mrpolynomial.h

#ifndef _SMQPALTARRAY_H_
#define _SMQPALTARRAY_H_

#include "../Polynomial/BPASRecursivePolynomial.hpp"
#include "../Interval/interval.h"
#include "urpolynomial.h"
#include "../IntegerPolynomial/mzpolynomial.hpp"
#include "../RingPolynomial/upolynomial.h"
#include "SMQP_CppSupport-AA.hpp"
#include "SMQP_Support-AA.h"
#include "SMQP_Support_Test-AA.h"
#include "SMQP_Support_Recursive-AA.h"
#include <gmpxx.h>
#include "../ExpressionTree/ExpressionTree.hpp"
#include "../DataStructures/Factors.hpp"
#include "../TriangularSet/triangularset.hpp"
#include "../IntegerPolynomial/DUZP_Support.h"
#include "../ModularPolynomial/DUSP_Parallel.hpp"
#include "../Parser/bpas_parser.h"

class SparseMultivariateIntegerPolynomial;

/**
 * An element of the SLP of a rational number polynomial.
 */
class SLPRepresentation {

    public:
        union CoefOrInt {
            RationalNumber* c;
            int i;
        };

        int op;         // 0: '+'; 1: '*'.
        int type;       // 0: coef & variate;
                        // 1: coef & result;
                        // 2: variate & result;
                        // 3: result & result;
                        // 4: coef (constant);
        CoefOrInt a;    // variate or result or coef position
        int b;          // variate or result or coef position
        Interval res;   // Storing the result

        SLPRepresentation() {
            a.c = NULL;
        }

        SLPRepresentation(const SLPRepresentation& r) {
            op = r.op;
            type = r.type;
            b = r.b;
            res = r.res;
            if (type == 0 || type == 1 || type == 4) {
                a.c = new RationalNumber(*(r.a.c));
            } else {
                a.i = r.a.i;
            }
        }

        ~SLPRepresentation() {
            if (type == 0 || type == 1) {
                delete a.c;
            }
        }
};

/**
 * A multivariate polynomial with RationalNumber coefficients represented sparely.
 * Only non-zero coefficients are encoded.
 */
class SparseMultivariateRationalPolynomial: public BPASRecursivelyViewedPolynomial<RationalNumber,SparseMultivariateRationalPolynomial> {

    private:
        mutable AltArr_t* poly;
        int nvar;             //number of variables
        Symbol* names;   //list of strings representing the variables.
                              //names[0] is 1 if "system-generated" variables
                              //names[0] is 9 if "user-specified" variables;


        friend class SparseMultivariateIntegerPolynomial;

        /**
         * Determine if *this and b have compatible variable orderings.
         * xs returns the mapping of both *this and b to the compatible superset
         * returns true iff compatible.
         */
        bool isOrderedRing(const SparseMultivariateRationalPolynomial& b, std::vector<int>& xs) const;

        /**
         * Rearrange exponent vectors in place and then re-sort the polynomial.
         */
        void reorderVarsInPlace(int varmap[]);

        /**
         * Rearrange exponent vectors, expanding as needed, and then re-sort the polynomial.
         */
        void expandVarsInPlace(int vars, Symbol* newvars, int varmap[]);

        /**
         * Returns a copy of *this under the new variable ordering supplied.
         * varmap is such that this.names[i] = newvars[varmap[i]]
         * Returns an SMQP equal to *this but expended to newvars.
         */
        SparseMultivariateRationalPolynomial expandVariables(int vars, Symbol* newvars, int varmap[]) const;

        void preparePolysForSRC(const SparseMultivariateRationalPolynomial& q, const Symbol& v, std::vector<Symbol>& superRing, bool sameRing, int* varMap, int& swapIdx, AltArrZ_t** ppZ, AltArrZ_t** qqZ) const;

    public:


        SparseMultivariateRationalPolynomial subresultantGCD (const SparseMultivariateRationalPolynomial& q) const;

        std::vector<SLPRepresentation> slp;

        /* Constructors */

        /**
         * Construct a multivariate polynomial
         *
         **/
        SparseMultivariateRationalPolynomial();

        /**
         * Construct a multivariate polynomial with specific number of variables
         *
         * @param v: Number of variables
         **/
        SparseMultivariateRationalPolynomial(int v);

        /**
         * Construct with a variable name such that f(x) = x;
         *
         * @param x: The variable name
         **/
        SparseMultivariateRationalPolynomial (const Symbol& x);

        /**
         * Construct a polynomial by parsing the string str.
         */
        SparseMultivariateRationalPolynomial (const std::string& str);

        /**
         * Construct an SMQP given the head Node*. The SMQP takes ownership
         * of the Node list.
         */
        SparseMultivariateRationalPolynomial(AltArr_t* aa, int vars, Symbol* varNames);

        /**
         * Copy Constructor.
         *
         * Does not reuse underlying memory allocated by b.
         *
         * @param b: A sparse multivariate polynomial
         **/
        SparseMultivariateRationalPolynomial(const SparseMultivariateRationalPolynomial& b);

        /**
         * Move Constructor.
         *
         * @params b: The r-value reference polynomial.
         */
        SparseMultivariateRationalPolynomial(SparseMultivariateRationalPolynomial&& b);

        /**
         * Create a SMZP from an SMQP.
         */
        SparseMultivariateRationalPolynomial(const SparseMultivariateIntegerPolynomial& b);

        /**
         * Create a SMQP from an Integer.
         */
        SparseMultivariateRationalPolynomial(const Integer& r, int nvar = 0);

        /**
         * Create a SMQP from a RationalNumber.
         */
        SparseMultivariateRationalPolynomial(const RationalNumber& r, int nvar = 0);

        /**
         * Create a SMQP from a univariate rational polynomial.
         *
         * @param p: A SUQP polynomial.
         **/
        SparseMultivariateRationalPolynomial (const DenseUnivariateRationalPolynomial& p);

        /**
         * Construct from a SUP<SMQP> polynomial such that the resulting SMQP
         * has main variable equal to the variable of s.
         *
         * @param s: The SUP<SMQP> polynomial
         **/
        SparseMultivariateRationalPolynomial (const SparseUnivariatePolynomial<SparseMultivariateRationalPolynomial>& s);

        /**
         * Destroy the polynomial and underlying node memory.
         **/
        ~SparseMultivariateRationalPolynomial();


        /* BPASRing interface */
        // static bool isPrimeField;
        // static bool isSmallPrimeField;
        // static bool isComplexField;

        /**
         * Is this polynomial zero.
         * returns true iff this polynomial encodes 0.
         */
        bool isZero() const;

        /**
         * Set this polynomial to zero. Maintains existing variable ordering.
         */
        void zero();

        /**
         * Is this polynomial one.
         * return true iff this polynomial encodes 1.
         */
        bool isOne() const;

        /**
         * Sets this polynomial to one. Maintains existing variable ordering.
         */
        void one();

        /**
         * Is this polynomial negative one.
         * returns true iff this polynomial encodes -1.
         */
        bool isNegativeOne() const;

        /**
         * Sets this polynomial to -1. Maintains existing variable ordering.
         */
        void negativeOne();

        /**
         * Determine if this polynomial encodes a constant.
         * returns 0 if not a constant, 1 if a constant >= 0, -1 if a negative contstant.
         */
        int isConstant() const;

        /**
         * Obtain the unit normal (a.k.a canonical associate) of an element.
         * If either parameters u, v, are non-NULL then the units are returned such that
         * b = ua, v = u^-1. Where b is the unit normal of a, and is the returned value.
         */
        inline SparseMultivariateRationalPolynomial unitCanonical(SparseMultivariateRationalPolynomial* u, SparseMultivariateRationalPolynomial* v) const {
            RationalNumber lead = leadingCoefficient();
            RationalNumber leadInv = lead.inverse();
            if (u != NULL) {
                *u = leadInv;
            }
            if (v != NULL) {
                *v = lead;
            }
            return (*this * leadInv);
        }

        /* BPASPolynomial interface */

        /**
         * Assign this polynomail to equal the specified.
         */
        SparseMultivariateRationalPolynomial& operator= (const SparseMultivariateRationalPolynomial& b);

        /**
         * Movement assignment: move b to be this polynomail.
         */
        SparseMultivariateRationalPolynomial& operator= (SparseMultivariateRationalPolynomial&& b);

        /**
         * Assign this polynomial to be equal to the constant ring element.
         */
        SparseMultivariateRationalPolynomial& operator= (const RationalNumber& r);

        /**
         * Add two SMQP polynomials together, *this and the specified.
         */
        SparseMultivariateRationalPolynomial operator+ (const SparseMultivariateRationalPolynomial& b) const;
        SparseMultivariateRationalPolynomial operator+ (SparseMultivariateRationalPolynomial&& b) const;

        // /**
        //  * Add the polynomails a and b and return the sum.
        //  */
        // friend SparseMultivariateRationalPolynomial operator+ (SparseMultivariateRationalPolynomial&& a, const SparseMultivariateRationalPolynomial& b);

        /**
         * Update *this by adding the specified polynomail to it.
         */
        SparseMultivariateRationalPolynomial& operator+= (const SparseMultivariateRationalPolynomial& b);

        /**
         * Subtract the specified polynomail from *this
         */
        SparseMultivariateRationalPolynomial operator- (const SparseMultivariateRationalPolynomial& b) const;
        SparseMultivariateRationalPolynomial operator- (SparseMultivariateRationalPolynomial&& b) const;

        // /**
        //  * Subtract the polynomial b from a and return the difference.
        //  */
        // friend SparseMultivariateRationalPolynomial operator- (SparseMultivariateRationalPolynomial&& a, const SparseMultivariateRationalPolynomial& b);

        /**
         * Unary operation, return  *this * -1.
         */
        SparseMultivariateRationalPolynomial operator- () const;

        /**
         * Update *this by subtracting the specified polynomial from it.
         */
        SparseMultivariateRationalPolynomial& operator-= (const SparseMultivariateRationalPolynomial& b);

        /**
         * Multiply *this by the specified polynomail.
         */
        SparseMultivariateRationalPolynomial operator* (const SparseMultivariateRationalPolynomial& b) const;
        SparseMultivariateRationalPolynomial operator* (SparseMultivariateRationalPolynomial&& b) const;

        // /**
        //  * Multiply the polynomials a and b, returning their product.
        //  */
        // friend SparseMultivariateRationalPolynomial operator* (SparseMultivariateRationalPolynomial&& a, const SparseMultivariateRationalPolynomial& b);

        /**
         * Update this by multiplying by the specified polynomail.
         */
        SparseMultivariateRationalPolynomial& operator*= (const SparseMultivariateRationalPolynomial& b);

        /**
         * Divide *this by the specified polynomial.
         */
        SparseMultivariateRationalPolynomial operator/ (const SparseMultivariateRationalPolynomial& b) const;
        SparseMultivariateRationalPolynomial operator/ (SparseMultivariateRationalPolynomial&& b) const;

        // friend SparseMultivariateRationalPolynomial operator/ (SparseMultivariateRationalPolynomial&& a, const SparseMultivariateRationalPolynomial& b);

        /**
         * Update *this by dividing by the specified polynomial.
         */
        SparseMultivariateRationalPolynomial& operator/= (const SparseMultivariateRationalPolynomial& b);

        /**
         * Exponentiate *this by the input exponent integer.
         * Treats negative exponents as positive.
         */
        SparseMultivariateRationalPolynomial operator^ (long long int e) const;

        /**
         * Update *this by exponentiating this to the input integer.
         * Treats negative exponents as positive.
         */
        SparseMultivariateRationalPolynomial& operator^= (long long int e);

        /**
         * Determine if *this is equal to the specified polynomial.
         * This takes into account the variable ordering on both poylnomials
         * in such a way that the same polynomial under different variable orderings
         * are NOT equal.
         */
        bool operator== (const SparseMultivariateRationalPolynomial& b) const;

        /**
         * Determine if *this is not equal to the specified polynomial.
         * This takes into account the variable ordering on both poylnomials
         * in such a way that the same polynomial under different variable orderings
         * are NOT equal.
         */
        bool operator!= (const SparseMultivariateRationalPolynomial& b) const;

        /**
         * Output the string representation of *this to the input ostream.
         */
        void print(std::ostream& os) const;

        /**
         * Parse polynomial from in stream. Exactly one line is parsed.
         */
        friend std::istream& operator>>(std::istream& in, SparseMultivariateRationalPolynomial& p);

        /**
         * Parse a polynomial from string str and place in *this.
         */
        void fromString(const std::string& str);

        /**
         * Given a polynomial q, compute the subresultant chain between this and q, viewing both polynomial
         * recursively with main variable v.
         *
         * @note, if this and q do not exist in the same ambient space, the space of p will define the ordering of the subresultants.
         *
         * @param q: the other polynomial for which to compute the subresultant chain.
         * @param v: the main variable to be used when computing the subresultant chain.
         * @param filled: if false, degenerative cases are not returned and so indices in the returned chain do not necessarily match subresultant degrees
         *
         * @return a vector containing the subresultant chain whereby index 0 is resultant and subresultant degrees increase with index.
         *
         */
        std::vector<SparseMultivariateRationalPolynomial> subresultantChain(const SparseMultivariateRationalPolynomial& q, const Symbol& v, bool filled = 1) const;

        /**
         * Subresultant Chain with respect to the leaving variable of this.
         * Return the list of subresultants
         **/
        std::vector<SparseMultivariateRationalPolynomial> subresultantChain (const SparseMultivariateRationalPolynomial& q, int filled = 1) const;

        /**
         * Given a polynomial q, compute the subresultant of index idx between this and q, viewing both polynomial
         * recursively with main variable v. This function in fact computes the subresultant of index idx and idx+1
         * and so both are returned, unless idx+1 does not exist (e.g. idx = this->degree(v)).
         *
         * Optionally, this function also returns the principleCoefs of the subresultants of index i to this->degree(v).
         *
         * @note, if the subresultant of index idx+1 is degenerative then many subresultants will be returned until the first non-degenerative one is found.
         * @note, if this and q do not exist in the same ambient space, the space of p will define the ordering of the subresultants.
         *
         * @param q: the other polynomial for which to compute the subresultant chain.subresultantInit
         * @param v: the main variable to be used when computing the subresultant chain.
         *
         * @return two successive subresultants: S_{i}, S_{i+1} s.t. deg(S_{i}) <= idx and deg(S_{i+1} > idx
         *
         */
        std::vector<SparseMultivariateRationalPolynomial> subresultantAtIdx (const SparseMultivariateRationalPolynomial& q, const Symbol& v, int idx = 0, specSRC_AAZ **lazyInfo = NULL) const;

        /**
         * Compute the initials of two successive subresultants: S_{i}, S_{i+1} s.t. deg(S_{i}) <= idx and deg(S_{i+1} > idx.
         * This returns init(S_{i},v) and init(S_{i+1}, v) as a vector.
         * The degrees degree(S_{i},v) and degree(S_{i+1}, v) are returned in mdegIdx and mdegIdx1, respectively.
         * These degrees can be used to determine if certain subresultants are defective, i.e. degree(S_{i}, v) != i.
         */
        std::vector<SparseMultivariateRationalPolynomial> subresultantInitialAtIdx (const SparseMultivariateRationalPolynomial& q, const Symbol& v, int idx, Integer& mdegIdx, Integer& mdegIdx1, specSRC_AAZ **lazyInfo = NULL) const;

        /**
         * Compute the initials of two successive subresultants: S_{i}, S_{i+1} s.t. deg(S_{i}) <= idx and deg(S_{i+1} > idx.
         * This returns init(S_{i},v) and init(S_{i+1}, v) as a vector.
         */
        std::vector<SparseMultivariateRationalPolynomial> subresultantInitialAtIdx (const SparseMultivariateRationalPolynomial& q, const Symbol& v, int idx = 0, specSRC_AAZ **lazyInfo = NULL) const {
            Integer d1, d2;
            return this->subresultantInitialAtIdx(q, v, idx, d1, d2, lazyInfo);
        }

        /**
         * Extended Subresultant Chain
         * Return the list of subresultants with Besout Coefficients
         */
        std::vector<std::vector<SparseMultivariateRationalPolynomial>> exSubresultantChain (const SparseMultivariateRationalPolynomial& q, const Symbol& v) const;

        SparseMultivariateRationalPolynomial resultant (const SparseMultivariateRationalPolynomial& q, const Symbol& v) const;

        /**
         * Get GCD between *this and b.
         */
        SparseMultivariateRationalPolynomial gcd(const SparseMultivariateRationalPolynomial& b) const;

        /**
         * Get the GCD between *this and b as a primitive polynomial.
         */
        SparseMultivariateRationalPolynomial primitiveGCD(const SparseMultivariateRationalPolynomial& b) const;

        /**
         * Compute squarefree factorization of *this with respect to all of its variables.
         */
        Factors<SparseMultivariateRationalPolynomial> squareFree() const;

        /**
         * Compute squarefree factorization of *this with respect to the list of variables, vars.
         */
        Factors<SparseMultivariateRationalPolynomial> squareFree(const std::vector<Symbol>& vars) const;

        /**
         * Computes the square free part of this polynomail. That is, the polynomial of this
         * divided by all square factors. This is with respect to all variables.
         */
        SparseMultivariateRationalPolynomial squareFreePart() const;

        /**
         * Computes the square free part of this polynomail. That is, the polynomial of this
         * divided by all square factors. This is with respect to all variables.
         */
        SparseMultivariateRationalPolynomial squareFreePart(std::vector<Symbol>& vars) const;

        /**
         * Get the content with respect to all variables. That is, a single
         * rational number. The content here is one such that this/content is
         * an integer polynomial with content of 1.
         *
         * Moreover, the content is one such that the leading coefficient
         * of the corresponding primitive part is positive.
         */
        RationalNumber content() const;

        /**
         * Get the content of this polynomial with respect to the variables in
         * the input vector v.
         *
         * Moreover, the content is one such that the leading coefficient
         * of the corresponding primitive part is positive.
         */
        SparseMultivariateRationalPolynomial content(const std::vector<Symbol>& v) const;

        /**
         * Get the primitive part with respect to all variables, returned as an SMZP.
         * This is equivalent to this / content();
         */
        SparseMultivariateIntegerPolynomial primitivePartSMZP() const;

        /**
         * Get the primitive part with respect to all variables, returned as an SMZP.
         * This is equivalent to this / content();
         *
         * Simultaneously returns the rational number content in the parameter content.
         */
        SparseMultivariateIntegerPolynomial primitivePartSMZP(RationalNumber& content) const;

        /**
         * Get the primitive part with respect to all variables.
         * This is equivalent to this / content().
         */
        SparseMultivariateRationalPolynomial primitivePart() const;

        /**
         * Get the primitive part with respect to the variable s.
         * This is equivalent to this / content(s).
         */
        SparseMultivariateRationalPolynomial primitivePart(const Symbol& s) const;

        /**
         * Get the primitive part with respect to all variables.
         * This is equivalent to this / content().
         *
         * Simultaneously returns the rational number content in the parameter content.
         */
        SparseMultivariateRationalPolynomial primitivePart(RationalNumber& content) const;

        /**
         * Get the primitive part with respect to the variables in the vector v.
         *
         * returns the primitive part.
         */
        SparseMultivariateRationalPolynomial primitivePart(const std::vector<Symbol>& v) const;

        /**
         * Get the primitive part with respect to the variables in the vector v.
         * Returns the corresponding content in the content reference.
         * returns the primitive part.
         */
        SparseMultivariateRationalPolynomial primitivePart(const std::vector<Symbol>& v, SparseMultivariateRationalPolynomial& content) const;

        SparseMultivariateRationalPolynomial mainPrimitivePart() const;

        SparseMultivariateRationalPolynomial mainPrimitivePart(SparseMultivariateRationalPolynomial& content) const;

        /**
         * Get the leading coefficient of *this with respect to the main variable.
         *
         * returns the initial.
         */
        SparseMultivariateRationalPolynomial initial() const;

        /**
         * Get the main variable. That is, the highest-ordered variable with
         * positive degree.
         *
         * returns the main variable.
         */
        inline Symbol mainVariable() const {
            return leadingVariable();
        }

        /**
         * Get the degree of the main variable.
         *
         * returns the degree.
         */
        int mainDegree() const;

        /**
         * Get the rank of this polynomial.
         * That is, the main variable raised to the main degree.
         *
         * returns the rank.
         */
        SparseMultivariateRationalPolynomial rank() const;

        /**
         * Get the head of this polynomial. That is, the initial multiplied
         * by the rank.
         *
         * returns the head.
         */
        SparseMultivariateRationalPolynomial head() const;

        /**
         * Get the tail of this polynomial. That is, This - this.head().
         *
         * returns the tail.
         */
        SparseMultivariateRationalPolynomial tail() const;

        SparseMultivariateRationalPolynomial separant() const;

        /* BPASMultivariatePolynomial interface */

        /**
         * Get the number of variables in this polynomial.
         */
        int numberOfVariables() const;

        /**
         * Get the number of variables in this polynomial ring.
         */
        inline int numberOfRingVariables() const {
            return ringVariables().size();
        }

        /**
         * Get the number of non-zero terms
         */
        Integer numberOfTerms() const;

        /**
         * Total degree.
         */
        Integer degree() const;

        /**
         * Get the degree of a variable
         */
        Integer degree(const Symbol&) const;

        /**
         * Get the leading coefficient
         */
        RationalNumber leadingCoefficient() const;

        /**
         * Get the trailing coefficient.
         */
        RationalNumber trailingCoefficient() const;

        /**
         * Get a coefficient, given the exponent of each variable
         */
        RationalNumber coefficient(int, const int*) const;

        inline RationalNumber coefficient(const std::vector<int>& v) const {
            return coefficient(v.size(), v.data());
        }

        /**
         * Set a coefficient, given the exponent of each variable
         */
        void setCoefficient(int, const int*, const RationalNumber&);

        inline void setCoefficient(const std::vector<int>& v, const RationalNumber& r) {
            setCoefficient(v.size(), v.data(), r);
        }


        /**
         * Set variables' names.
         *
         * This method can be used to shrink, expand, re-order, and re-name the variables
         * of the polynomial ring in which this polynomial belongs.
         *
         * Any symbol in the input vector that matches the current variables
         * is considered a re-order. Non-matching symbols are considered a re-name
         * and this renames is done in the order they appear by increasing index.
         * For example: Q[x,y,z] -> Q[y,s,t] will have y reordered to be the
         * first variable, x renamed to s, and z renamed to t.
         *
         * If the size of the input vector is greater than the current number of variables
         * then the polynomial ring is being expanded. Matching variables from this
         * polynomial are re-ordered as they appear in the input vector. Current variables
         * that have no match in the input vector are re-named in order of increasing index
         * of unused variables from the input vector.
         * For example: Q[x,y,z] -> Q[s,t,z,x] will have y named to s, t a new variable,
         * and z and x re-ordered accordingly.
         *
         * If the size of the input vector is less than the current number of variables
         * then the polynomial ring is shrink to remove variables. Variables in the
         * input vector that are also found to be in the current polynomial ring
         * are matched and re-ordered as necessary.
         *
         * It is invalid to remove a variable which is non-zero in this polynomial.
         */
        void setRingVariables (const std::vector<Symbol>&);

        /**
         * Get variable names of all variables available to this polynomial,
         * even those that have zero degree.
         */
        std::vector<Symbol> ringVariables() const;

        /**
         * Get variable names of variables with non-zero degree;
         */
        std::vector<Symbol> variables() const;


        /* SMQP-specific */

        /**
         * Determine if *this is equal to b.
         * This takes into account the variable ordering on both poylnomials
         * in such a way that the same polynomial under different variable orderings
         * are NOT equal.
         */
        bool isEqual(const SparseMultivariateRationalPolynomial& b) const;

        /**
         * Convert current object to its k-th derivative
         *
         * @param s: Symbol to differentiate with respect to
         * @param k: Order of the derivative, k > 0
         **/
        inline void differentiate(const Symbol& s, int k) {
            *this = this->derivative(s, k);
        }

        /**
         * Convert current object to its derivative
         *
         * @param s: Symbol to differentiate with respect to
         **/
        inline void differentiate(const Symbol& s) {
            this->differentiate(s,1);
        }

        /**
         * Return k-th derivative
         *
         * @param s: Symbol to differentiate with respect to
         * @param k: Order of the k-th derivative, k > 0
         **/
        SparseMultivariateRationalPolynomial derivative(const Symbol& s, int k) const;

        /**
         * Compute derivative
         *
         * @param s: Symbol to differentiate with respect to
         **/
        inline SparseMultivariateRationalPolynomial derivative(const Symbol& s) const {
            return this->derivative(s,1);
        }

        /**
         * Convert current object to its k-th integral
         *
         * @param s: Symbol to differentiate with respect to
         * @param k: Order of the derivative, k > 0
         **/
        inline void integrate(const Symbol& s, int k) {
            *this = this->integral(s, k);
        }

        /**
         * Convert current object to its derivative
         *
         * @param s: Symbol to differentiate with respect to
         **/
        inline void integrate(const Symbol& s) {
            this->integrate(s,1);
        }

        /**
         * Return k-th integral. If Symbol s is not a symbol contained in this
         * polynomial then it becomes the main variable of the result and
         * integration proceeds as expected.
         *
         * @param s: Symbol to differentiate with respect to
         * @param k: Order of the k-th derivative, k > 0
         **/
        SparseMultivariateRationalPolynomial integral(const Symbol& s, int k) const;

        /**
         * Compute integral
         *
         * @param s: Symbol to differentiate with respect to
         **/
        inline SparseMultivariateRationalPolynomial integral(const Symbol& s) const {
            return this->integral(s,1);
        }

        /**
         * Evaluate f(x)
         *
         * @param syms: Array of Symbols to evaluate at corresponding xs
         * @param xs: Evaluation points
         **/
        inline SparseMultivariateRationalPolynomial evaluate(int n, const Symbol* syms, const RationalNumber* xs) const {
            std::vector<Symbol> vecSyms;
            std::vector<RationalNumber> vecRats;
            vecSyms.reserve(n);
            vecRats.reserve(n);
            for (int i = 0; i < n; ++i) {
                vecSyms.push_back(syms[i]);
                vecRats.push_back(xs[i]);
            }
            return evaluate(vecSyms, vecRats);
        }

        /**
         * Evaluate *this polynomial given the variables and their values in vars, and values.
         * vars must be a list of variables which are a (not necessarily proper) subset.
         * vars and values should have matching indices. i.e. the values of vars[i] is values[i].
         *
         * returns a new SMQP where all variables in vars have been evaluated using the values given.
         */
        SparseMultivariateRationalPolynomial evaluate(const std::vector<Symbol>& vars, const std::vector<RationalNumber>& values) const;

        /**
         * Find the interpolating polynomial for the points and values specified by each vector, respectively.
         * The points vector is a vector of vectors, where each element of the outer vector represents a multi-dimensional point.
         * Each multi-dimensional point should have the same number of dimensions and in the same order.
         *
         * returns the interpolating polynomial.
         */
        static SparseMultivariateRationalPolynomial interpolate(const std::vector<std::vector<RationalNumber>>& points, const std::vector<RationalNumber>& vals);

        /**
         * Divide this by polynomial b, returning the quotient and remainder in q and r, respectively.
         *
         * returns a boolean indicating if the division was exact.
         */
        bool divide(const SparseMultivariateRationalPolynomial& b, SparseMultivariateRationalPolynomial& q, SparseMultivariateRationalPolynomial& r) const;

        /**
         * Get the remainder of *this divided by b.
         */
        SparseMultivariateRationalPolynomial operator% (const SparseMultivariateRationalPolynomial& b) const;

        /**
         * Update *this by setting it to the remainder of *this divided by b.
         */
        SparseMultivariateRationalPolynomial& operator%= (const SparseMultivariateRationalPolynomial& b);


        inline SparseMultivariateRationalPolynomial operator% (const Integer& i) const {
            SparseMultivariateRationalPolynomial ret = *this;
            ret %= i;
            return ret;
        }

        inline SparseMultivariateRationalPolynomial& operator%= (const Integer& i) {
            if (!this->isZero()) {
                applyModuloSymmetricPrimPart_AA_inp(this->poly, i.get_mpz_t());
            }
            return *this;
        }

        /**
         * Pseudo divide this by b. The remainder is returned.
         * if parameter quo is not null then the quotient is returned in quo.
         * if parameter mult is not null then the multiplier is set to the initial of b
         * raised to the power of degree(c, mvar(c)) - degree(b, mvar(c)) + 1.
         *
         * returns the pseudo remainder.
         */
        SparseMultivariateRationalPolynomial pseudoDivide(const SparseMultivariateRationalPolynomial& b, SparseMultivariateRationalPolynomial* quo = NULL, SparseMultivariateRationalPolynomial* mult = NULL, bool lazy = 0) const;

        /**
         * Pseudo divide this by b. The remainder is returned.
         * This function assumes that both this and b are primitive, and thus
         * can be viewed as integer polynomials. The psuedo-division is then performed
         * over the integers and the quotient and remainder returned also so they are primitive.
         *
         * If parameter quo is not null then the quotient is returned in quo.
         * If parameter mult is not null then the multiplier is set to the initial of b
         * raised to the power of degree(c, mvar(c)) - degree(b, mvar(c)) + 1.
         *
         * @return the pseudo remainder.
         */
        SparseMultivariateRationalPolynomial pseudoDivideBySMZP(const SparseMultivariateRationalPolynomial& b, SparseMultivariateRationalPolynomial* quo = NULL, SparseMultivariateRationalPolynomial* mult = NULL, bool lazy = 0) const;


        /**
         * Add *this and a ratNum_t.
         */
        inline SparseMultivariateRationalPolynomial operator+ (const ratNum_t& r) const {
            return (*this + RationalNumber(r));
        }

        /**
         * Add *this and the RationalNumber r.
         */
        SparseMultivariateRationalPolynomial operator+ (const RationalNumber& r) const;

        /**
         * Add ratNum_t r and SMQP b.
         */
        inline friend SparseMultivariateRationalPolynomial operator+ (const ratNum_t& r, const SparseMultivariateRationalPolynomial& b) {
            return (b + r);
        }

        /**
         * Update *this by adding r
         */
        inline SparseMultivariateRationalPolynomial& operator+= (const ratNum_t& r) {
            return (*this += RationalNumber(r));
        }

        /**
         * Add RationalNumber r and SMQP b.
         */
        inline friend SparseMultivariateRationalPolynomial operator+ (const RationalNumber& r, const SparseMultivariateRationalPolynomial& b) {
            return (b + r);
        }

        /**
         * Update *this by adding the RationalNumber r to it.
         */
        SparseMultivariateRationalPolynomial& operator+= (const RationalNumber& r);

        /**
         * Subtract the ratNum_t r from *this.
         */
        inline SparseMultivariateRationalPolynomial operator- (const ratNum_t& r) const {
            return (*this - RationalNumber(r));
        }

        /**
         * Subtract the RationalNumber r from *this.
         */
        SparseMultivariateRationalPolynomial operator- (const RationalNumber& r) const;

        /**
         * Subtract SMQP b from ratNum_t r.
         */
        inline friend SparseMultivariateRationalPolynomial operator- (const ratNum_t& r, const SparseMultivariateRationalPolynomial& b) {
            return (-b + r);
        }

        /**
         * Subtract SMQP b from ratNum_t r.
         */
        inline friend SparseMultivariateRationalPolynomial operator- (const RationalNumber& r, const SparseMultivariateRationalPolynomial& b) {
            return (-b + r);
        }

        /**
         * Update *this by subtracting ratNum_t r.
         */
        inline SparseMultivariateRationalPolynomial& operator-= (const ratNum_t& r) {
            return (*this -= RationalNumber(r));
        }

        /**
         * Update *this by subtracting RationalNumber r.
         */
        SparseMultivariateRationalPolynomial& operator-= (const RationalNumber& r);

        /**
         * Multiply *this by ratNum_t r.
         */
        inline SparseMultivariateRationalPolynomial operator* (const ratNum_t& r) const {
            return (*this * RationalNumber(r));
        }

        /**
         * Multiply *this by RationalNumber r.
         */
        SparseMultivariateRationalPolynomial operator* (const RationalNumber& r) const;

        /**
         * Multiply ratNum_t r and SMQP b.
         */
        inline friend SparseMultivariateRationalPolynomial operator* (const ratNum_t& r, const SparseMultivariateRationalPolynomial& b) {
            return (b * r);
        }

        /**
         * Update *this by multiplying by ratNum_t r.
         */
        inline SparseMultivariateRationalPolynomial& operator*= (const ratNum_t& r) {
            return (*this *= RationalNumber(r));
        }

        /**
         * Multiply RationalNumber r and SMQP b.
         */
        inline friend SparseMultivariateRationalPolynomial operator* (const RationalNumber& r, const SparseMultivariateRationalPolynomial& b) {
            return (b * r);
        }

        /**
         * Update *this by multiplying by RationalNumber r.
         */
        SparseMultivariateRationalPolynomial& operator*= (const RationalNumber& r);

        /**
         * Divide *this by ratNum_t r.
         */
        inline SparseMultivariateRationalPolynomial operator/ (const ratNum_t& r) const {
            return (*this / RationalNumber(r));
        }

        /**
         * Divide *this by RationalNumber r.
         */
        SparseMultivariateRationalPolynomial operator/ (const RationalNumber& r) const;

        /**
         * Divide ratNum_t r by SMQP b.
         */
        friend SparseMultivariateRationalPolynomial operator/ (const ratNum_t& r, const SparseMultivariateRationalPolynomial& b) ;

        /**
         * Update *this by dividing by ratNum_t r.
         */
        inline SparseMultivariateRationalPolynomial& operator/= (const ratNum_t& r) {
            return (*this /= RationalNumber(r));
        }
        /**
         * Divide RationalNumber r by SMQP b.
         */
        inline friend SparseMultivariateRationalPolynomial operator/ (const RationalNumber& r, const SparseMultivariateRationalPolynomial& b) {
            ratNum_t t;
            mpq_init(t);
            mpq_set(t, r.get_mpq_t());
            SparseMultivariateRationalPolynomial ret = (t / b);
            mpq_clear(t);
            return ret;
        }

        /**
         * Update *this by dividing by RationalNumber r.
         */
        SparseMultivariateRationalPolynomial& operator/= (const RationalNumber& r);

        /**
         * Get the polynomial term at index. Returns 0 if index is beyond the
         * number of terms in this polynomial.
         */
        SparseMultivariateRationalPolynomial operator[] (int index) const;

        /**
         * Get the leading variable, that is, the highest-order variable with positive degree
         * of this polynomial.
         * returns the leading variable or the empty string if this polynomial has zero variables.
         */
        Symbol leadingVariable() const;

        /**
         * Get the degree of this polynomial w.r.t the leading variable.
         */
        Integer leadingVariableDegree() const;

        /**
         * Is the contant term zero.
         */
        bool isConstantTermZero() const;

        /**
         * Get the leading coefficient w.r.t the input variable 'x'.
         * Returns the leading exponent as e.
         *
         * returns the coefficient as a new SMQP.
         */
        SparseMultivariateRationalPolynomial leadingCoefficientInVariable (const Symbol& x, int* e = NULL) const;

        /**
         * Convert to a SUP<SMQP> given the variable 'x'.
         *
         * returns the SUP<SMQP>.
         */
        SparseUnivariatePolynomial<SparseMultivariateRationalPolynomial> convertToSUP(const Symbol& x) const;

        /**
         * Negate all the coefficients of *this. Note, that due to the
         * sharing nature of underling nodes, this may alter the Nodes of
         * other SMQP.
         */
        void negate();

        /**
         * Get a copy of this such that all underlying memory is NOT shared.
         * Note, any following arithmetic operations on the returned result
         * will result in possibly making the underlying memory shared again.
         */
        SparseMultivariateRationalPolynomial deepCopy() const;

        /**
         * Factors this polynomial into irreducible factors.
         * The Factors may include a common numerical (rational) factor.
         * See also, Factors.
         */
        Factors<SparseMultivariateRationalPolynomial> factor() const;

        /**
         * SLP representation of the polynomial
         **/
        void straightLineProgram();

        /**
         * Print SLP representation
         **/
        void printSLP(std::ostream& out = std::cout) const;

        /* Real Root Isolation */
    private:
        int isIntervalsMatchable(Intervals*, Intervals*, DenseUnivariateRationalPolynomial*, DenseUnivariateRationalPolynomial*, lfixq);
        int refineSleeveUnivariateIntervals(Intervals*, Intervals*, Intervals*, DenseUnivariateRationalPolynomial*, DenseUnivariateRationalPolynomial*, lfixq);
        void sleeveBoundURPolynomials(DenseUnivariateRationalPolynomial*, DenseUnivariateRationalPolynomial*, Intervals&, int, int);
    public:
        /**
         * Given one real root for x_1, .., x_{n-1},
         * isolate positive roots for x_n
         *
         * @param mpIs: Roots of x_n (Output)
         * @param apIs: A root of previous polynomials
         * @param s: deal with 0: positive roots; 1: negative roots
         * @param check: 1: check the leading or tail coefficient; 0: do not
         * @param width: Interval's right - left < width
         * @param ts: Taylor Shift option
         *
         * Return
         * 1: Need to refine preious polynomials
         * 0: Found positive real roots
         **/
        int positiveRealRootIsolation(Intervals* pIs, Intervals& apIs, mpq_class width, int ts=-1, bool s=0, bool check=1);

        /**
         * Change *this to be a random non-zero polynomial.
         * numvar: number of variables
         * nterms: number of terms
         * coefBound: limit on the number of bits encoding the coefficients.
         * sparsity: succesive terms are at most sparsity away from each other
         * includeNeg: a bool to say if coefs can be randomly negative or all positive
         */
        void randomPolynomial(int numvar, int nterms, unsigned long int coefBound, degree_t sparsity, bool includeNeg);

        /**
         * Change *this to be a random non-zero polynomial. The number of variables will
         * be equal to the size of the maxDegs vector. The term whose monomial
         * has exponents equal to those in maxDegs is guaranteed to exist in the
         * resulting polynomial.
         * A sparsity of 0 produces a dense polynomial. A sparsity of 1 produces only
         * one term; one whose monomial has exponents equal to maxDegs.
         *
         * coefBound: limit on the number of bits encoding the coefficients.
         * sparsity: a percentage of zero-terms between term with maxDegs and the constant term.
         * includeNeg: a bool to say if coefs can be randomly negative or all positive
         */
        void randomPolynomial(std::vector<int> maxDegs, unsigned long int coefBound, float sparsity, bool includeNeg);

        /**
         * Convert *this to an ExpressionTree.
         *
         * returns the new ExpressionTree.
         */
        ExpressionTree convertToExpressionTree() const;


/****************
* Multi-Divisor Division
*****************/

/**
 * Normal Form (or Multi-Divisor Division (MDD))
 * Given the dividend, f, and a divisor-set of polynomials of size s,
 * G[s] = {g_0, ..., g_{s-1}} to compute the reduce polynomial (remainder) r with respect to the G[s],
 * Return (by reference) the remainder and the quotient set Q[s] = {q_0, ..., q_{s-1}},
 * such that f = q_0*g_0 + ... + q_{s-1}*g_{s-1} + r.
 */
    SparseMultivariateRationalPolynomial lexNormalForm(const std::vector<Symbol>& superNames,
                               const std::vector<SparseMultivariateRationalPolynomial>& ts, std::vector<SparseMultivariateRationalPolynomial>* quoSet = NULL) const;

  SparseMultivariateRationalPolynomial lexNormalizeDim0 (const std::vector<Symbol>& superNames,
                                                         const std::vector<SparseMultivariateRationalPolynomial>& ts, SparseMultivariateRationalPolynomial* A) const;

  /**
 * Multi-Divisor Division (MDD) where the divisor-set is a Triangular Set
 * Given the  dividend, f, and a divisor-set of polynomials of size s,
 * G[s] = {g_0, ..., g_{s-1}} to compute the reduce polynomial (remainder) r with respect to the G[s],
 * Return (by reference) the remainder and the quotient set Q[s] = {q_0, ..., q_{s-1}}.
 */
    SparseMultivariateRationalPolynomial triangularSetNormalForm(const TriangularSet<RationalNumber,SparseMultivariateRationalPolynomial>& ts, std::vector<SparseMultivariateRationalPolynomial>* quoSet) const;

/**
 * Do the pseudo division of c by the triangular-set (divisor set) of B in the naive principle
 * such that hPow*f = quoSet_0 * B_0 + ... + quoSet_{nSet-1} * B_{nSet-1} + rem.
 * Quotients are returned in quoSet, and remainder in rem.
 * If hPow is not NULL then *hPow is set to the initial of b to the power of
 * the exact number of division steps which occurred..
 * nvar : size of exponent vectors
 * nSet : the size of the divisor set
 */
    SparseMultivariateRationalPolynomial triangularSetPseudoDivide (const TriangularSet<RationalNumber,SparseMultivariateRationalPolynomial>& ts,
                                    std::vector<SparseMultivariateRationalPolynomial>* quoSet, SparseMultivariateRationalPolynomial* h) const;

/**
 * Specialized Normal Form where the divisor-set is a Triangular Set
 * Given the  dividend, f, and a divisor-set of polynomials of size s,
 * G[s] = {g_0, ..., g_{s-1}} to compute the reduce polynomial (remainder) r with respect to the G[s].
 */
SparseMultivariateRationalPolynomial triangularSetOnlyNormalForm (const TriangularSet<RationalNumber, SparseMultivariateRationalPolynomial>& ts) const;








/*

***/

Factors<SparseMultivariateRationalPolynomial> Factoring (SparseMultivariateRationalPolynomial& a);
Factors<SparseMultivariateRationalPolynomial> AlgebricFactorization(SparseMultivariateRationalPolynomial& minimal,SparseMultivariateRationalPolynomial*polyTotalcontent);


};





#endif //_SMQPLINKEDLIST_H_