UnivariatePolynomial.tpp

Go to the documentation of this file.

/** 
 * @file:   UnivariatePolynomial.tpp
 * @author: Sebastian Junges
 *
 * @since August 26, 2013
 */
 
#pragma once
 
#include <carl-common/debug/debug.h>
#include <carl-common/meta/platform.h>
#include <carl-common/meta/SFINAE.h>
#include <carl-logging/carl-logging.h>
#include "MultivariatePolynomial.h"
#include <carl-arith/core/Sign.h>
 
#include "functions/Derivative.h"
#include "functions/Division.h"
 
#include <algorithm>
#include <iomanip>
 
namespace carl
{
 
template<typename Coeff>
UnivariatePolynomial<Coeff>::UnivariatePolynomial(const UnivariatePolynomial& p):
    mMainVar(p.mMainVar), mCoefficients(p.mCoefficients)
{
    assert(this->is_consistent());
}
 
template<typename Coeff>
UnivariatePolynomial<Coeff>::UnivariatePolynomial(UnivariatePolynomial&& p) noexcept:
    mMainVar(p.mMainVar), mCoefficients()
{
    mCoefficients = std::move(p.mCoefficients);
    assert(is_consistent());
}
 
template<typename Coeff>
UnivariatePolynomial<Coeff>& UnivariatePolynomial<Coeff>::operator=(const UnivariatePolynomial& p) {
    mMainVar = p.mMainVar;
    mCoefficients = p.mCoefficients;
    assert(is_consistent());
    return *this;
}
 
template<typename Coeff>
UnivariatePolynomial<Coeff>& UnivariatePolynomial<Coeff>::operator=(UnivariatePolynomial&& p) noexcept {
    mMainVar = p.mMainVar;
    mCoefficients = std::move(p.mCoefficients);
    assert(is_consistent());
    return *this;
}
 
template<typename Coeff>
UnivariatePolynomial<Coeff>::UnivariatePolynomial(Variable mainVar)
: mMainVar(mainVar), mCoefficients()
{
    assert(this->is_consistent());
}
template<typename Coeff>
UnivariatePolynomial<Coeff>::UnivariatePolynomial(Variable mainVar, const Coeff& coeff, std::size_t degree) :
mMainVar(mainVar),
mCoefficients(degree+1, Coeff(0)) // We would like to use 0 here, but Coeff(0) is not always constructable (some methods need more parameter)
{
    if(coeff != Coeff(0))
    {
        mCoefficients[degree] = coeff;
    }
    else
    {
        mCoefficients.clear();
    }
    strip_leading_zeroes();
    assert(is_consistent());
}
 
template<typename Coeff>
UnivariatePolynomial<Coeff>::UnivariatePolynomial(Variable mainVar, std::initializer_list<Coeff> coefficients)
: mMainVar(mainVar), mCoefficients(coefficients)
{
    this->strip_leading_zeroes();
    assert(this->is_consistent());
}
 
template<typename Coeff>
template<typename C, DisableIf<std::is_same<C, typename UnderlyingNumberType<C>::type>>>
UnivariatePolynomial<Coeff>::UnivariatePolynomial(Variable mainVar, std::initializer_list<typename UnderlyingNumberType<C>::type> coefficients)
: mMainVar(mainVar), mCoefficients()
{
    for (auto c: coefficients) {
        this->mCoefficients.push_back(Coeff(c));
    }
    this->strip_leading_zeroes();
    assert(this->is_consistent());
}
 
template<typename Coeff>
UnivariatePolynomial<Coeff>::UnivariatePolynomial(Variable mainVar, const std::vector<Coeff>& coefficients)
: mMainVar(mainVar), mCoefficients(coefficients)
{
    this->strip_leading_zeroes();
    assert(this->is_consistent());
}
 
template<typename Coeff>
UnivariatePolynomial<Coeff>::UnivariatePolynomial(Variable mainVar, std::vector<Coeff>&& coefficients)
: mMainVar(mainVar), mCoefficients(coefficients)
{
    this->strip_leading_zeroes();
    assert(this->is_consistent());
}
 
template<typename Coeff>
UnivariatePolynomial<Coeff>::UnivariatePolynomial(Variable mainVar, const std::map<uint, Coeff>& coefficients)
: mMainVar(mainVar)
{
    mCoefficients.reserve(coefficients.rbegin()->first);
    for (const auto& expAndCoeff : coefficients)
    {
        if(expAndCoeff.first != mCoefficients.size())
        {
            mCoefficients.resize(expAndCoeff.first, Coeff(0));
        }
        mCoefficients.push_back(expAndCoeff.second);
    }
    this->strip_leading_zeroes();
    assert(this->is_consistent());
}
 
template<typename Coeff>
Coeff UnivariatePolynomial<Coeff>::evaluate(const Coeff& value) const 
{
    Coeff result(0);
    Coeff var(1);
    for(const Coeff& coeff : mCoefficients)
    {
        result += (coeff * var);
        var *= value;
    }
    return result;
}
 
template<typename Coeff>
bool UnivariatePolynomial<Coeff>::is_normal() const
{
    return unit_part() == Coeff(1);
}
 
template<typename Coefficient>
UnivariatePolynomial<Coefficient>& UnivariatePolynomial<Coefficient>::mod(const Coefficient& modulus)
{
    for(Coefficient& coeff : mCoefficients)
    {
        coeff = carl::mod(coeff, modulus);
    }
    return *this;
}
 
template<typename Coefficient>
UnivariatePolynomial<Coefficient> UnivariatePolynomial<Coefficient>::mod(const Coefficient& modulus) const
{
    UnivariatePolynomial<Coefficient> result;
    result.mCoefficients.reserve(mCoefficients.size());
    for(const Coefficient& coeff : mCoefficients)
    {
        result.mCoefficients.push_back(mod(coeff, modulus));
    }
    result.strip_leading_zeroes();
    return result;
}
 
template<typename Coeff>
UnivariatePolynomial<Coeff> UnivariatePolynomial<Coeff>::pow(std::size_t exp) const
{
    if (exp == 0) return UnivariatePolynomial(mMainVar, constant_one<Coeff>::get());
    if (carl::is_zero(*this)) return UnivariatePolynomial(mMainVar);
    UnivariatePolynomial<Coeff> res(mMainVar, constant_one<Coeff>::get());
    UnivariatePolynomial<Coeff> mult(*this);
    while(exp > 0) {
        if ((exp & 1) != 0) res *= mult;
        exp /= 2;
        if(exp > 0) mult = mult * mult;
    }
    return res;
}
 
template<typename Coeff>
template<typename C, DisableIf<is_number_type<C>>>
UnivariatePolynomial<typename UnivariatePolynomial<Coeff>::NumberType> UnivariatePolynomial<Coeff>::toNumberCoefficients() const {
    std::vector<NumberType> coeffs;
    coeffs.reserve(this->mCoefficients.size());
    for (auto c: this->mCoefficients) {
        assert(c.is_constant());
        coeffs.push_back(c.constant_part());
    }
    return UnivariatePolynomial<NumberType>(this->mMainVar, coeffs);
}
 
template<typename Coeff>
template<typename NewCoeff>
UnivariatePolynomial<NewCoeff> UnivariatePolynomial<Coeff>::convert() const {
    std::vector<NewCoeff> coeffs;
    coeffs.resize(this->mCoefficients.size());
    for (std::size_t i = 0; i < this->mCoefficients.size(); i++) {
        coeffs[i] = NewCoeff(this->mCoefficients[i]);
    }
    return UnivariatePolynomial<NewCoeff>(this->mMainVar, coeffs);
}
 
template<typename Coeff>
template<typename NewCoeff>
UnivariatePolynomial<NewCoeff> UnivariatePolynomial<Coeff>::convert(const std::function<NewCoeff(const Coeff&)>& f) const {
    std::vector<NewCoeff> coeffs;
    coeffs.resize(this->mCoefficients.size());
    for (std::size_t i = 0; i < this->mCoefficients.size(); i++) {
        coeffs[i] = f(this->mCoefficients[i]);
    }
    return UnivariatePolynomial<NewCoeff>(this->mMainVar, coeffs);
}
 
template<typename Coeff>
UnivariatePolynomial<Coeff> UnivariatePolynomial<Coeff>::normalized() const
{
    if(carl::is_zero(*this))
    {
        return *this;
    }
    return *this/unit_part();
}
 
template<typename Coeff>
Coeff UnivariatePolynomial<Coeff>::unit_part() const {
    if constexpr (is_field_type<Coeff>::value) {
        // Coeffs from a field
        return lcoeff();
    } else if constexpr (is_number_type<Coeff>::value) {
        // Coeffs from a number ring
        if (carl::is_zero(*this) || lcoeff() > Coeff(0)) {
            return Coeff(1);
        } else {
            return Coeff(-1);
        }
    } else {
        // Coeffs from a polynomial ring
        if (carl::is_zero(*this) || carl::is_zero(lcoeff()) || lcoeff().lcoeff() > NumberType(0)) {
            return Coeff(1);
        } else {
            return Coeff(-1);
        }
    }
}
 
template<typename Coeff>
template<typename C, EnableIf<is_subset_of_rationals_type<C>>>
Coeff UnivariatePolynomial<Coeff>::coprime_factor() const
{
    assert(!carl::is_zero(*this));
    auto it = mCoefficients.begin();
    IntNumberType num = get_num(*it);
    IntNumberType den = get_denom(*it);
    for (++it; it != mCoefficients.end(); ++it) {
        num = carl::gcd(num, get_num(*it));
        den = carl::lcm(den, get_denom(*it));
    }
    return Coeff(den)/Coeff(num);
}
 
template<typename Coeff>
template<typename C, DisableIf<is_subset_of_rationals_type<C>>>
typename UnderlyingNumberType<Coeff>::type UnivariatePolynomial<Coeff>::coprime_factor() const
{
    assert(!carl::is_zero(*this));
    auto it = mCoefficients.begin();
    typename UnderlyingNumberType<Coeff>::type factor = it->coprime_factor();
    for (++it; it != mCoefficients.end(); ++it) {
        factor = carl::lcm(factor, it->coprime_factor());
    }
    return factor;
}
 
template<typename Coeff>
template<typename C, EnableIf<is_subset_of_rationals_type<C>>>
UnivariatePolynomial<typename IntegralType<Coeff>::type> UnivariatePolynomial<Coeff>::coprime_coefficients() const
{
    CARL_LOG_TRACE("carl.core", *this << " .coprime_coefficients()");
    // Notice that even if factor is 1, we create a new polynomial
    UnivariatePolynomial<typename IntegralType<Coeff>::type> result(mMainVar);
    if (carl::is_zero(*this)) {
        return result;
    }
    result.mCoefficients.reserve(mCoefficients.size());
    Coeff factor = this->coprime_factor();
    for (const Coeff& coeff: mCoefficients) {
        assert(get_denom(coeff * factor) == 1);
        result.mCoefficients.push_back(get_num(coeff * factor));
    }
    return result;
}
 
template<typename Coeff>
template<typename C, DisableIf<is_subset_of_rationals_type<C>>>
UnivariatePolynomial<Coeff> UnivariatePolynomial<Coeff>::coprime_coefficients() const {
    if (carl::is_zero(*this)) return *this;
    UnivariatePolynomial<Coeff> result(mMainVar);
    result.mCoefficients.reserve(mCoefficients.size());
    auto factor = this->coprime_factor();
    for (const Coeff& c: mCoefficients) {
        result.mCoefficients.push_back(factor * c);
    }
    return result;
}
 
template<typename Coeff>
template<typename C, EnableIf<is_subset_of_rationals_type<C>>>
UnivariatePolynomial<typename IntegralType<Coeff>::type> UnivariatePolynomial<Coeff>::coprime_coefficients_sign_preserving() const
{
    CARL_LOG_TRACE("carl.core", *this << " .coprime_coefficients_sign_preserving()");
    // Notice that even if factor is 1, we create a new polynomial
    UnivariatePolynomial<typename IntegralType<Coeff>::type> result(mMainVar);
    if (carl::is_zero(*this)) {
        return result;
    }
    result.mCoefficients.reserve(mCoefficients.size());
    Coeff factor = carl::abs(this->coprime_factor());
    for (const Coeff& coeff: mCoefficients) {
        assert(get_denom(coeff * factor) == 1);
        result.mCoefficients.push_back(get_num(coeff * factor));
    }
    return result;
}
 
template<typename Coeff>
template<typename C, DisableIf<is_subset_of_rationals_type<C>>>
UnivariatePolynomial<Coeff> UnivariatePolynomial<Coeff>::coprime_coefficients_sign_preserving() const {
    if (this->is_zero()) return *this;
    UnivariatePolynomial<Coeff> result(mMainVar);
    result.mCoefficients.reserve(mCoefficients.size());
    auto factor = carl::abs(this->coprime_factor());
    for (const Coeff& c: mCoefficients) {
        result.mCoefficients.push_back(factor * c);
    }
    return result;
}
 
template<typename Coeff>
bool UnivariatePolynomial<Coeff>::divides(const UnivariatePolynomial& divisor) const
{
    ///@todo Is this correct?
    return carl::is_zero(divisor.divideBy(*this).remainder);
}
 
template<typename Coeff>
template<typename C, EnableIf<is_instantiation_of<GFNumber, C>>>
UnivariatePolynomial<typename IntegralType<Coeff>::type> UnivariatePolynomial<Coeff>::to_integer_domain() const
{
    UnivariatePolynomial<typename IntegralType<Coeff>::type> res(mMainVar);
    res.mCoefficients.reserve(mCoefficients.size());
    for(const Coeff& c : mCoefficients)
    {
        assert(carl::is_integer(c));
        res.mCoefficients.push_back(c.representing_integer());
    }
    res.strip_leading_zeroes();
    return res;
}
 
template<typename Coeff>
template<typename C, DisableIf<is_instantiation_of<GFNumber, C>>>
UnivariatePolynomial<typename IntegralType<Coeff>::type> UnivariatePolynomial<Coeff>::to_integer_domain() const
{
    UnivariatePolynomial<typename IntegralType<Coeff>::type> res(mMainVar);
    res.mCoefficients.reserve(mCoefficients.size());
    for(const Coeff& c : mCoefficients)
    {
        assert(carl::is_integer(c));
        res.mCoefficients.push_back(get_num(c));
    }
    res.strip_leading_zeroes();
    return res;
}
 
template<typename Coeff>
//template<typename T = Coeff, EnableIf<!std::is_same<IntegralType<Coeff>, bool>::value>>
UnivariatePolynomial<GFNumber<typename IntegralType<Coeff>::type>> UnivariatePolynomial<Coeff>::toFiniteDomain(const GaloisField<typename IntegralType<Coeff>::type>* galoisField) const
{
    UnivariatePolynomial<GFNumber<typename IntegralType<Coeff>::type>> res(mMainVar);
    res.mCoefficients.reserve(mCoefficients.size());
    for(const Coeff& c : mCoefficients)
    {
        assert(carl::is_integer(c));
        res.mCoefficients.push_back(GFNumber<typename IntegralType<Coeff>::type>(c,galoisField));
    }
    res.strip_leading_zeroes();
    return res;
    
}
 
template<typename Coeff>
template<typename N, EnableIf<is_subset_of_rationals_type<N>>>
typename UnivariatePolynomial<Coeff>::NumberType UnivariatePolynomial<Coeff>::numeric_content() const
{
    if (carl::is_zero(*this)) return NumberType(0);
    // Obtain main denominator for all coefficients.
    IntNumberType mainDenom = this->main_denom();
    
    // now, some coefficient * mainDenom is always integral.
    // we convert such a product to an integral data type by get_num()
    UnivariatePolynomial<Coeff>::NumberType c = this->numeric_content(0) * mainDenom;
    assert(get_denom(c) == 1);
    IntNumberType res = get_num(c);
    for (std::size_t i = 1; i < this->mCoefficients.size(); i++) {
        c = this->numeric_content(i) * mainDenom;
        assert(get_denom(c) == 1);
        res = carl::gcd(get_num(c), res);
    }
    return NumberType(res) / mainDenom;
}
 
template<typename Coeff>
template<typename C, EnableIf<is_number_type<C>>>
typename UnivariatePolynomial<Coeff>::IntNumberType UnivariatePolynomial<Coeff>::main_denom() const
{
    IntNumberType denom = 1;
    for (const auto& c: mCoefficients) {
        denom = carl::lcm(denom, get_denom(c));
    }
    CARL_LOG_TRACE("carl.core", "mainDenom of " << *this << " is " << denom);
    return denom;
}
template<typename Coeff>
template<typename C, DisableIf<is_number_type<C>>>
typename UnivariatePolynomial<Coeff>::IntNumberType UnivariatePolynomial<Coeff>::main_denom() const
{
    IntNumberType denom = 1;
    for (const auto& c: mCoefficients) {
        denom = carl::lcm(denom, c.main_denom());
    }
    return denom;
}
 
template<typename Coeff>
Coeff UnivariatePolynomial<Coeff>::synthetic_division(const Coeff& zeroOfDivisor)
{
    if(coefficients().empty()) return Coeff(0);
    if(coefficients().size() == 1) return coefficients().back();
    std::vector<Coeff> secondRow;
    secondRow.reserve(coefficients().size());
    secondRow.push_back(Coeff(0));
    std::vector<Coeff> thirdRow(coefficients().size(), Coeff(0));
    size_t posThirdRow = coefficients().size()-1; 
    auto coeff = coefficients().rbegin();
    thirdRow[posThirdRow] = (*coeff) + secondRow.front();
    ++coeff;
    while(coeff != coefficients().rend())
    {
        secondRow.push_back(zeroOfDivisor*thirdRow[posThirdRow]);
        --posThirdRow;
        thirdRow[posThirdRow] = (*coeff) + secondRow.back();
        ++coeff;
    }
    assert(posThirdRow == 0);
    CARL_LOG_TRACE("carl.core.upoly", "UnivSynDiv: (" << *this << ")[x -> " << zeroOfDivisor << "]  =  " << thirdRow.front());
    if(thirdRow.front() == 0)
    {
        thirdRow.erase(thirdRow.begin());
        this->mCoefficients.swap(thirdRow);
        CARL_LOG_TRACE("carl.core.upoly", "UnivSynDiv: reduced by ((" << carl::abs(get_denom(thirdRow.front())) << ")*" << main_var() << " + (" << (thirdRow.front()<0 ? "-" : "") << carl::abs(get_num(thirdRow.front())) << "))  ->  " << *this);
        return Coeff(0);
    }
    return thirdRow.front();
}
 
template<typename Coeff>
UnivariatePolynomial<Coeff> UnivariatePolynomial<Coeff>::operator -() const
{
    UnivariatePolynomial result(mMainVar);
    result.mCoefficients.reserve(mCoefficients.size());
    for(const auto& c : mCoefficients) {
        result.mCoefficients.push_back(-c);
    }
 
    return result;       
}
 
template<typename Coefficient>
UnivariatePolynomial<Coefficient>& UnivariatePolynomial<Coefficient>::operator+=(const Coefficient& rhs)
{
    if(rhs == Coefficient(0)) return *this;
    if(mCoefficients.empty())
    {
        // Adding non-zero rhs to zero.
        mCoefficients.resize(1, rhs);
    }
    else
    {
        mCoefficients.front() += rhs;
        if(mCoefficients.size() == 1 && mCoefficients.front() == Coefficient(0)) 
        {
            // Result is zero.
            mCoefficients.clear();
        }
    }
    return *this;
}
 
template<typename Coeff>
UnivariatePolynomial<Coeff>& UnivariatePolynomial<Coeff>::operator+=(const UnivariatePolynomial& rhs)
{
    assert(mMainVar == rhs.mMainVar);
    
    if(carl::is_zero(rhs))
    {
        return *this;
    }
    
    if(mCoefficients.size() < rhs.mCoefficients.size())
    {
        for(std::size_t i = 0; i < mCoefficients.size(); ++i)
        {
            mCoefficients[i] += rhs.mCoefficients[i];
        }
        mCoefficients.insert(mCoefficients.end(), rhs.mCoefficients.end() - sint(rhs.mCoefficients.size() - mCoefficients.size()), rhs.mCoefficients.end());
    }
    else
    {
        for(std::size_t i = 0; i < rhs.mCoefficients.size(); ++i)
        {
            mCoefficients[i] += rhs.mCoefficients[i]; 
        }
    }
    strip_leading_zeroes();
    return *this;
}
 
template<typename C>
UnivariatePolynomial<C> operator+(const UnivariatePolynomial<C>& lhs, const UnivariatePolynomial<C>& rhs)
{
    UnivariatePolynomial<C> res(lhs);
    res += rhs;
    return res;
}
 
template<typename C>
UnivariatePolynomial<C> operator+(const UnivariatePolynomial<C>& lhs, const C& rhs)
{
    UnivariatePolynomial<C> res(lhs);
    res += rhs;
    return res;
}
 
template<typename C>
UnivariatePolynomial<C> operator+(const C& lhs, const UnivariatePolynomial<C>& rhs)
{
    return rhs + lhs;
}
    
 
template<typename Coefficient>
UnivariatePolynomial<Coefficient>& UnivariatePolynomial<Coefficient>::operator-=(const Coefficient& rhs)
{
//  CARL_LOG_INEFFICIENT();
    return *this += -rhs;
}
 
template<typename Coeff>
UnivariatePolynomial<Coeff>& UnivariatePolynomial<Coeff>::operator-=(const UnivariatePolynomial& rhs)
{
//  CARL_LOG_INEFFICIENT();
    return *this += -rhs;
}
 
 
template<typename C>
UnivariatePolynomial<C> operator-(const UnivariatePolynomial<C>& lhs, const UnivariatePolynomial<C>& rhs)
{
    UnivariatePolynomial<C> res(lhs);
    res -= rhs;
    return res;
}
 
template<typename C>
UnivariatePolynomial<C> operator-(const UnivariatePolynomial<C>& lhs, const C& rhs)
{
    UnivariatePolynomial<C> res(lhs);
    res -= rhs;
    return res;
}
 
template<typename C>
UnivariatePolynomial<C> operator-(const C& lhs, const UnivariatePolynomial<C>& rhs)
{
    return rhs - lhs;
}
 
template<typename Coefficient>
template<typename C, EnableIf<is_number_type<C>>>
UnivariatePolynomial<Coefficient>& UnivariatePolynomial<Coefficient>::operator*=(Variable rhs) {
    if (rhs == this->mMainVar) {
        this->mCoefficients.insert(this->mCoefficients.begin(), Coefficient(0));
        return *this;
    }
    assert(!carl::is_number_type<Coefficient>::value);
    return *this;
}
template<typename Coefficient>
template<typename C, DisableIf<is_number_type<C>>>
UnivariatePolynomial<Coefficient>& UnivariatePolynomial<Coefficient>::operator*=(Variable rhs) {
    if (rhs == this->mMainVar) {
        this->mCoefficients.insert(this->mCoefficients.begin(), Coefficient(0));
        return *this;
    }
    assert(!carl::is_number_type<Coefficient>::value);
    for (auto& c: this->mCoefficients) c *= rhs;
    return *this;
}
 
template<typename Coefficient>
UnivariatePolynomial<Coefficient>& UnivariatePolynomial<Coefficient>::operator*=(const Coefficient& rhs)
{
    if(rhs == Coefficient(0))
    {
        mCoefficients.clear();
        return *this;
    }
    for(Coefficient& c : mCoefficients)
    {
        c *= rhs;
    }
    
    if(is_finite_type<Coefficient>::value)
    {
        strip_leading_zeroes();
    }
    
    return *this;       
}
 
 
template<typename Coeff>
template<typename I, DisableIf<std::is_same<Coeff, I>>...>
UnivariatePolynomial<Coeff>& UnivariatePolynomial<Coeff>::operator*=(const typename IntegralType<Coeff>::type& rhs)
{
    static_assert(std::is_same<Coeff, I>::value, "Do not provide template parameters");
    if(rhs == I(0))
    {
        mCoefficients.clear();
        return *this;
    }
    for(Coeff& c : mCoefficients)
    {
        c *= rhs;
    }
    return *this;       
}
 
template<typename Coeff>
UnivariatePolynomial<Coeff>& UnivariatePolynomial<Coeff>::operator*=(const UnivariatePolynomial& rhs)
{
    assert(mMainVar == rhs.mMainVar);
    if(carl::is_zero(rhs))
    {
        mCoefficients.clear();
        return *this;
    }
    
    std::vector<Coeff> newCoeffs; 
    newCoeffs.reserve(mCoefficients.size() + rhs.mCoefficients.size());
    for(std::size_t e = 0; e < mCoefficients.size() + rhs.degree(); ++e)
    {
        newCoeffs.push_back(Coeff(0));
        for(std::size_t i = 0; i < mCoefficients.size() && i <= e; ++i)
        {
            if(e - i < rhs.mCoefficients.size())
            {
                newCoeffs.back() += mCoefficients[i] * rhs.mCoefficients[e-i];
            }
        }
    }
    mCoefficients.swap(newCoeffs);
    strip_leading_zeroes();
    return *this;
}
 
 
template<typename C>
UnivariatePolynomial<C> operator*(const UnivariatePolynomial<C>& lhs, const UnivariatePolynomial<C>& rhs)
{
    UnivariatePolynomial<C> res(lhs);
    res *= rhs;
    return res;
}
 
template<typename C>
UnivariatePolynomial<C> operator*(const UnivariatePolynomial<C>& lhs, Variable rhs) {
    return std::move(UnivariatePolynomial<C>(lhs) *= rhs);
}
template<typename C>
UnivariatePolynomial<C> operator*(Variable lhs, const UnivariatePolynomial<C>& rhs) {
    return std::move(UnivariatePolynomial<C>(rhs) *= lhs);
}
 
template<typename C>
UnivariatePolynomial<C> operator*(const UnivariatePolynomial<C>& lhs, const C& rhs)
{
    UnivariatePolynomial<C> res(lhs);
    res *= rhs;
    return res;
}
 
template<typename C>
UnivariatePolynomial<C> operator*(const C& lhs, const UnivariatePolynomial<C>& rhs)
{
    return rhs * lhs;
}
 
template<typename C>
UnivariatePolynomial<C> operator*(const UnivariatePolynomial<C>& lhs, const IntegralTypeIfDifferent<C>& rhs)
{
    UnivariatePolynomial<C> res(lhs);
    res *= rhs;
    return res;
}
 
template<typename C>
UnivariatePolynomial<C> operator*(const IntegralTypeIfDifferent<C>& lhs, const UnivariatePolynomial<C>& rhs)
{
    return rhs * lhs;
}
 
 
 
template<typename Coeff>
template<typename C, EnableIf<is_field_type<C>>>
UnivariatePolynomial<Coeff>& UnivariatePolynomial<Coeff>::operator/=(const Coeff& rhs)
{
    assert(rhs != Coeff(0));
    for(Coeff& c : mCoefficients)
    {
        c /= rhs;
    }
    return *this;       
}
 
template<typename Coeff>
template<typename C, DisableIf<is_field_type<C>>>
UnivariatePolynomial<Coeff>& UnivariatePolynomial<Coeff>::operator/=(const Coeff& rhs)
{
    assert(rhs != Coeff(0));
    for(Coeff& c : mCoefficients)
    {
        c = quotient(c, rhs);
    }
    /// TODO not fully sure whether this is necessary
    this->strip_leading_zeroes();
    return *this;       
}
 
 
 
template<typename C>
UnivariatePolynomial<C> operator/(const UnivariatePolynomial<C>& lhs, const C& rhs)
{
    assert(rhs != C(0));
    if(carl::is_zero(lhs)) return lhs;
    UnivariatePolynomial<C> res(lhs);
    return res /= rhs;
}
 
template<typename C>
bool operator==(const UnivariatePolynomial<C>& lhs, const UnivariatePolynomial<C>& rhs)
{
    assert(lhs.is_consistent());
    assert(rhs.is_consistent());
    if(lhs.mMainVar == rhs.mMainVar)
    {
        return lhs.mCoefficients == rhs.mCoefficients;
    }
    else
    {
        // in different variables, polynomials can still be equal if constant.
        if(carl::is_zero(lhs) && carl::is_zero(rhs)) return true;
        if ((lhs.mCoefficients.size() == 1) && (rhs.mCoefficients.size() == 1) && (lhs.mCoefficients == rhs.mCoefficients)) return true;
        // Convert to multivariate and compare that.
        return MultivariatePolynomial<typename carl::UnderlyingNumberType<C>::type>(lhs) == MultivariatePolynomial<typename carl::UnderlyingNumberType<C>::type>(rhs);
    }
}
template<typename C>
bool operator==(const UnivariatePolynomialPtr<C>& lhs, const UnivariatePolynomialPtr<C>& rhs)
{
    if (lhs == nullptr && rhs == nullptr) return true;
    if (lhs == nullptr || rhs == nullptr) return false;
    return *lhs == *rhs;
}
 
template<typename C>
bool operator==(const UnivariatePolynomial<C>& lhs, const C& rhs)
{   
    if (lhs.coefficients().size() == 0) {
        return carl::is_zero(rhs);
    }
    return (lhs.coefficients().size() == 1) && lhs.lcoeff() == rhs;
}
 
template<typename C>
bool operator==(const C& lhs, const UnivariatePolynomial<C>& rhs)
{
    return rhs == lhs;
}
 
 
template<typename C>
bool operator!=(const UnivariatePolynomial<C>& lhs, const UnivariatePolynomial<C>& rhs)
{
    return !(lhs == rhs);
}
template<typename C>
bool operator!=(const UnivariatePolynomialPtr<C>& lhs, const UnivariatePolynomialPtr<C>& rhs)
{
    if (lhs == nullptr && rhs == nullptr) return false;
    if (lhs == nullptr || rhs == nullptr) return true;
    return *lhs != *rhs;
}
 
template<typename C>
bool UnivariatePolynomial<C>::less(const UnivariatePolynomial<C>& rhs, const PolynomialComparisonOrder& order) const {
    switch (order) {
        case PolynomialComparisonOrder::CauchyBound: /*{
            C a = this->cauchyBound();
            C b = rhs.cauchyBound();
            if (a < b) return true;
            return *this < rhs;
        }*/
        case PolynomialComparisonOrder::LowDegree:
            if (carl::is_zero(rhs)) return false;
            if (carl::is_zero(*this) || this->degree() < rhs.degree()) return true;
            return *this < rhs;
        case PolynomialComparisonOrder::Memory:
            return this < &rhs;
    }
    return false;
}
 
template<typename C>
bool operator<(const UnivariatePolynomial<C>& lhs, const UnivariatePolynomial<C>& rhs)
{
    if(lhs.mMainVar == rhs.mMainVar)
    {
        if(lhs.coefficients().size() == rhs.coefficients().size())
        {
            auto iterLhs = lhs.coefficients().rbegin();
            auto iterRhs = rhs.coefficients().rbegin();
            while(iterLhs != lhs.coefficients().rend())
            {
                assert(iterRhs != rhs.coefficients().rend());
                if(*iterLhs == *iterRhs)
                {
                    ++iterLhs;
                    ++iterRhs;
                }
                else
                {
                    return *iterLhs < *iterRhs;
                }
            }
        }
        return lhs.coefficients().size() < rhs.coefficients().size();
    }
    return lhs.mMainVar < rhs.mMainVar;
}
 
template<typename C>
std::ostream& operator<<(std::ostream& os, const UnivariatePolynomial<C>& rhs)
{
    if(carl::is_zero(rhs)) return os << "0";
    for(size_t i = 0; i < rhs.mCoefficients.size()-1; ++i )
    {
        const C& c = rhs.mCoefficients[rhs.mCoefficients.size()-i-1];
        if(!(c == 0))
        {
            if (!(c == 1)) os << "(" << c << ")*";
            os << rhs.mMainVar << "^" << rhs.mCoefficients.size()-i-1 << " + ";
        }
    }
    os << rhs.mCoefficients[0];
    return os;
}
 
template<typename Coefficient>
template<typename C, EnableIf<is_number_type<C>>>
bool UnivariatePolynomial<Coefficient>::is_consistent() const {
    if (!mCoefficients.empty()) {
        assert(!carl::is_zero(lcoeff()));
    }
    return true;
}
 
template<typename Coefficient>
template<typename C, DisableIf<is_number_type<C>>>
bool UnivariatePolynomial<Coefficient>::is_consistent() const {
    if (!mCoefficients.empty()) {
        assert(!carl::is_zero(lcoeff()));
    }
    for (const auto& c: mCoefficients) {
        assert(!c.has(main_var()));
        assert(c.is_consistent());
    }
    return true;
}
 
}