TriangularSet< Field, RecursivePoly > Class Template Reference

A triangular set templated by a multivariate polynomial over a field. More...

#include <triangularset.hpp>

The $\texttt{TriangularSet}$ class is a container class for handing triangular sets, i.e. , sets of polynomials of $\mathbb{K}[x_1,\ldots,x_n]$, with $ $\mathbb{K}$ $ a field, and variable ordering $x_1<\cdots x_n$, with pairwise distinct main variables. This class is intended mainly to support the regular chain classes. For the notions of a regular chain and a triangular set see https://en.wikipedia.org/wiki/Regular_chain and https://en.wikipedia.org/wiki/Wu%27s_method_of_characteristic_set#Triangular_set and as the references therein. More... (pdf)

Simplified semantic inheritance diagram for TriangularSet< Field, RecursivePoly >:

 Full inheritance diagram for TriangularSet< Field, RecursivePoly >:

Public Member Functions
	TriangularSet ()
	Default constructor: creates an empty triangular set of variable size with empty list of transcendentals. More...
	TriangularSet (const std::vector< Symbol > &xs)
	Construct an empty triangular set of fixed size in the s decreasingly ordered variables given by xs with empty list of transcendentals. More...
	TriangularSet (const std::vector< Symbol > &xs, const std::vector< Symbol > &ts)
	Construct an empty triangular set of fixed size in the s decreasingly ordered variables given by xs and list of transcendentals given by ts. More...
	TriangularSet (const RecursivePoly &p)
	Construct a variable triangular set containing p, such that the variables of p are treated as (potentially algebraic) variables, with empty list of transcendentals. More...
	TriangularSet (const RecursivePoly &p, const std::vector< Symbol > &ts)
	Construct a variable triangular set containing p, such that the variables in ts are treated as transcendental, while any remaining variables of p are treated as (potentially algebraic) variables. More...
	TriangularSet (const TriangularSet< Field, RecursivePoly > &a)
	Copy constructor. More...
	TriangularSet (TriangularSet< Field, RecursivePoly > &&a)
	Copy constructor. More...
	TriangularSet (const std::vector< Symbol > &&vs, const std::vector< Symbol > &&avs, const std::vector< Symbol > &&tvs, const std::vector< RecursivePoly > &&ts, TriangularSetMode tsm, const mpz_class &c)
	Move constructor. More...
	~TriangularSet ()
	Deconstructor. More...
bool	isEmpty () const
	Copy an object derived from abstract BPASTriangularSet class to type of current object. More...
bool	isConstantPolynomial (const RecursivePoly &p) const
	Tests if the polynomial is constant relative to the TriangularSet, i.e., whether it is and element of the Field or its only variables are transcendental. More...
TriangularSet< Field, RecursivePoly > &	operator= (const TriangularSet< Field, RecursivePoly > &a)
	Assignment operator =. More...
BPASTriangularSet< Field, RecursivePoly > &	operator= (const BPASTriangularSet< Field, RecursivePoly > &a) override
	Assignment operator =. More...
TriangularSet< Field, RecursivePoly > &	operator= (TriangularSet< Field, RecursivePoly > &&a)
	Move assignment operator =. More...
BPASTriangularSet< Field, RecursivePoly > &	operator= (BPASTriangularSet< Field, RecursivePoly > &&a) override
	Move assignment operator =. More...
TriangularSet< Field, RecursivePoly >	operator+ (const RecursivePoly &p)
	Add operator +. More...
TriangularSet< Field, RecursivePoly > &	operator+= (const RecursivePoly &p)
	Add assignment operator +=. More...
bool	operator== (const TriangularSet< Field, RecursivePoly > &a) const
	Identity operator ==. More...
bool	operator!= (const TriangularSet< Field, RecursivePoly > &a) const
	Negated identity operator !=. More...
int	numberOfVariables () const
	Get the number of variables. More...
int	size () const
	Get the size of the triangular set. More...
int	numberOfAlgebraicVariables () const
	Get the number of algebraic variables. More...
int	numberOfTranscendentalVariables () const
	Get the number of transcendental variables. More...
std::vector< Symbol >	variables () const
	Get the variable names in decreasing order. More...
std::vector< Symbol >	mainVariables () const
	Get the algebraic variables. More...
std::vector< Symbol >	transcendentalVariables () const
	Get the transcendentalVariables variables. More...
std::vector< Symbol >	allVariables () const
	Get the list of variables followed by the transcendental variables. More...
bool	isAlgebraic (const Symbol &s) const
	Determine if the input variable s is algebraic, i.e., if the triangular set contains a polynomial with s its as leading variable. More...
bool	isStronglyNormalized () const
	Return true if the triangular set is strongly normalized, i.e., the initals of all polynomials are in the Field; return false otherwise. More...
std::vector< RecursivePoly >	polynomials () const
	Get the vector of polynoials in the triangular set. More...
int	dimension () const
	Return the dimension of the triangular set (understood in terms of the space of (potentially algebraic) variables). More...
int	dimensionLower (Symbol v) const
	Return the dimension of the triangular set lower(v) (understood in terms of the space of (potentially algebraic) variables). More...
int	codimension () const
	Return the codimension of the triangular set (understood in terms of the space of (potentially algebraic) variables). More...
bool	canComputeInDimensionZero (const RecursivePoly &p, bool excludeMainVariable=false) const
	Test to determine whether the triangular set can be treated as zero dimensional, i.e., whether the triangular set becomes zero dimensional when all non-algebraic variables are removed and whether the polynomial p contains only algebraic variables. More...
bool	isZeroDimensionalMathematically () const
	Test to determine if only algebraic variables (aside from transcendentals) appear in the polynomials of the triangular set. More...
RecursivePoly	select (const Symbol &s) const
	Select a polynomial given the leading variable; if no such polynomial, 0 is returned. More...
void	makePrimitive ()
	Replace each polynomial of the triangular set with its primitive part. More...
void	lower (const Symbol &s, BPASTriangularSet< Field, RecursivePoly > &ts) const
	Returns the triangular set consisting of polynomials with main variable strictly less than s. More...
void	upper (const Symbol &s, BPASTriangularSet< Field, RecursivePoly > &ts) const
	Returns the triangular set consisting of polynomials with main variable strictly greater than s. More...
void	cutChain (const TriangularSet< Field, RecursivePoly > &T, const Symbol &v, TriangularSet< Field, RecursivePoly > &Tlv, RecursivePoly &Tv, TriangularSet< Field, RecursivePoly > &Tgv) const
	Cut an input triangular set at the symbol v, returning the subchain below v, the polynomial with main variable v and the subchain above v. More...
void	cutChain (const Symbol &v, RecursivePoly &Tv, TriangularSet< Field, RecursivePoly > &Tgv) const
	Cut the current object at the symbol v, returning the polynomial with main variable v and the subchain above v. More...
void	cutChain (const Symbol &v, TriangularSet< Field, RecursivePoly > &Tlv, RecursivePoly &Tv) const
	Cut the current object at the symbol v, returning the subchain below v and the polynomial with main variable v. More...
RecursivePoly	pseudoDivide (const RecursivePoly &p, std::vector< RecursivePoly > *quo=NULL, RecursivePoly *c=NULL) const
	Pseudo division: return the pseudo-remainder, the pseudo-quotients and c such that c*p = ∑(q_i T_i) + r. More...
RecursivePoly	normalForm (const RecursivePoly &p, std::vector< RecursivePoly > *Q=NULL) const
	Return the normalForm of the input polynomial modulo the triangular set in the sense of Groebner basis. More...
RecursivePoly	reduce (const RecursivePoly &p) const
	Reduce the input polynomial modulo the triangular set. More...
RecursivePoly	reduce (const RecursivePoly &p, RecursivePoly &c, bool takeMainPrimitivePart=false, bool onlyInDimZero=false) const
	returns r such that c*r = p modulo sat(T) such that c has no algebraic variables, and c is returned as an input parameter. More...
RecursivePoly	randomTriangularSetPolynomial (std::vector< Symbol > variables, int algVar, std::vector< Symbol > transcendentalVariables, int nTerms, unsigned long int coefBound, int pSparsity, bool includeNeg)
	Generate a random triangular set polynomial based on its number of terms. More...
RecursivePoly	randomTriangularSetPolynomial (std::vector< Symbol > variables, int algVar, std::vector< Symbol > transcendentalVariables, std::vector< int > maxDegs, unsigned long int coefBound, double pSparsity, bool includeNeg)
	Generate a random triangular set polynomial based on its maximum degrees in its variables. More...
void	randomTriangularSet (int nVars, int nAlgVars, int nTrcVars, int nTerms, unsigned long int coefBound, int pSparsity, bool includeNeg)
	Generate a random triangular set based on the number of terms of its polynomials. More...
void	randomStronglyNormalizedTriangularSet (int nVars, int nAlgVars, int nTrcVars, int nTerms, unsigned long int coefBound, int pSparsity, bool includeNeg)
	Generate a random strongly normalized triangular set based on the number of terms in its polynomials. More...
void	display ()
	Display the triangular set. More...
ExpressionTree	convertToExpressionTree () const
	Convert a triangular set to an expression tree (array of its polynomials). More...

Friends
std::ostream &	operator<< (std::ostream &out, const TriangularSet< Field, RecursivePoly > &a)
	Overload stream operator << for triangular sets. More...

Detailed Description

template<class Field, class RecursivePoly>
class TriangularSet< Field, RecursivePoly >

A triangular set templated by a multivariate polynomial over a field.

The field should be a BPASField and the multivariate polynomial should be recursively viewed, as in BPASRecursivelyViewedPolynomial.

Constructor & Destructor Documentation

TriangularSet() [1/8]

template<class Field, class RecursivePoly>

TriangularSet< Field, RecursivePoly >::TriangularSet

(

)

Default constructor: creates an empty triangular set of variable size with empty list of transcendentals.

Parameters

TriangularSet() [2/8]

template<class Field, class RecursivePoly>

TriangularSet< Field, RecursivePoly >::TriangularSet

(

const std::vector< Symbol > &

xs

)

Construct an empty triangular set of fixed size in the s decreasingly ordered variables given by xs with empty list of transcendentals.

Parameters

xs	The variable names

TriangularSet() [3/8]

template<class Field, class RecursivePoly>

TriangularSet< Field, RecursivePoly >::TriangularSet	(	const std::vector< Symbol > &	xs,
		const std::vector< Symbol > &	ts
	)

Construct an empty triangular set of fixed size in the s decreasingly ordered variables given by xs and list of transcendentals given by ts.

Parameters

xs	The variable names
ts	The transcendental variable names

TriangularSet() [4/8]

template<class Field, class RecursivePoly>

TriangularSet< Field, RecursivePoly >::TriangularSet

(

const RecursivePoly &

p

)

Construct a variable triangular set containing p, such that the variables of p are treated as (potentially algebraic) variables, with empty list of transcendentals.

Parameters

p	The recursively viewed polynomial to add

TriangularSet() [5/8]

template<class Field, class RecursivePoly>

TriangularSet< Field, RecursivePoly >::TriangularSet	(	const RecursivePoly &	p,
		const std::vector< Symbol > &	ts
	)

Construct a variable triangular set containing p, such that the variables in ts are treated as transcendental, while any remaining variables of p are treated as (potentially algebraic) variables.

Parameters

p	The recursively viewed polynomial to add
ts	The transcendental variable names

TriangularSet() [6/8]

template<class Field, class RecursivePoly>

TriangularSet< Field, RecursivePoly >::TriangularSet

(

const TriangularSet< Field, RecursivePoly > &

a

)

Copy constructor.

Parameters

a	A triangular set

TriangularSet() [7/8]

template<class Field, class RecursivePoly>

TriangularSet< Field, RecursivePoly >::TriangularSet

(

TriangularSet< Field, RecursivePoly > &&

a

)

Copy constructor.

Parameters

a	A regular chain Copy constructor
a	A zero dimensional regular chain Move constructor.
a	An r-value triangular set

TriangularSet() [8/8]

template<class Field, class RecursivePoly>

TriangularSet< Field, RecursivePoly >::TriangularSet	(	const std::vector< Symbol > &&	vs,
		const std::vector< Symbol > &&	avs,
		const std::vector< Symbol > &&	tvs,
		const std::vector< RecursivePoly > &&	ts,
		TriangularSetMode	tsm,
		const mpz_class &	c
	)

Move constructor.

Parameters

a	An r-value regular chain Move constructor
a	An r-value zero dimensional regular chain Computational constructor: creates a triangular set given all the data.
vs	rvalue reference to variables of the triangular set
avs	rvalue reference to algebraic variables of the triangular set
tvs	rvalue reference to transcendental variables of the triangular set
polys	rvalue reference to polynomials of the triangular set
tsm	whether the triangular set is variable or fixed
c	characteristic of the triangular set

~TriangularSet()

template<class Field, class RecursivePoly>

TriangularSet< Field, RecursivePoly >::~TriangularSet

(

)

Deconstructor.

Parameters

Member Function Documentation

allVariables()

template<class Field, class RecursivePoly>

std::vector<Symbol> TriangularSet< Field, RecursivePoly >::allVariables

(

)

const

Get the list of variables followed by the transcendental variables.

Parameters

canComputeInDimensionZero()

template<class Field, class RecursivePoly>

bool TriangularSet< Field, RecursivePoly >::canComputeInDimensionZero	(	const RecursivePoly &	p,
		bool	excludeMainVariable = false
	)		const

Test to determine whether the triangular set can be treated as zero dimensional, i.e., whether the triangular set becomes zero dimensional when all non-algebraic variables are removed and whether the polynomial p contains only algebraic variables.

Parameters

p	a recursively viewed polynomial
p	(optional) flag to exclude the main variable of p when determining whether it contains only algebraic variables (default false)

codimension()

template<class Field, class RecursivePoly>

int TriangularSet< Field, RecursivePoly >::codimension ( ) const

inline

Return the codimension of the triangular set (understood in terms of the space of (potentially algebraic) variables).

Parameters

convertToExpressionTree()

template<class Field, class RecursivePoly>

ExpressionTree TriangularSet< Field, RecursivePoly >::convertToExpressionTree ( ) const

inline

Convert a triangular set to an expression tree (array of its polynomials).

Parameters

cutChain() [1/3]

template<class Field, class RecursivePoly>

void TriangularSet< Field, RecursivePoly >::cutChain	(	const TriangularSet< Field, RecursivePoly > &	T,
		const Symbol &	v,
		TriangularSet< Field, RecursivePoly > &	Tlv,
		RecursivePoly &	Tv,
		TriangularSet< Field, RecursivePoly > &	Tgv
	)		const

Cut an input triangular set at the symbol v, returning the subchain below v, the polynomial with main variable v and the subchain above v.

Parameters

T	a triangular set
v	a symbol
Tlv	the subchain of T below v
Tv	the recursively viewed polynomial with main variable v, if it exists
Tgv	the subchain of T above v

cutChain() [2/3]

template<class Field, class RecursivePoly>

void TriangularSet< Field, RecursivePoly >::cutChain	(	const Symbol &	v,
		RecursivePoly &	Tv,
		TriangularSet< Field, RecursivePoly > &	Tgv
	)		const

Cut the current object at the symbol v, returning the polynomial with main variable v and the subchain above v.

Parameters

v	a symbol
Tv	the recursively viewed polynomial with main variable v, if it exists
Tgv	the subchain of the current object above v

cutChain() [3/3]

template<class Field, class RecursivePoly>

void TriangularSet< Field, RecursivePoly >::cutChain	(	const Symbol &	v,
		TriangularSet< Field, RecursivePoly > &	Tlv,
		RecursivePoly &	Tv
	)		const

Cut the current object at the symbol v, returning the subchain below v and the polynomial with main variable v.

Parameters

v	a symbol
Tlv	the subchain of the current object below v
Tv	the recursively viewed polynomial with main variable v, if it exists

dimension()

template<class Field, class RecursivePoly>

int TriangularSet< Field, RecursivePoly >::dimension ( ) const

inline

Return the dimension of the triangular set (understood in terms of the space of (potentially algebraic) variables).

Parameters

dimensionLower()

template<class Field, class RecursivePoly>

int TriangularSet< Field, RecursivePoly >::dimensionLower ( Symbol  v ) const

inline

Return the dimension of the triangular set lower(v) (understood in terms of the space of (potentially algebraic) variables).

Parameters

display()

template<class Field, class RecursivePoly>

void TriangularSet< Field, RecursivePoly >::display

(

)

Display the triangular set.

Parameters

isAlgebraic()

template<class Field, class RecursivePoly>

bool TriangularSet< Field, RecursivePoly >::isAlgebraic ( const Symbol &  s ) const

inline

Determine if the input variable s is algebraic, i.e., if the triangular set contains a polynomial with s its as leading variable.

Parameters

s	the input variable

isConstantPolynomial()

template<class Field, class RecursivePoly>

bool TriangularSet< Field, RecursivePoly >::isConstantPolynomial

(

const RecursivePoly &

p

)

const

Tests if the polynomial is constant relative to the TriangularSet, i.e., whether it is and element of the Field or its only variables are transcendental.

Parameters

p	a recursively viewed polynomial

isEmpty()

template<class Field, class RecursivePoly>

bool TriangularSet< Field, RecursivePoly >::isEmpty

(

)

const

Copy an object derived from abstract BPASTriangularSet class to type of current object.

Parameters

ts	triangular set to copy Tests if the TriangularSet is empty.

isStronglyNormalized()

template<class Field, class RecursivePoly>

bool TriangularSet< Field, RecursivePoly >::isStronglyNormalized ( ) const

inline

Return true if the triangular set is strongly normalized, i.e., the initals of all polynomials are in the Field; return false otherwise.

Parameters

isZeroDimensionalMathematically()

template<class Field, class RecursivePoly>

bool TriangularSet< Field, RecursivePoly >::isZeroDimensionalMathematically

(

)

const

Test to determine if only algebraic variables (aside from transcendentals) appear in the polynomials of the triangular set.

Parameters

lower()

template<class Field, class RecursivePoly>

void TriangularSet< Field, RecursivePoly >::lower ( const Symbol &  s, BPASTriangularSet< Field, RecursivePoly > &  ts  ) const

virtual

Returns the triangular set consisting of polynomials with main variable strictly less than s.

Parameters

s	symbol of the main variable of specified element of the triangular set
ts	The returned triangular set

Implements BPASTriangularSet< Field, RecursivePoly >.

mainVariables()

template<class Field, class RecursivePoly>

std::vector<Symbol> TriangularSet< Field, RecursivePoly >::mainVariables ( ) const

inline

Get the algebraic variables.

Parameters

makePrimitive()

template<class Field, class RecursivePoly>

void TriangularSet< Field, RecursivePoly >::makePrimitive

(

)

Replace each polynomial of the triangular set with its primitive part.

Parameters

normalForm()

template<class Field, class RecursivePoly>

RecursivePoly TriangularSet< Field, RecursivePoly >::normalForm ( const RecursivePoly &  p, std::vector< RecursivePoly > *  Q = NULL  ) const

virtual

Return the normalForm of the input polynomial modulo the triangular set in the sense of Groebner basis.

Parameters

p	innput recursively viewed polynomial
Q	(optional) the array of quotient

Implements BPASTriangularSet< Field, RecursivePoly >.

numberOfAlgebraicVariables()

template<class Field, class RecursivePoly>

int TriangularSet< Field, RecursivePoly >::numberOfAlgebraicVariables ( ) const

inline

Get the number of algebraic variables.

Parameters

numberOfTranscendentalVariables()

template<class Field, class RecursivePoly>

int TriangularSet< Field, RecursivePoly >::numberOfTranscendentalVariables ( ) const

inline

Get the number of transcendental variables.

Parameters

numberOfVariables()

template<class Field, class RecursivePoly>

int TriangularSet< Field, RecursivePoly >::numberOfVariables ( ) const

inlinevirtual

Get the number of variables.

Parameters

Implements BPASTriangularSet< Field, RecursivePoly >.

operator!=()

template<class Field, class RecursivePoly>

bool TriangularSet< Field, RecursivePoly >::operator!=

(

const TriangularSet< Field, RecursivePoly > &

a

)

const

Negated identity operator !=.

Parameters

a	A triangular set

operator+()

template<class Field, class RecursivePoly>

TriangularSet<Field,RecursivePoly> TriangularSet< Field, RecursivePoly >::operator+

(

const RecursivePoly &

p

)

Add operator +.

Adds a polynomial to a triangular set and returns a new triangular set.

Parameters

p	A recursively viewed polynomial

operator+=()

template<class Field, class RecursivePoly>

TriangularSet<Field,RecursivePoly>& TriangularSet< Field, RecursivePoly >::operator+=

(

const RecursivePoly &

p

)

Add assignment operator +=.

Adds a polynomial to a triangular set.

Parameters

p	A recursively viewed polynomial

operator=() [1/4]

template<class Field, class RecursivePoly>

TriangularSet<Field,RecursivePoly>& TriangularSet< Field, RecursivePoly >::operator=

(

const TriangularSet< Field, RecursivePoly > &

a

)

Assignment operator =.

Parameters

a	A triangular set

operator=() [2/4]

template<class Field, class RecursivePoly>

BPASTriangularSet<Field,RecursivePoly>& TriangularSet< Field, RecursivePoly >::operator= ( const BPASTriangularSet< Field, RecursivePoly > &  a )

overridevirtual

Assignment operator =.

Parameters

a	A triangular set

Implements BPASTriangularSet< Field, RecursivePoly >.

operator=() [3/4]

template<class Field, class RecursivePoly>

TriangularSet<Field,RecursivePoly>& TriangularSet< Field, RecursivePoly >::operator=

(

TriangularSet< Field, RecursivePoly > &&

a

)

Move assignment operator =.

Parameters

a	A triangular set

operator=() [4/4]

template<class Field, class RecursivePoly>

BPASTriangularSet<Field,RecursivePoly>& TriangularSet< Field, RecursivePoly >::operator= ( BPASTriangularSet< Field, RecursivePoly > &&  a )

overridevirtual

Move assignment operator =.

Parameters

a	A triangular set

Implements BPASTriangularSet< Field, RecursivePoly >.

operator==()

template<class Field, class RecursivePoly>

bool TriangularSet< Field, RecursivePoly >::operator==

(

const TriangularSet< Field, RecursivePoly > &

a

)

const

Identity operator ==.

Parameters

a	A triangular set

polynomials()

template<class Field, class RecursivePoly>

std::vector<RecursivePoly> TriangularSet< Field, RecursivePoly >::polynomials ( ) const

inline

Get the vector of polynoials in the triangular set.

Parameters

pseudoDivide()

template<class Field, class RecursivePoly>

RecursivePoly TriangularSet< Field, RecursivePoly >::pseudoDivide ( const RecursivePoly &  p, std::vector< RecursivePoly > *  quo = NULL, RecursivePoly *  c = NULL  ) const

virtual

Pseudo division: return the pseudo-remainder, the pseudo-quotients and c such that c*p = ∑(q_i T_i) + r.

Parameters

p	an input recursively viewed polynomial
quo	(optional) the array of quotients
c	(optional) the constant multiplied to the input polynomial

Implements BPASTriangularSet< Field, RecursivePoly >.

randomStronglyNormalizedTriangularSet()

template<class Field, class RecursivePoly>

void TriangularSet< Field, RecursivePoly >::randomStronglyNormalizedTriangularSet	(	int	nVars,
		int	nAlgVars,
		int	nTrcVars,
		int	nTerms,
		unsigned long int	coefBound,
		int	pSparsity,
		bool	includeNeg
	)

Generate a random strongly normalized triangular set based on the number of terms in its polynomials.

Parameters

nVars	number of variables
nAlgVars	number of algebraic variables
nTrcVars	number of transcendental variables
nTerms	number of terms in the polynomial
coefBound	maximum size of the coefficients
pSparsity	sparsity of the polynomial
includeNeg	whether to include negative coefficients

randomTriangularSet()

template<class Field, class RecursivePoly>

void TriangularSet< Field, RecursivePoly >::randomTriangularSet	(	int	nVars,
		int	nAlgVars,
		int	nTrcVars,
		int	nTerms,
		unsigned long int	coefBound,
		int	pSparsity,
		bool	includeNeg
	)

Generate a random triangular set based on the number of terms of its polynomials.

Parameters

nVars	number of variables
nAlgVars	number of algebraic variables
nTrcVars	number of transcendental variables
nTerms	number of terms in the polynomial
coefBound	maximum size of the coefficients
pSparsity	sparsity of the polynomial
includeNeg	whether to include negative coefficients

randomTriangularSetPolynomial() [1/2]

template<class Field, class RecursivePoly>

RecursivePoly TriangularSet< Field, RecursivePoly >::randomTriangularSetPolynomial	(	std::vector< Symbol >	variables,
		int	algVar,
		std::vector< Symbol >	transcendentalVariables,
		int	nTerms,
		unsigned long int	coefBound,
		int	pSparsity,
		bool	includeNeg
	)

Generate a random triangular set polynomial based on its number of terms.

Parameters

variables	variables of the triangular set
algVar	index of the algebraic variable
transcendentalVariables	transcendental variables of the triangular set
nTerms	number of terms in the polynomial
coefBound	maximum size of the coefficients
pSparsity	sparsity of the polynomial
includeNeg	whether to include negative coefficients

randomTriangularSetPolynomial() [2/2]

template<class Field, class RecursivePoly>

RecursivePoly TriangularSet< Field, RecursivePoly >::randomTriangularSetPolynomial	(	std::vector< Symbol >	variables,
		int	algVar,
		std::vector< Symbol >	transcendentalVariables,
		std::vector< int >	maxDegs,
		unsigned long int	coefBound,
		double	pSparsity,
		bool	includeNeg
	)

Generate a random triangular set polynomial based on its maximum degrees in its variables.

Parameters

variables	variables of the triangular set
algVar	index of the algebraic variable
transcendentalVariables	transcendental variables of the triangular set
maxDegs	vector of maximum degrees among the set of variables
coefBound	maximum size of the coefficients
pSparsity	proportional sparsity of the polynomial
includeNeg	whether to include negative coefficients

reduce() [1/2]

template<class Field, class RecursivePoly>

RecursivePoly TriangularSet< Field, RecursivePoly >::reduce

(

const RecursivePoly &

p

)

const

Reduce the input polynomial modulo the triangular set.

Parameters

p	input recursively viewed polynomial

reduce() [2/2]

template<class Field, class RecursivePoly>

RecursivePoly TriangularSet< Field, RecursivePoly >::reduce	(	const RecursivePoly &	p,
		RecursivePoly &	c,
		bool	takeMainPrimitivePart = false,
		bool	onlyInDimZero = false
	)		const

returns r such that c*r = p modulo sat(T) such that c has no algebraic variables, and c is returned as an input parameter.

Parameters

p	input recursively viewed polynomial
c	returned value of the content of p modulo sat(T)
usePrimitiveFactorization	(optional) whether to use primitive factorization to compute c (default true)
onlyInDimZero	(optional) only perform the reduction if the canComputeInDimensionZero(p) is true (default false)

select()

template<class Field, class RecursivePoly>

RecursivePoly TriangularSet< Field, RecursivePoly >::select ( const Symbol &  s ) const

virtual

Select a polynomial given the leading variable; if no such polynomial, 0 is returned.

Parameters

x	The leading variable name

Implements BPASTriangularSet< Field, RecursivePoly >.

size()

template<class Field, class RecursivePoly>

int TriangularSet< Field, RecursivePoly >::size ( ) const

inline

Get the size of the triangular set.

Parameters

transcendentalVariables()

template<class Field, class RecursivePoly>

std::vector<Symbol> TriangularSet< Field, RecursivePoly >::transcendentalVariables ( ) const

inline

Get the transcendentalVariables variables.

Parameters

upper()

template<class Field, class RecursivePoly>

void TriangularSet< Field, RecursivePoly >::upper ( const Symbol &  s, BPASTriangularSet< Field, RecursivePoly > &  ts  ) const

virtual

Returns the triangular set consisting of polynomials with main variable strictly greater than s.

Parameters

s	symbol of the main variable of specified element of the triangular set
ts	The returned triangular set

Implements BPASTriangularSet< Field, RecursivePoly >.

variables()

template<class Field, class RecursivePoly>

std::vector<Symbol> TriangularSet< Field, RecursivePoly >::variables ( ) const

inlinevirtual

Get the variable names in decreasing order.

Parameters

Implements BPASTriangularSet< Field, RecursivePoly >.

Friends And Related Function Documentation

operator<<

template<class Field, class RecursivePoly>

std::ostream& operator<< ( std::ostream &  out, const TriangularSet< Field, RecursivePoly > &  a  )

friend

Overload stream operator << for triangular sets.

Parameters

out	Stream object
a	A triangular set

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The documentation for this class was generated from the following file:

• include/TriangularSet/triangularset.hpp