, vector<bool> > >.
• myStandardPairs() -- Computes the standard pairs of I. it returns vector<pair<PPMonoidElem, vector<bool> > >.
• myJNormalForm(elem) -- Computes the involutive normal form of elem and returns a RingElem.
• myJDivisor(elem) -- Computes the involutive divisor of LPP(elem). If there is an involutive divisor it returns it as RingElem if not the method returns 0.

## Maintainer documentation for JBDatastructure, JBSets, JBAlgorithm, JBMill

The basic datastructures to deal with Janet basis are implemented in JBDatastructure.C. Everything of the following lives in the namespace CoCoA::Involutive.

### JanetDatastructure

#### JanetTriple

The JanetTriple is nothing else than a polynomial with some extra informations. In addition to the polynomial myPolynom it has a data member myAncestor which is usually the LPP of myPolynom and the already prolonged variables (myAlreadyProlongedVars). If the JanetTriple arises from a prolongation x_i * myP^\prime the ancestor is the LPP of myP^\prime.

#### JanetNodeBase, JanetLeafNodeImpl, JanetInternalNodeImpl, JanetHandle, JanetTree

The JanetTree is the basic data structure to compute and deal efficiently with a Janet basis. It is a binary tree. A Janet tree contains the Janet basis in its leaf nodes. Therefore we distinguish between internal nodes (JanetInternalNodes) and leaf nodes (JanetLeafNodes). The tree is designed as a nested set of lists. A node basically consists of the distance to the next variable (the distance to next node to the right) and the next degree (the distance to next node to the left). An internal node contains a list of JanetHandles additionally, which represents the following tree to the right. A leaf node contains, beside the distance information, a JanetTriple. The JanetTriple is not a direct data member of a leaf node. It is stored in a list. JanetLeafNodeImpl only gets an iterator from this list. The JanetHandle handles the distinction between the JanetLeafNodeImpl and the JanetInternalNodeImpl because a stl-container cannot reasonable handle different classes even if they have the same base class.

##### Problems&Ideals related to JanetNodeBase, JanetLeafNodeImpl, JanetInternalNodeImpl, JanetHandle, JanetTree

The JanetTree only works with a list of JanetTriple's. It would be useful if it would work with a list of polynomials as well.

#### JanetContainer

The last part of the previous paragraph shows a strong connection between the list of JanetTriple which shall represents the Janet basis and the JanetTree which is another representation of the Janet basis. This could lead to strange situations which has as a consequence invalidate iterators. To avoid this during the normal usage of these two datastructure we introduce a JanetContainer. JanetContainer couples these two datastructures. It contains a list of JanetTriple's and a JanetTree which leaf nodes consists of iterators to this list. With this coupling the user can deal with a Janet basis safely. But for computing a Janet basis we do not use this class for efficiency reasons.

#### JanetIterator

The task of JanetIterator is to offer a way to navigate through the JanetTree. Basically the JanetIterator consists of a pointer to the specific JanetTree, pointer to the current in the tree and an iterator to a specific position in this list. The JanetIterator provides access (if possible) to the underlying JanetTriple, provides the possibility to move forward in the tree, provides some informations of the current position in the tree and provides the functionality to add a new node in the JanetTree behind the current position. For knowing the way from the beginning of the tree to the current position it consists of a vector of longs which stores the specific degrees and the current variable.

### JBSets

The most important algorithm to compute Janet basis is the TQ-Algorithm. There are two variants of it: the basic TQDegree strategy and the more advanced TQBlock strategy. The TQDegree strategy deals with a set T and Q. In short, through the computation the algorithm moves elements mainly from Q to T and vica versa. To deal efficiently with it we introduced the class TQSets. It consists of the sets T (mySetT) and Q (mySetQ) which are ordered. Both are represented as std::multiset. They contain JanetTriple and ordered by the LPP's of them (Because these LPP's are not unique during the computation we choosing std::multiset). The JanetTriple's are not contained directly in the set T and Q itself, as it is very expensive to move them from one set to the other. Therefore there is a third set (myBasicSet) which is implemented as list of JanetTriple's which contains the JanetTriple's itself. The sets T and Q only contain an iterator to a specific position of these sets.

For applying the BlockTQ algorithm we need a third set P (mySetP) which is implemented like T and Q. Due to the similarity we introduced a subclass of TQSets which is called TQPSets. In addition to the new set P it introduces a strategy flag which influences the way how we move elements from Q to P.

In addition to the above mentioned sets TQSets consists of a SparsePolyRing, a ReductionCog and a bitset<3> (myCriteria). myCriteria regulates which involutive criteria shall be applied during the computation. Every bit stands for one single involutive criteria.

#### Problems&Ideas related to JBSets

Again the construction of the sets T,Q and myBasicSet is dangerous. There could be invalid iterators in the set T and Q. In addition to that it can happen (it really happens!!!!) that we can modify an element in myBasicSet in such a way that the ordering in T or Q would be change. But T and Q does not realizing this change. Therefore we getting again an invalid state. A solution for the second problem could be to store T and Q simply as a list of iterators of JanetTriple's and sort the list manually every time we want to have a sorted list. Maybe this solution is even faster than the current one!

### JBAlgorithm

#### JBAlgorithm

This class provides an interface for computing Janet bases. It defines a method to compute a Janet basis for a given input, and a method to get a JanetContainer which should contain the computed Janet basis. Also it contains as basic data the polynomial ring and the PPMonoid. Every class which computes a Janet basis has to be a subclass of this class.

#### TQAlgorithm

This class is a subclass of JBAlgorithm but is again purely virtual. It acts as an interface for all algorithms which using the TQ strategy. In addition to the data members of the base class it defines amongst other things a JanetTree (myJTree). All TQAlgorithm subclasses deal with the class TQSets or a subclass of it. To get a unique access to the specific data member (which is defined in the subclasses) we implemented a purely virtual function myGetSets which returns a reference to the specific data members. With this construction we are able to initialize the specific set in the class TQAlgorithm via the method myInitialization. In addition to that TQAlgorithm contains a method to return the ideal which is generated by 1.

#### DegreeTQ

This class is a subclass of TQAlgorithm. It defines the data member mySets (a TQSets instance) additionally. In addition to that it implements the purely virtual methods myGetSets and myComputer.

#### BlockTQ

This class is a subclass of TQAlgorithm. It defines the data member mySets (a TQPSets instance) additionally. In addition to that it implements the purely virtual methods myGetSets and myComputer.

#### CompletionGB

This class defines another approach to compute Janet basis, than the TQ approach. Here we first compute a reduced Groebner basis and complete it to the minimal Janet basis. It is a subclass of JBAlgorithm. The class implements the purely virtual methods myComputer and myOutputResult and defines a JanetTree and a list of JanetTriple's as data members. In addition to that it implements several methods to compute the completion.

### JBMill

#### JBMill

This class defines the representation of a Janet basis accessible by the user. As data members it contains a JanetContainer (myBasis), a SparsePolyRing (myPolyRing) and a PPMonoid (myPPMValue). The class defines several methods to work with the Janet basis. For example the user can compute the multiplicative variables, the Groebner basis or some invariants like the hilbert polynomial. In addition to that it acts as a base class for the PBMill, which is the representation of a Pommaret basis.

##### Problems&Ideas related to JBMill

Maybe introduce typedefs or structs for complicated objects like a complementary decomposition. Add several methods to check different stability position.

#### JBMill::Builder

This class is designed to construct a Janet basis. The goal of this class is to separate the construction of the JBMill from its representation. The 'Gang of Four' (Gamma, Helm, Johnson, Vlissides - Design Patterns) served as template for the construction. The corresponding pattern is called Building Pattern. To construct a JBMill out of the builder object the user can call a constructor of JBMill with a configured builder object.