tpl_polynomial.H

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/*
                          Aleph_w
 
  Data structures & Algorithms
  version 2.0.0b
  https://github.com/lrleon/Aleph-w
 
  This file is part of Aleph-w library
 
  Copyright (c) 2002-2026 Leandro Rabindranath Leon
 
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  of this software and associated documentation files (the "Software"), to deal
  in the Software without restriction, including without limitation the rights
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*/
 
#ifndef TPL_POLYNOMIAL_H
#define TPL_POLYNOMIAL_H
 
#include <cmath>
#include <initializer_list>
#include <limits>
#include <sstream>
#include <string>
#include <type_traits>
#include <utility>
#include <numeric>  // std::gcd
#include <random>   // std::mt19937_64
 
#include <ah-errors.H>
#include <tpl_dynMapTree.H>
#include <tpl_array.H>
#include <htlist.H>
#include <modular_arithmetic.H>  // mod_inv, mod_mul
 
namespace Aleph {
 
namespace polynomial_detail {
 
template <typename C>
C power(C base, size_t exp)
{
  C result = C(1);
  while (exp > 0)
    {
      if (exp & 1)
        result = result * base;
      base = base * base;
      exp >>= 1;
    }
  return result;
}
 
template <typename Int>
  requires std::is_integral_v<Int>
[[nodiscard]] uint64_t abs_to_u64(Int value) noexcept
{
  if constexpr (std::is_signed_v<Int>)
    {
      using Unsigned = std::make_unsigned_t<Int>;
      const Unsigned magnitude =
        value < Int{} ? Unsigned{} - static_cast<Unsigned>(value) : static_cast<Unsigned>(value);
      return static_cast<uint64_t>(magnitude);
    }
  else
    return static_cast<uint64_t>(value);
}
 
template <typename Int>
  requires std::is_integral_v<Int>
[[nodiscard]] uint64_t normalize_mod_u64(Int value, const uint64_t modulus) noexcept
{
  if (modulus == 0)
    return 0;
 
  if constexpr (std::is_signed_v<Int>)
    {
      const auto mod_i64 = static_cast<int64_t>(modulus);
      auto rem = static_cast<int64_t>(value % static_cast<Int>(mod_i64));
      if (rem < 0)
        rem += mod_i64;
      return static_cast<uint64_t>(rem);
    }
  else
    return static_cast<uint64_t>(value % static_cast<Int>(modulus));
}
 
template <typename Int>
  requires std::is_integral_v<Int>
[[nodiscard]] Int centered_from_mod_u64(const uint64_t value, const uint64_t modulus) noexcept
{
  if constexpr (std::is_signed_v<Int>)
    {
      const uint64_t canonical = modulus == 0 ? 0 : value % modulus;
      const uint64_t half = modulus / 2;
      if (canonical > half)
        return static_cast<Int>(static_cast<int64_t>(canonical) - static_cast<int64_t>(modulus));
      return static_cast<Int>(canonical);
    }
  else
    return static_cast<Int>(modulus == 0 ? 0 : value % modulus);
}
 
}  // end namespace polynomial_detail
 
template <typename Coefficient = double>
class Gen_Polynomial
{
public:
  using Coeff = Coefficient;
 
private:
  using Term = std::pair<size_t, Coefficient>;
  DynMapTree<size_t, Coefficient> coeffs;
 
  // --- Epsilon-based zero detection for floating-point types ----------
 
  static constexpr Coefficient epsilon() noexcept
  {
    if constexpr (std::is_floating_point_v<Coefficient>)
      return Coefficient(128) * std::numeric_limits<Coefficient>::epsilon();
    else
      return Coefficient{};
  }
 
  static bool coeff_is_zero(const Coefficient &c) noexcept
  {
    if constexpr (std::is_floating_point_v<Coefficient>)
      return std::abs(c) <= epsilon();
    else
      return c == Coefficient{};
  }
 
  void remove_zeros()
  {
    DynList<size_t> zeros;
    for (auto it = coeffs.get_it(); it.has_curr(); it.next_ne())
      {
        auto &p = it.get_curr();
        if (coeff_is_zero(p.second))
          zeros.append(p.first);
      }
    zeros.for_each([this](size_t e)
      {
        coeffs.remove_key(e);
      });
  }
 
  Coefficient coeff_at(size_t exp) const
  {
    auto *p = coeffs.search(exp);
    return p != nullptr ? p->second : Coefficient{};
  }
 
  Coefficient *coeff_ptr(size_t exp) noexcept
  {
    auto *p = coeffs.search(exp);
    return p != nullptr ? &p->second : nullptr;
  }
 
  const Coefficient *coeff_ptr(size_t exp) const noexcept
  {
    auto *p = coeffs.search(exp);
    return p != nullptr ? &p->second : nullptr;
  }
 
  void add_to_coeff(size_t exp, const Coefficient &delta)
  {
    if (coeff_is_zero(delta))
      return;
 
    if (auto *slot = coeff_ptr(exp); slot != nullptr)
      {
        *slot = *slot + delta;
        if (coeff_is_zero(*slot))
          coeffs.remove_key(exp);
      }
    else
      coeffs.insert(exp, delta);
  }
 
  void add_scaled_shifted(const Gen_Polynomial &src, const Coefficient &scale, size_t shift = 0)
  {
    if (coeff_is_zero(scale) or src.is_zero())
      return;
 
    src.for_each_term([this, &scale, shift](size_t exp, const Coefficient &c)
      {
        add_to_coeff(exp + shift, c * scale);
      });
  }
 
  void scale_inplace(const Coefficient &s)
  {
    if (coeffs.is_empty())
      return;
 
    if (coeff_is_zero(s))
      {
        coeffs = DynMapTree<size_t, Coefficient>();
        return;
      }
 
    if (s == Coefficient(1))
      return;
 
    DynList<size_t> zeros;
    for (auto it = coeffs.get_it(); it.has_curr(); it.next_ne())
      {
        auto &p = it.get_curr();
        p.second = p.second * s;
        if (coeff_is_zero(p.second))
          zeros.append(p.first);
      }
 
    zeros.for_each([this](size_t exp)
      {
        coeffs.remove_key(exp);
      });
  }
 
  void divide_scalar_inplace(const Coefficient &s)
  {
    ah_domain_error_if(coeff_is_zero(s)) << "polynomial division by zero scalar";
 
    if (coeffs.is_empty())
      return;
 
    if (s == Coefficient(1))
      return;
 
    DynList<size_t> zeros;
    for (auto it = coeffs.get_it(); it.has_curr(); it.next_ne())
      {
        auto &p = it.get_curr();
        p.second = p.second / s;
        if (coeff_is_zero(p.second))
          zeros.append(p.first);
      }
 
    zeros.for_each([this](size_t exp)
      {
        coeffs.remove_key(exp);
      });
  }
 
  template <class Op>
  void for_each_term_desc(Op &&op) const
  {
    Array<const Term *> desc_terms(coeffs.size());
    for (auto it = coeffs.get_it(); it.has_curr(); it.next_ne())
      {
        const auto &p = it.get_curr();
        desc_terms.append(&p);
      }
 
    for (size_t i = desc_terms.size(); i > 0; --i)
      {
        const Term &term = *desc_terms(i - 1);
        op(term.first, term.second);
      }
  }
 
  [[nodiscard]] Gen_Polynomial multiply_by_monic_linear(const Coefficient &c) const
  {
    if (is_zero())
      return Gen_Polynomial();
 
    Gen_Polynomial result;
    for (auto it = coeffs.get_it(); it.has_curr(); it.next_ne())
      {
        auto &p = it.get_curr();
        result.add_to_coeff(p.first, p.second * c);
        result.add_to_coeff(p.first + 1, p.second);
      }
    return result;
  }
 
public:
  // =================================================================
  //  Construction
  // =================================================================
 
  Gen_Polynomial() = default;
 
  explicit Gen_Polynomial(const Coefficient &c)
  {
    if (not coeff_is_zero(c))
      coeffs[0] = c;
  }
 
  Gen_Polynomial(const Coefficient &c, size_t exp)
  {
    if (not coeff_is_zero(c))
      coeffs[exp] = c;
  }
 
  Gen_Polynomial(std::initializer_list<Coefficient> il)
  {
    size_t exp = 0;
    for (auto &c : il)
      {
        if (not coeff_is_zero(c))
          coeffs[exp] = c;
        ++exp;
      }
  }
 
  explicit Gen_Polynomial(const DynList<Coefficient> &l)
  {
    size_t exp = 0;
    l.for_each([this, &exp](const Coefficient &c)
      {
        if (not coeff_is_zero(c))
          coeffs[exp] = c;
        ++exp;
      });
  }
 
  explicit Gen_Polynomial(const DynList<std::pair<size_t, Coefficient>> &term_list)
  {
    term_list.for_each([this](const Term &t)
      {
        if (not coeff_is_zero(t.second))
          add_to_coeff(t.first, t.second);
      });
  }
 
  // =================================================================
  //  Static factory methods
  // =================================================================
 
  [[nodiscard]] static Gen_Polynomial zero()
  {
    return Gen_Polynomial();
  }
 
  [[nodiscard]] static Gen_Polynomial one()
  {
    return Gen_Polynomial(Coefficient(1));
  }
 
  [[nodiscard]] static Gen_Polynomial x_to(size_t n)
  {
    return Gen_Polynomial(Coefficient(1), n);
  }
 
  [[nodiscard]] static Gen_Polynomial from_roots(const DynList<Coefficient> &roots)
  {
    Gen_Polynomial result(Coefficient(1));  // start with constant 1
    roots.for_each([&result](const Coefficient &r)
      {
        result = result.multiply_by_monic_linear(-r);  // (x - r)
      });
    return result;
  }
 
  [[nodiscard]] static Gen_Polynomial interpolate(
    const DynList<std::pair<Coefficient, Coefficient>> &points)
  {
    ah_domain_error_if(points.is_empty()) << "interpolate requires at least one point";
 
    const size_t n = points.size();
    Array<Coefficient> xs;
    Array<Coefficient> dd;
    xs.putn(n);
    dd.putn(n);
 
    size_t idx = 0;
    points.for_each([&](const std::pair<Coefficient, Coefficient> &pt)
      {
        xs[idx] = pt.first;
        dd[idx] = pt.second;
        ++idx;
      });
 
    for (size_t i = 0; i < n; ++i)
      for (size_t j = i + 1; j < n; ++j)
        ah_domain_error_if(coeff_is_zero(xs[i] - xs[j])) << "interpolate: duplicate x-values";
 
    for (size_t order = 1; order < n; ++order)
      for (size_t i = n - 1; i >= order; --i)
        {
          Coefficient denom = xs[i] - xs[i - order];
          dd[i] = (dd[i] - dd[i - 1]) / denom;
          if (i == order)
            break;
        }
 
    Gen_Polynomial result(dd[0]);
    Gen_Polynomial basis(Coefficient(1));
    for (size_t i = 1; i < n; ++i)
      {
        basis = basis.multiply_by_monic_linear(-xs[i - 1]);
        if (not coeff_is_zero(dd[i]))
          result.add_scaled_shifted(basis, dd[i]);
      }
 
    return result;
  }
 
  // =================================================================
  //  Properties
  // =================================================================
 
  [[nodiscard]] bool is_zero() const noexcept
  {
    return coeffs.is_empty();
  }
 
  [[nodiscard]] size_t degree() const noexcept
  {
    if (coeffs.is_empty())
      return 0;
    return coeffs.max().first;
  }
 
  [[nodiscard]] size_t num_terms() const noexcept
  {
    return coeffs.size();
  }
 
  [[nodiscard]] bool is_constant() const noexcept
  {
    return coeffs.is_empty() or (coeffs.size() == 1 and coeffs.min().first == 0);
  }
 
  [[nodiscard]] bool is_monomial() const noexcept
  {
    return coeffs.size() == 1;
  }
 
  [[nodiscard]] bool is_monic() const noexcept
  {
    return not coeffs.is_empty() and coeff_is_zero(coeffs.max().second - Coefficient(1));
  }
 
  // =================================================================
  //  Coefficient access
  // =================================================================
 
  [[nodiscard]] Coefficient operator [] (size_t exp) const
  {
    return coeff_at(exp);
  }
 
  [[nodiscard]] Coefficient leading_coeff() const noexcept
  {
    if (coeffs.is_empty())
      return Coefficient{};
    return coeffs.max().second;
  }
 
  [[nodiscard]] bool has_term(size_t exp) const noexcept
  {
    return coeffs.has(exp);
  }
 
  // =================================================================
  //  Evaluation
  // =================================================================
 
  [[nodiscard]] Coefficient horner_eval(const Coefficient &x) const
  {
    if (coeffs.is_empty())
      return Coefficient{};
 
    if (x == Coefficient{})
      return coeff_at(0);
 
    Coefficient result = Coefficient{};
    size_t prev_exp = 0;
    bool first = true;
 
    for_each_term_desc([&](size_t exp, const Coefficient &c)
      {
        if (first)
          {
            result = c;
            prev_exp = exp;
            first = false;
            return;
          }
 
        result = result * polynomial_detail::power(x, prev_exp - exp) + c;
        prev_exp = exp;
      });
 
    if (prev_exp > 0)
      result = result * polynomial_detail::power(x, prev_exp);
 
    return result;
  }
 
  [[nodiscard]] Coefficient sparse_eval(const Coefficient &x) const
  {
    Coefficient result = Coefficient{};
    for (auto it = coeffs.get_it(); it.has_curr(); it.next_ne())
      {
        auto &p = it.get_curr();
        result = result + p.second * polynomial_detail::power(x, p.first);
      }
    return result;
  }
 
  [[nodiscard]] Coefficient eval(const Coefficient &x) const
  {
    if (coeffs.is_empty())
      return Coefficient{};
 
    if (x == Coefficient{})
      return coeff_at(0);
 
    if (num_terms() <= 2)
      return sparse_eval(x);
 
    return horner_eval(x);
  }
 
  [[nodiscard]] Coefficient operator () (const Coefficient &x) const
  {
    return eval(x);
  }
 
  [[nodiscard]] DynList<Coefficient> multi_eval(const DynList<Coefficient> &points) const
  {
    DynList<Coefficient> results;
    points.for_each([this, &results](const Coefficient &x)
      {
        results.append(eval(x));
      });
    return results;
  }
 
  // =================================================================
  //  Arithmetic operators
  // =================================================================
 
  [[nodiscard]] Gen_Polynomial operator - () const
  {
    Gen_Polynomial result;
    for (auto it = coeffs.get_it(); it.has_curr(); it.next_ne())
      {
        auto &p = it.get_curr();
        result.coeffs[p.first] = -p.second;
      }
    return result;
  }
 
  [[nodiscard]] Gen_Polynomial operator + (const Gen_Polynomial &q) const
  {
    Gen_Polynomial result = *this;
    result += q;
    return result;
  }
 
  Gen_Polynomial &operator += (const Gen_Polynomial &q)
  {
    if (this == &q)
      {
        scale_inplace(Coefficient(1) + Coefficient(1));
        return *this;
      }
 
    for (auto it = q.coeffs.get_it(); it.has_curr(); it.next_ne())
      {
        auto &p = it.get_curr();
        add_to_coeff(p.first, p.second);
      }
    return *this;
  }
 
  [[nodiscard]] Gen_Polynomial operator + (const Coefficient &c) const
  {
    Gen_Polynomial result = *this;
    result += c;
    return result;
  }
 
  Gen_Polynomial &operator += (const Coefficient &c)
  {
    add_to_coeff(0, c);
    return *this;
  }
 
  [[nodiscard]] Gen_Polynomial operator - (const Gen_Polynomial &q) const
  {
    Gen_Polynomial result = *this;
    result -= q;
    return result;
  }
 
  Gen_Polynomial &operator -= (const Gen_Polynomial &q)
  {
    if (this == &q)
      {
        coeffs = DynMapTree<size_t, Coefficient>();
        return *this;
      }
 
    for (auto it = q.coeffs.get_it(); it.has_curr(); it.next_ne())
      {
        auto &p = it.get_curr();
        add_to_coeff(p.first, -p.second);
      }
    return *this;
  }
 
  [[nodiscard]] Gen_Polynomial operator - (const Coefficient &c) const
  {
    Gen_Polynomial result = *this;
    result -= c;
    return result;
  }
 
  Gen_Polynomial &operator -= (const Coefficient &c)
  {
    add_to_coeff(0, -c);
    return *this;
  }
 
  [[nodiscard]] Gen_Polynomial operator * (const Gen_Polynomial &q) const
  {
    if (is_zero() or q.is_zero())
      return Gen_Polynomial();
 
    Gen_Polynomial result;
    const Gen_Polynomial *outer = this;
    const Gen_Polynomial *inner = &q;
 
    if (q.num_terms() < num_terms())
      {
        outer = &q;
        inner = this;
      }
 
    for (auto it1 = outer->coeffs.get_it(); it1.has_curr(); it1.next_ne())
      {
        auto &t1 = it1.get_curr();
        for (auto it2 = inner->coeffs.get_it(); it2.has_curr(); it2.next_ne())
          {
            auto &t2 = it2.get_curr();
            result.add_to_coeff(t1.first + t2.first, t1.second * t2.second);
          }
      }
    return result;
  }
 
  Gen_Polynomial &operator *= (const Gen_Polynomial &q)
  {
    *this = *this * q;
    return *this;
  }
 
  [[nodiscard]] Gen_Polynomial operator * (const Coefficient &s) const
  {
    if (coeff_is_zero(s))
      return Gen_Polynomial();
    Gen_Polynomial result;
    for (auto it = coeffs.get_it(); it.has_curr(); it.next_ne())
      {
        auto &p = it.get_curr();
        Coefficient prod = p.second * s;
        if (not coeff_is_zero(prod))
          result.coeffs[p.first] = prod;
      }
    return result;
  }
 
  Gen_Polynomial &operator *= (const Coefficient &s)
  {
    scale_inplace(s);
    return *this;
  }
 
  [[nodiscard]] Gen_Polynomial operator / (const Coefficient &s) const
  {
    ah_domain_error_if(coeff_is_zero(s)) << "polynomial division by zero scalar";
    Gen_Polynomial result;
    for (auto it = coeffs.get_it(); it.has_curr(); it.next_ne())
      {
        auto &p = it.get_curr();
        Coefficient q = p.second / s;
        if (not coeff_is_zero(q))
          result.coeffs[p.first] = q;
      }
    return result;
  }
 
  Gen_Polynomial &operator /= (const Coefficient &s)
  {
    divide_scalar_inplace(s);
    return *this;
  }
 
  // =================================================================
  //  Polynomial division
  // =================================================================
 
  [[nodiscard]] std::pair<Gen_Polynomial, Gen_Polynomial> divmod(const Gen_Polynomial &d) const
  {
    ah_domain_error_if(d.is_zero()) << "polynomial division by zero";
 
    if (is_zero() or degree() < d.degree())
      return {Gen_Polynomial(), *this};
 
    Gen_Polynomial q;          // quotient
    Gen_Polynomial r = *this;  // remainder
    const size_t d_degree = d.degree();
    const Coefficient d_lc = d.leading_coeff();
 
    while (not r.is_zero())
      {
        const size_t r_degree = r.degree();
        if (r_degree < d_degree)
          break;
 
        const Coefficient scale = r.leading_coeff() / d_lc;
        ah_domain_error_if(coeff_is_zero(scale))
          << "divmod: inexact coefficient division (leading_coeff=" << r.leading_coeff()
          << " / divisor_lc=" << d_lc << " = 0). "
          << "For integral/non-field coefficients, use pseudo_divmod() instead.";
        const size_t shift = r_degree - d_degree;
 
        q.add_to_coeff(shift, scale);
        r.add_scaled_shifted(d, -scale, shift);
      }
 
    return {q, r};
  }
 
  [[nodiscard]] Gen_Polynomial operator / (const Gen_Polynomial &d) const
  {
    return divmod(d).first;
  }
 
  [[nodiscard]] Gen_Polynomial operator % (const Gen_Polynomial &d) const
  {
    return divmod(d).second;
  }
 
  Gen_Polynomial &operator /= (const Gen_Polynomial &d)
  {
    *this = divmod(d).first;
    return *this;
  }
 
  Gen_Polynomial &operator %= (const Gen_Polynomial &d)
  {
    *this = divmod(d).second;
    return *this;
  }
 
  // =================================================================
  //  Calculus
  // =================================================================
 
  [[nodiscard]] Gen_Polynomial derivative() const
  {
    Gen_Polynomial result;
    for (auto it = coeffs.get_it(); it.has_curr(); it.next_ne())
      {
        auto &p = it.get_curr();
        if (p.first == 0)
          continue;  // derivative of constant is 0
        Coefficient dc = p.second * Coefficient(p.first);
        if (not coeff_is_zero(dc))
          result.coeffs[p.first - 1] = dc;
      }
    return result;
  }
 
  [[nodiscard]] Gen_Polynomial nth_derivative(size_t n) const
  {
    Gen_Polynomial result = *this;
    for (size_t i = 0; i < n and not result.is_zero(); ++i)
      result = result.derivative();
    return result;
  }
 
  [[nodiscard]] Gen_Polynomial integral(const Coefficient &C = Coefficient{}) const
  {
    Gen_Polynomial result;
    if (not coeff_is_zero(C))
      result.coeffs[0] = C;
    for (auto it = coeffs.get_it(); it.has_curr(); it.next_ne())
      {
        auto &p = it.get_curr();
        Coefficient ic = p.second / Coefficient(p.first + 1);
        if (not coeff_is_zero(ic))
          result.coeffs[p.first + 1] = ic;
      }
    return result;
  }
 
  [[nodiscard]] Coefficient definite_integral(const Coefficient &a, const Coefficient &b) const
  {
    Coefficient result = Coefficient{};
 
    for (auto it = coeffs.get_it(); it.has_curr(); it.next_ne())
      {
        auto &p = it.get_curr();
        const size_t next_exp = p.first + 1;
        const Coefficient scale = p.second / Coefficient(next_exp);
        if (coeff_is_zero(scale))
          continue;
 
        result = result +
          scale * (polynomial_detail::power(b, next_exp) - polynomial_detail::power(a, next_exp));
      }
 
    return result;
  }
 
  // =================================================================
  //  Composition and power
  // =================================================================
 
  [[nodiscard]] Gen_Polynomial compose(const Gen_Polynomial &q) const
  {
    if (is_zero())
      return Gen_Polynomial();
 
    if (q.is_constant())
      return Gen_Polynomial(eval(q[0]));
 
    Gen_Polynomial result;
    size_t prev_exp = 0;
    bool first = true;
 
    for_each_term_desc([&](size_t exp, const Coefficient &c)
      {
        if (first)
          {
            result = Gen_Polynomial(c);
            prev_exp = exp;
            first = false;
            return;
          }
 
        const size_t gap = prev_exp - exp;
        if (gap == 1)
          result *= q;
        else if (gap > 1)
          result *= q.pow(gap);
 
        result.add_to_coeff(0, c);
        prev_exp = exp;
      });
 
    if (prev_exp == 1)
      result *= q;
    else if (prev_exp > 1)
      result *= q.pow(prev_exp);
 
    return result;
  }
 
  [[nodiscard]] Gen_Polynomial pow(size_t n) const
  {
    Gen_Polynomial result(Coefficient(1));
    Gen_Polynomial base = *this;
    while (n > 0)
      {
        if (n & 1)
          result *= base;
        base *= base;
        n >>= 1;
      }
    return result;
  }
 
  // =================================================================
  //  GCD
  // =================================================================
 
  [[nodiscard]] static Gen_Polynomial gcd(Gen_Polynomial a, Gen_Polynomial b)
  {
    while (not b.is_zero())
      {
        Gen_Polynomial r = a % b;
        a = b;
        b = r;
      }
    // Make monic
    if (not a.is_zero())
      a /= a.leading_coeff();
    return a;
  }
 
  struct Xgcd_Result
  {
    Gen_Polynomial g;  
    Gen_Polynomial s;  
    Gen_Polynomial t;  
  };
 
  [[nodiscard]] static Xgcd_Result xgcd(Gen_Polynomial a, Gen_Polynomial b)
  {
    Gen_Polynomial s0(Coefficient(1)), s1;
    Gen_Polynomial t0, t1(Coefficient(1));
 
    while (not b.is_zero())
      {
        auto [q, r] = a.divmod(b);
        a = b;
        b = r;
        Gen_Polynomial s2 = s0 - q * s1;
        Gen_Polynomial t2 = t0 - q * t1;
        s0 = s1;
        s1 = s2;
        t0 = t1;
        t1 = t2;
      }
 
    if (not a.is_zero())
      {
        Coefficient lc = a.leading_coeff();
        a /= lc;
        s0 /= lc;
        t0 /= lc;
      }
 
    return {a, s0, t0};
  }
 
  [[nodiscard]] static Gen_Polynomial lcm(const Gen_Polynomial &a, const Gen_Polynomial &b)
  {
    if (a.is_zero() or b.is_zero())
      return Gen_Polynomial();
    Gen_Polynomial g = gcd(a, b);
    return a / g * b;
  }
 
  [[nodiscard]] std::pair<Gen_Polynomial, Gen_Polynomial> pseudo_divmod(const Gen_Polynomial &d) const
  {
    ah_domain_error_if(d.is_zero()) << "pseudo-division by zero polynomial";
 
    if (is_zero() or degree() < d.degree())
      return {Gen_Polynomial(), *this};
 
    const Coefficient lc_d = d.leading_coeff();
    const size_t d_degree = d.degree();
    size_t delta = degree() - d_degree + 1;
 
    Gen_Polynomial q;
    Gen_Polynomial r = *this;
 
    while (not r.is_zero())
      {
        const size_t r_degree = r.degree();
        if (r_degree < d_degree)
          break;
 
        const Coefficient lc_r = r.leading_coeff();
        const size_t exp = r_degree - d_degree;
 
        // Multiply q and r by lc(d) to avoid division
        q.scale_inplace(lc_d);
        r.scale_inplace(lc_d);
        --delta;
 
        q.add_to_coeff(exp, lc_r);
        r.add_scaled_shifted(d, -lc_r, exp);
      }
 
    // Multiply by remaining powers of lc(d)
    if (delta > 0)
      {
        const Coefficient scale = polynomial_detail::power(lc_d, delta);
        q.scale_inplace(scale);
        r.scale_inplace(scale);
      }
 
    return {q, r};
  }
 
  // =================================================================
  //  Algebraic transformations
  // =================================================================
 
  [[nodiscard]] Gen_Polynomial to_monic() const
  {
    ah_domain_error_if(is_zero()) << "cannot make zero polynomial monic";
    return *this / leading_coeff();
  }
 
  [[nodiscard]] Gen_Polynomial reverse() const
  {
    if (is_zero())
      return Gen_Polynomial();
    size_t d = degree();
    Gen_Polynomial result;
    for (auto it = coeffs.get_it(); it.has_curr(); it.next_ne())
      {
        auto &p = it.get_curr();
        result.coeffs[d - p.first] = p.second;
      }
    return result;
  }
 
  [[nodiscard]] Gen_Polynomial negate_arg() const
  {
    Gen_Polynomial result;
    for (auto it = coeffs.get_it(); it.has_curr(); it.next_ne())
      {
        auto &p = it.get_curr();
        result.coeffs[p.first] = (p.first % 2 == 0) ? p.second : -p.second;
      }
    return result;
  }
 
  [[nodiscard]] Gen_Polynomial shift(const Coefficient &k) const
  {
    return compose(Gen_Polynomial({k, Coefficient(1)}));
  }
 
  [[nodiscard]] Gen_Polynomial shift_up(size_t k) const
  {
    if (k == 0)
      return *this;
    Gen_Polynomial result;
    for (auto it = coeffs.get_it(); it.has_curr(); it.next_ne())
      {
        auto &p = it.get_curr();
        result.coeffs[p.first + k] = p.second;
      }
    return result;
  }
 
  [[nodiscard]] Gen_Polynomial shift_down(size_t k) const
  {
    if (k == 0)
      return *this;
    Gen_Polynomial result;
    for (auto it = coeffs.get_it(); it.has_curr(); it.next_ne())
      {
        auto &p = it.get_curr();
        if (p.first >= k)
          result.coeffs[p.first - k] = p.second;
      }
    return result;
  }
 
  [[nodiscard]] Gen_Polynomial truncate(size_t n) const
  {
    Gen_Polynomial result;
    for (auto it = coeffs.get_it(); it.has_curr(); it.next_ne())
      {
        auto &p = it.get_curr();
        if (p.first < n)
          result.coeffs[p.first] = p.second;
      }
    return result;
  }
 
  [[nodiscard]] Array<Coefficient> to_dense() const
  {
    if (is_zero())
      return Array<Coefficient>();
    size_t d = degree();
    Array<Coefficient> result(d + 1, Coefficient{});
    for (auto it = coeffs.get_it(); it.has_curr(); it.next_ne())
      {
        auto &p = it.get_curr();
        result(p.first) = p.second;
      }
    return result;
  }
 
  [[nodiscard]] Gen_Polynomial square_free() const
  {
    if (is_zero() or is_constant())
      return *this;
    Gen_Polynomial g = gcd(*this, derivative());
    return *this / g;
  }
 
  // =================================================================
  //  Root analysis (floating-point coefficients)
  // =================================================================
 
  [[nodiscard]] Coefficient cauchy_bound() const
  {
    ah_domain_error_if(is_zero()) << "cauchy_bound: zero polynomial has no roots";
    if (is_constant())
      return Coefficient{};
 
    const size_t d = degree();
    const Coefficient lc = leading_coeff();
    if constexpr (std::is_floating_point_v<Coefficient>)
      {
        Coefficient max_ratio = Coefficient{};
        for (auto it = coeffs.get_it(); it.has_curr(); it.next_ne())
          {
            auto &p = it.get_curr();
            if (p.first == d)
              continue;
            const Coefficient ratio = std::abs(p.second / lc);
            if (ratio > max_ratio)
              max_ratio = ratio;
          }
        return Coefficient(1) + max_ratio;
      }
    else
      {
        long double max_ratio_ld = 0.0L;
        for (auto it = coeffs.get_it(); it.has_curr(); it.next_ne())
          {
            auto &p = it.get_curr();
            if (p.first == d)
              continue;
            const long double ratio =
              std::fabs(static_cast<long double>(p.second) / static_cast<long double>(lc));
            if (ratio > max_ratio_ld)
              max_ratio_ld = ratio;
          }
        return Coefficient(1) + static_cast<Coefficient>(std::ceil(max_ratio_ld));
      }
  }
 
  [[nodiscard]] size_t sign_variations() const
  {
    if (coeffs.size() < 2)
      return 0;
 
    size_t changes = 0;
    bool have_prev = false;
    bool prev_positive = false;
 
    for (auto it = coeffs.get_it(); it.has_curr(); it.next_ne())
      {
        auto &p = it.get_curr();
        bool positive;
        if constexpr (std::is_floating_point_v<Coefficient>)
          positive = p.second > epsilon();
        else
          positive = p.second > Coefficient{};
 
        bool is_neg;
        if constexpr (std::is_floating_point_v<Coefficient>)
          is_neg = p.second < -epsilon();
        else
          is_neg = p.second < Coefficient{};
 
        if (not positive and not is_neg)
          continue;  // skip zero coefficients
 
        if (have_prev and positive != prev_positive)
          ++changes;
        prev_positive = positive;
        have_prev = true;
      }
    return changes;
  }
 
  [[nodiscard]] DynList<Gen_Polynomial> sturm_chain() const
  {
    DynList<Gen_Polynomial> chain;
    chain.append(*this);
    if (is_zero() or is_constant())
      return chain;
 
    Gen_Polynomial p0 = *this;
    Gen_Polynomial p1 = derivative();
    chain.append(p1);
 
    while (not p1.is_zero())
      {
        Gen_Polynomial r = -(p0 % p1);
        if (r.is_zero())
          break;
        chain.append(r);
        p0 = p1;
        p1 = r;
      }
 
    return chain;
  }
 
  [[nodiscard]] static size_t sturm_sign_changes(const DynList<Gen_Polynomial> &chain,
                                                 const Coefficient &x)
  {
    size_t changes = 0;
    bool have_prev = false;
    bool prev_positive = false;
 
    chain.for_each([&](const Gen_Polynomial &p)
      {
        Coefficient val = p(x);
        if (coeff_is_zero(val))
          return;  // skip zeros in Sturm evaluation
 
        bool positive = val > Coefficient{};
        if (have_prev and positive != prev_positive)
          ++changes;
        prev_positive = positive;
        have_prev = true;
      });
 
    return changes;
  }
 
  [[nodiscard]] size_t count_real_roots(const Coefficient &a, const Coefficient &b) const
  {
    if (b < a)
      return count_real_roots(b, a);
 
    if (is_zero())
      return 0;
 
    auto chain = sturm_chain();
    size_t sa = sturm_sign_changes(chain, a);
    size_t sb = sturm_sign_changes(chain, b);
    size_t count = sa > sb ? sa - sb : 0;
 
    // Sturm's theorem counts roots in (a, b]; add root at a if present
    if (coeff_is_zero((*this)(a)))
      ++count;
 
    return count;
  }
 
  [[nodiscard]] size_t count_all_real_roots() const
  {
    if (is_zero() or is_constant())
      return 0;
    Coefficient bound = cauchy_bound();
    return count_real_roots(-bound, bound);
  }
 
  [[nodiscard]] Coefficient bisect_root(Coefficient a,
                                        Coefficient b,
                                        Coefficient tol = Coefficient(1e-12),
                                        size_t max_iter = 200) const
  {
    if (b < a)
      std::swap(a, b);
 
    Coefficient fa = eval(a), fb = eval(b);
    if (coeff_is_zero(fa))
      return a;
    if (coeff_is_zero(fb))
      return b;
 
    ah_domain_error_if((fa > Coefficient{}) == (fb > Coefficient{}) and not coeff_is_zero(fa) and
                       not coeff_is_zero(fb))
      << "bisect_root: f(a) and f(b) must have opposite signs";
 
    for (size_t i = 0; i < max_iter; ++i)
      {
        Coefficient mid = (a + b) / Coefficient(2);
        if (b - a < tol)
          return mid;
        Coefficient fm = eval(mid);
        if (coeff_is_zero(fm))
          return mid;
        if (fa > Coefficient{} == fm > Coefficient{})
          {
            a = mid;
            fa = fm;
          }
        else
          {
            b = mid;
            fb = fm;
          }
      }
    return (a + b) / Coefficient(2);
  }
 
  [[nodiscard]] Coefficient newton_root(Coefficient x0,
                                        Coefficient tol = Coefficient(1e-12),
                                        size_t max_iter = 100) const
    requires std::is_floating_point_v<Coefficient>
  {
    Gen_Polynomial dp = derivative();
 
    for (size_t i = 0; i < max_iter; ++i)
      {
        Coefficient fx = eval(x0);
        if (coeff_is_zero(fx))
          return x0;
        Coefficient dfx = dp.eval(x0);
        ah_domain_error_if(coeff_is_zero(dfx)) << "newton_root: derivative is zero at x = " << x0;
        Coefficient x1 = x0 - fx / dfx;
        Coefficient delta = x1 - x0;
        if constexpr (std::is_floating_point_v<Coefficient>)
          {
            if (std::abs(delta) < tol)
              return x1;
          }
        else
          {
            if (delta == Coefficient{})
              return x1;
          }
        x0 = x1;
      }
    return x0;
  }
 
  // =================================================================
  //  Comparison
  // =================================================================
 
  [[nodiscard]] bool operator == (const Gen_Polynomial &q) const noexcept
  {
    if (this == &q)
      return true;
 
    auto it1 = coeffs.get_it();
    auto it2 = q.coeffs.get_it();
 
    while (it1.has_curr() and it2.has_curr())
      {
        auto &lhs = it1.get_curr();
        auto &rhs = it2.get_curr();
 
        if (lhs.first == rhs.first)
          {
            if (not coeff_is_zero(lhs.second - rhs.second))
              return false;
            it1.next_ne();
            it2.next_ne();
            continue;
          }
 
        if (lhs.first < rhs.first)
          {
            if (not coeff_is_zero(lhs.second))
              return false;
            it1.next_ne();
            continue;
          }
 
        if (not coeff_is_zero(rhs.second))
          return false;
        it2.next_ne();
      }
 
    while (it1.has_curr())
      {
        if (not coeff_is_zero(it1.get_curr().second))
          return false;
        it1.next_ne();
      }
 
    while (it2.has_curr())
      {
        if (not coeff_is_zero(it2.get_curr().second))
          return false;
        it2.next_ne();
      }
 
    return true;
  }
 
  [[nodiscard]] bool operator != (const Gen_Polynomial &q) const noexcept
  {
    return not (*this == q);
  }
 
  // =================================================================
  //  Iteration
  // =================================================================
 
  template <class Op>
  void for_each_term(Op &&op) const
  {
    for (auto it = coeffs.get_it(); it.has_curr(); it.next_ne())
      {
        auto &p = it.get_curr();
        op(p.first, p.second);
      }
  }
 
  [[nodiscard]] DynList<std::pair<size_t, Coefficient>> terms_list() const
  {
    DynList<std::pair<size_t, Coefficient>> result;
    for (auto it = coeffs.get_it(); it.has_curr(); it.next_ne())
      {
        auto &p = it.get_curr();
        result.append({p.first, p.second});
      }
    return result;
  }
 
  [[nodiscard]] DynList<size_t> exponents() const
  {
    return coeffs.keys();
  }
 
  [[nodiscard]] DynList<Coefficient> coefficients() const
  {
    return coeffs.values();
  }
 
  // =================================================================
  //  String representation
  // =================================================================
 
  [[nodiscard]] std::string to_str(const std::string &var = "x") const
  {
    if (coeffs.is_empty())
      return "0";
 
    std::ostringstream oss;
    bool first = true;
 
    for_each_term_desc([&](size_t exp, const Coefficient &c)
      {
        bool negative = false;
        if constexpr (requires { c < Coefficient{}; })
          negative = c < Coefficient{};
 
        Coefficient ac = negative ? -c : c;
 
        // Sign
        if (first)
          {
            if (negative)
              oss << "-";
          }
        else
          oss << (negative ? " - " : " + ");
 
        // Coefficient and variable
        if (exp == 0)
          oss << ac;
        else
          {
            if (not (ac == Coefficient(1)))
              oss << ac;
            oss << var;
            if (exp > 1)
              oss << "^" << exp;
          }
 
        first = false;
      });
 
    return oss.str();
  }
 
  // =================================================================
  //  Layer 5 — Exact Univariate Factorization
  // =================================================================
 
  struct SfdTerm
  {
    Gen_Polynomial factor;
    size_t multiplicity;
  };
 
  [[nodiscard]] Coefficient get_coeff(size_t exp) const noexcept
  {
    return coeff_at(exp);
  }
 
  void set_coeff(size_t exp, const Coefficient &c)
  {
    if (auto *slot = coeff_ptr(exp); slot != nullptr)
      {
        if (coeff_is_zero(c))
          coeffs.remove_key(exp);
        else
          *slot = c;
      }
    else if (not coeff_is_zero(c))
      coeffs.insert(exp, c);
  }
 
  [[nodiscard]] Coefficient content() const
    requires std::is_integral_v<Coefficient>
  {
    if (coeffs.is_empty())
      return Coefficient{};
 
    Coefficient result = Coefficient{};
    for (auto it = coeffs.get_it(); it.has_curr(); it.next_ne())
      {
        auto &p = it.get_curr();
        result = std::gcd(result, p.second);
      }
 
    // Ensure sign follows leading coefficient
    const Coefficient lc = leading_coeff();
    if (lc < Coefficient{} and result > Coefficient{})
      result = -result;
 
    return result;
  }
 
  [[nodiscard]] Gen_Polynomial primitive_part() const
    requires std::is_integral_v<Coefficient>
  {
    if (is_zero())
      return Gen_Polynomial();
 
    Coefficient c = content();
    if (c == Coefficient{} or c == Coefficient(1) or c == Coefficient(-1))
      {
        // Already primitive; ensure leading coefficient is positive
        Gen_Polynomial result = *this;
        if (leading_coeff() < Coefficient{})
          result.scale_inplace(Coefficient(-1));
        return result;
      }
 
    Gen_Polynomial result;
    for (auto it = coeffs.get_it(); it.has_curr(); it.next_ne())
      {
        auto &p = it.get_curr();
        result.coeffs.insert(p.first, p.second / c);
      }
 
    // Ensure leading coefficient is positive
    if (result.leading_coeff() < Coefficient{})
      result.scale_inplace(Coefficient(-1));
 
    return result;
  }
 
  static Gen_Polynomial integer_gcd(Gen_Polynomial a, Gen_Polynomial b)
    requires std::is_integral_v<Coefficient>
  {
    if (a.is_zero())
      return b.primitive_part();
    if (b.is_zero())
      return a.primitive_part();
 
    a = a.primitive_part();
    b = b.primitive_part();
 
    while (not b.is_zero())
      {
        auto [quo, rem] = a.pseudo_divmod(b);
        a = b;
        b = rem.is_zero() ? rem : rem.primitive_part();
      }
 
    return a.primitive_part();
  }
 
  [[nodiscard]] DynList<SfdTerm> yun_sfd() const
    requires std::is_integral_v<Coefficient>
  {
    DynList<SfdTerm> result;
 
    if (is_constant())
      return result;
 
    const Gen_Polynomial f = primitive_part();
    const Gen_Polynomial fp = f.derivative();
 
    Gen_Polynomial g = integer_gcd(f, fp);
    Gen_Polynomial h = _integer_exact_quot(f, g);
 
    size_t multiplicity = 1;
 
    while (not h.is_constant())
      {
        Gen_Polynomial h_prev = h;
        h = integer_gcd(g, h);
        Gen_Polynomial w = _integer_exact_quot(h_prev, h);
 
        if (not w.is_constant())
          result.append(SfdTerm{w.primitive_part(), multiplicity});
 
        g = _integer_exact_quot(g, h);
        multiplicity++;
      }
 
    // Final factor
    if (not g.is_constant())
      result.append(SfdTerm{g.primitive_part(), multiplicity});
 
    return result;
  }
 
  static Gen_Polynomial _integer_exact_quot(const Gen_Polynomial &a, const Gen_Polynomial &b)
    requires std::is_integral_v<Coefficient>
  {
    ah_domain_error_if(b.is_zero()) << "_integer_exact_quot: zero divisor";
    if (b.is_constant())
      {
        Coefficient d = b.leading_coeff();
        Gen_Polynomial r;
        for (auto it = a.coeffs.get_it(); it.has_curr(); it.next_ne())
          {
            auto &p = it.get_curr();
            ah_domain_error_if(not coeff_is_zero(p.second % d))
              << "_integer_exact_quot: inexact constant division for coefficient " << p.second
              << " and divisor " << d;
            r.coeffs.insert(p.first, p.second / d);
          }
        return r;
      }
    auto [q, rem] = a.pseudo_divmod(b);
    ah_domain_error_if(not rem.is_zero())
      << "_integer_exact_quot: divisor does not divide dividend exactly";
    return q.is_zero() ? q : q.primitive_part();
  }
 
  [[nodiscard]] static Gen_Polynomial reduce_coeffs_mod(const Gen_Polynomial &f, Coefficient mod)
    requires std::is_integral_v<Coefficient>
  {
    Gen_Polynomial reduced;
    f.for_each_term([&reduced, mod](size_t exp, const Coefficient &coeff)
      {
        Coefficient c = coeff % mod;
        if (c < Coefficient{})
          c = c + mod;
        if (c != Coefficient{})
          reduced.set_coeff(exp, c);
      });
    return reduced;
  }
 
  [[nodiscard]] static uint64_t eval_mod_u64(const Gen_Polynomial &f, uint64_t x, uint64_t mod)
    requires std::is_integral_v<Coefficient>
  {
    if (mod == 0 or mod == 1)
      return 0;
 
    uint64_t result = 0;
    f.for_each_term([&result, x, mod](size_t exp, const Coefficient &coeff)
      {
        const uint64_t c = polynomial_detail::normalize_mod_u64(coeff, mod);
        const uint64_t xpow = mod_exp(x, exp, mod);
        const uint64_t term = mod_mul(c, xpow, mod);
        result = (result + term) % mod;
      });
    return result;
  }
 
  [[nodiscard]] static Gen_Polynomial divide_by_monic_linear_mod(const Gen_Polynomial &f,
                                                                 Coefficient root,
                                                                 Coefficient mod)
    requires std::is_integral_v<Coefficient>
  {
    ah_domain_error_if(f.degree() == 0)
      << "divide_by_monic_linear_mod: constant polynomial is not divisible by x - r";
 
    const size_t degree = f.degree();
    Array<Coefficient> synthetic(degree + 1, Coefficient{});
 
    synthetic(degree) = ((f.get_coeff(degree) % mod) + mod) % mod;
 
    for (size_t idx = degree; idx-- > 0;)
      {
        Coefficient term = (f.get_coeff(idx) % mod + mod) % mod;
        Coefficient next = (term + (synthetic(idx + 1) * root) % mod) % mod;
        synthetic(idx) = next;
      }
 
    ah_domain_error_if(synthetic(0) != Coefficient{})
      << "divide_by_monic_linear_mod: x - r does not divide polynomial modulo p";
 
    Gen_Polynomial quotient;
    for (size_t exp = 0; exp < degree; ++exp)
      if (synthetic(exp + 1) != Coefficient{})
        quotient.set_coeff(exp, synthetic(exp + 1));
 
    return quotient;
  }
 
  [[nodiscard]] static DynList<Coefficient> positive_divisors(Coefficient value)
    requires std::is_integral_v<Coefficient>
  {
    DynList<Coefficient> result;
 
    const uint64_t abs_value = polynomial_detail::abs_to_u64(value);
    if (abs_value == 0)
      return result;
 
    for (uint64_t d = 1; d * d <= abs_value; ++d)
      if (abs_value % d == 0)
        {
          result.append(static_cast<Coefficient>(d));
          const uint64_t other = abs_value / d;
          if (other != d)
            result.append(static_cast<Coefficient>(other));
        }
 
    return result;
  }
 
  [[nodiscard]] static bool try_exact_candidate_division(const Gen_Polynomial &f,
                                                         const Gen_Polynomial &candidate,
                                                         Gen_Polynomial &quotient)
    requires std::is_integral_v<Coefficient>
  {
    try
      {
        auto [q, r] = f.divmod(candidate);
        if (not r.is_zero())
          return false;
        quotient = std::move(q);
        return true;
      }
    catch (const std::domain_error &)
      {
        return false;
      }
  }
 
  [[nodiscard]] static bool try_divide_by_linear_factor(const Gen_Polynomial &f,
                                                        Coefficient a,
                                                        Coefficient b,
                                                        Gen_Polynomial &quotient)
    requires std::is_integral_v<Coefficient>
  {
    if (f.is_zero() or f.degree() == 0)
      return false;
 
    if (a == Coefficient{})
      return false;
 
    const size_t degree = f.degree();
    Array<Coefficient> q_coeffs(degree, Coefficient{});
 
    const Coefficient leading = f.get_coeff(degree);
    if (leading % a != Coefficient{})
      return false;
 
    q_coeffs(degree - 1) = leading / a;
 
    for (size_t k = degree - 1; k > 0; --k)
      {
        const Coefficient numerator = f.get_coeff(k) - b * q_coeffs(k);
        if (numerator % a != Coefficient{})
          return false;
        q_coeffs(k - 1) = numerator / a;
      }
 
    if (f.get_coeff(0) - b * q_coeffs(0) != Coefficient{})
      return false;
 
    quotient = Gen_Polynomial();
    for (size_t exp = 0; exp < degree; ++exp)
      if (q_coeffs(exp) != Coefficient{})
        quotient.set_coeff(exp, q_coeffs(exp));
 
    return true;
  }
 
  [[nodiscard]] static bool try_extract_rational_linear_factor(const Gen_Polynomial &f,
                                                               Gen_Polynomial &factor,
                                                               Gen_Polynomial &quotient)
    requires std::is_integral_v<Coefficient>
  {
    if (f.is_zero() or f.degree() == 0)
      return false;
 
    if (f.get_coeff(0) == Coefficient{})
      {
        if (try_divide_by_linear_factor(f, Coefficient(1), Coefficient{}, quotient))
          {
            factor = Gen_Polynomial(Coefficient(1), 1);
            return true;
          }
        return false;
      }
 
    DynList<Coefficient> numerator_divs = positive_divisors(f.get_coeff(0));
    DynList<Coefficient> denominator_divs = positive_divisors(f.leading_coeff());
 
    for (auto den_it = denominator_divs.get_it(); den_it.has_curr(); den_it.next_ne())
      {
        const Coefficient den = den_it.get_curr();
 
        for (auto num_it = numerator_divs.get_it(); num_it.has_curr(); num_it.next_ne())
          {
            const Coefficient num_abs = num_it.get_curr();
 
            for (Coefficient sign : {Coefficient(1), Coefficient(-1)})
              {
                const Coefficient num = sign * num_abs;
                if (std::gcd(polynomial_detail::abs_to_u64(num),
                             polynomial_detail::abs_to_u64(den)) != 1)
                  continue;
 
                if (try_divide_by_linear_factor(f, den, -num, quotient))
                  {
                    factor = Gen_Polynomial();
                    factor.set_coeff(0, -num);
                    factor.set_coeff(1, den);
                    return true;
                  }
              }
          }
      }
 
    return false;
  }
 
  [[nodiscard]] static bool try_extract_primitive_quadratic_factor(const Gen_Polynomial &f,
                                                                   Gen_Polynomial &factor,
                                                                   Gen_Polynomial &quotient)
    requires std::is_integral_v<Coefficient>
  {
    if (f.is_zero() or f.degree() < 2)
      return false;
 
    if (f.get_coeff(0) == Coefficient{})
      return false;
 
    DynList<Coefficient> leading_divs = positive_divisors(f.leading_coeff());
    DynList<Coefficient> constant_divs = positive_divisors(f.get_coeff(0));
    if (leading_divs.is_empty() or constant_divs.is_empty())
      return false;
 
    const double abs_lc = std::abs(static_cast<double>(f.leading_coeff()));
    double max_ratio = 0.0;
    for (auto it = f.coeffs.get_it(); it.has_curr(); it.next_ne())
      {
        const auto &term = it.get_curr();
        if (term.first == f.degree())
          continue;
        if (const double ratio = std::abs(static_cast<double>(term.second)) / abs_lc;
            ratio > max_ratio)
          max_ratio = ratio;
      }
 
    const double root_bound = 1.0 + max_ratio;
 
    for (auto a_it = leading_divs.get_it(); a_it.has_curr(); a_it.next_ne())
      {
        const Coefficient a = a_it.get_curr();
        const int64_t b_bound =
          static_cast<int64_t>(std::ceil(2.0 * std::abs(static_cast<double>(a)) * root_bound));
 
        for (auto c_it = constant_divs.get_it(); c_it.has_curr(); c_it.next_ne())
          {
            const Coefficient c_abs = c_it.get_curr();
 
            for (Coefficient sign : {Coefficient(1), Coefficient(-1)})
              {
                const Coefficient c = sign * c_abs;
 
                for (int64_t b_i = -b_bound; b_i <= b_bound; ++b_i)
                  {
                    const Coefficient b = static_cast<Coefficient>(b_i);
                    const uint64_t gcd_ab =
                      std::gcd(polynomial_detail::abs_to_u64(a), polynomial_detail::abs_to_u64(b));
                    if (std::gcd(gcd_ab, polynomial_detail::abs_to_u64(c)) != 1)
                      continue;
 
                    Gen_Polynomial candidate;
                    candidate.set_coeff(0, c);
                    if (b != Coefficient{})
                      candidate.set_coeff(1, b);
                    candidate.set_coeff(2, a);
 
                    Gen_Polynomial q;
                    if (try_exact_candidate_division(f, candidate, q))
                      {
                        factor = candidate.primitive_part();
                        quotient = q.is_zero() ? q : q.primitive_part();
                        return true;
                      }
                  }
              }
          }
      }
 
    return false;
  }
 
  [[nodiscard]] static bool try_extract_primitive_cubic_factor(const Gen_Polynomial &f,
                                                               Gen_Polynomial &factor,
                                                               Gen_Polynomial &quotient)
    requires std::is_integral_v<Coefficient>
  {
    if (f.is_zero() or f.degree() < 3)
      return false;
 
    if (f.get_coeff(0) == Coefficient{})
      return false;
 
    DynList<Coefficient> leading_divs = positive_divisors(f.leading_coeff());
    DynList<Coefficient> constant_divs = positive_divisors(f.get_coeff(0));
    if (leading_divs.is_empty() or constant_divs.is_empty())
      return false;
 
    const double abs_lc = std::abs(static_cast<double>(f.leading_coeff()));
    double l2_norm_sq = 0.0;
    for (auto it = f.coeffs.get_it(); it.has_curr(); it.next_ne())
      {
        const auto &term = it.get_curr();
        const double coeff_abs = std::abs(static_cast<double>(term.second));
        l2_norm_sq += coeff_abs * coeff_abs;
      }
 
    const int64_t coeff_bound = static_cast<int64_t>(std::ceil(8.0 * std::sqrt(l2_norm_sq) / abs_lc));
    if (coeff_bound <= 0)
      return false;
 
    for (auto a_it = leading_divs.get_it(); a_it.has_curr(); a_it.next_ne())
      {
        const Coefficient a = a_it.get_curr();
        if (polynomial_detail::abs_to_u64(a) > static_cast<uint64_t>(coeff_bound))
          continue;
 
        for (auto d_it = constant_divs.get_it(); d_it.has_curr(); d_it.next_ne())
          {
            const Coefficient d_abs = d_it.get_curr();
            if (polynomial_detail::abs_to_u64(d_abs) > static_cast<uint64_t>(coeff_bound))
              continue;
 
            for (Coefficient sign : {Coefficient(1), Coefficient(-1)})
              {
                const Coefficient d = sign * d_abs;
 
                for (int64_t b_i = -coeff_bound; b_i <= coeff_bound; ++b_i)
                  {
                    const Coefficient b = static_cast<Coefficient>(b_i);
                    const uint64_t gcd_ab =
                      std::gcd(polynomial_detail::abs_to_u64(a), polynomial_detail::abs_to_u64(b));
 
                    for (int64_t c_i = -coeff_bound; c_i <= coeff_bound; ++c_i)
                      {
                        const Coefficient c = static_cast<Coefficient>(c_i);
                        const uint64_t gcd_abc = std::gcd(gcd_ab, polynomial_detail::abs_to_u64(c));
                        if (std::gcd(gcd_abc, polynomial_detail::abs_to_u64(d)) != 1)
                          continue;
 
                        Gen_Polynomial candidate;
                        candidate.set_coeff(0, d);
                        if (c != Coefficient{})
                          candidate.set_coeff(1, c);
                        if (b != Coefficient{})
                          candidate.set_coeff(2, b);
                        candidate.set_coeff(3, a);
 
                        Gen_Polynomial q;
                        if (try_exact_candidate_division(f, candidate, q))
                          {
                            factor = candidate.primitive_part();
                            quotient = q.is_zero() ? q : q.primitive_part();
                            return true;
                          }
                      }
                  }
              }
          }
      }
 
    return false;
  }
 
  static DynList<Gen_Polynomial> factor_mod_p(const Gen_Polynomial &f, Coefficient p)
    requires std::is_integral_v<Coefficient>
  {
    ah_domain_error_if(p <= Coefficient(1)) << "factor_mod_p: modulus p must be > 1, got " << p;
 
    DynList<Gen_Polynomial> result;
 
    if (f.is_zero() or f.is_constant())
      return result;
 
    Gen_Polynomial remaining = reduce_coeffs_mod(f, p);
 
    for (Coefficient r = Coefficient{}; r < p; r = r + Coefficient(1))
      {
        while (not remaining.is_zero() and remaining.degree() > 0 and
               eval_mod_u64(remaining,
                            polynomial_detail::normalize_mod_u64(r, polynomial_detail::abs_to_u64(p)),
                            polynomial_detail::abs_to_u64(p)) == 0)
          {
            Gen_Polynomial factor;
            factor.set_coeff(0, ((-r) % p + p) % p);
            factor.set_coeff(1, Coefficient(1));
            result.append(factor);
            remaining = divide_by_monic_linear_mod(remaining, r, p);
          }
      }
 
    if (not remaining.is_zero() and not remaining.is_constant())
      result.append(remaining);
 
    return result;
  }
 
  [[nodiscard]] double mignotte_bound() const
    requires std::is_integral_v<Coefficient>
  {
    if (is_zero())
      return 0.0;
 
    const size_t d = degree();
    if (d == 0)
      return 0.0;
 
    Coefficient max_coeff = Coefficient{};
    for (auto it = coeffs.get_it(); it.has_curr(); it.next_ne())
      {
        auto &p = it.get_curr();
        Coefficient abs_c = p.second < Coefficient{} ? -p.second : p.second;
        if (abs_c > max_coeff)
          max_coeff = abs_c;
      }
 
    const auto d_real = static_cast<double>(d);
    const auto max_real = static_cast<double>(max_coeff);
    const double two_pow_d = polynomial_detail::power(2.0, d);
 
    return std::sqrt(d_real + 1.0) * two_pow_d * max_real;
  }
 
  static DynList<Gen_Polynomial> hensel_lift(const Gen_Polynomial &f,
                                             DynList<Gen_Polynomial> factors,
                                             Coefficient p,
                                             size_t levels)
    requires std::is_integral_v<Coefficient>
  {
    ah_domain_error_if(p <= Coefficient(1)) << "hensel_lift: modulus p must be > 1, got " << p;
    ah_domain_error_if(levels == 0) << "hensel_lift: levels must be >= 1, got " << levels;
    ah_domain_error_if(levels > 30)
      << "hensel_lift: levels must be <= 30 to avoid overflow, got " << levels;
 
    if (factors.is_empty())
      return factors;
 
    const size_t target_steps = static_cast<size_t>(1) << levels;
    const uint64_t p_u64 = polynomial_detail::abs_to_u64(p);
    const auto checked_mul_u64 = [](uint64_t lhs, uint64_t rhs, const char *name) -> uint64_t
      {
        ah_domain_error_if(rhs != 0 and lhs > std::numeric_limits<uint64_t>::max() / rhs)
          << "hensel_lift: overflow computing " << name << " (" << lhs << " * " << rhs << ")";
        return lhs * rhs;
      };
 
    uint64_t target_u64 = p_u64;
    for (size_t i = 1; i < target_steps; ++i)
      target_u64 = checked_mul_u64(target_u64, p_u64, "target_u64");
 
    ah_domain_error_if(target_u64 > static_cast<uint64_t>(std::numeric_limits<Coefficient>::max()))
      << "hensel_lift: target modulus " << target_u64 << " does not fit in coefficient type";
    const Coefficient p_power = static_cast<Coefficient>(target_u64);
    Gen_Polynomial derivative = f.derivative();
 
    DynList<Gen_Polynomial> result;
    for (auto it = factors.get_it(); it.has_curr(); it.next_ne())
      {
        const Gen_Polynomial &factor = it.get_curr();
        bool lifted_linear_factor = false;
 
        if (factor.degree() == 1 and factor.get_coeff(1) == Coefficient(1))
          {
            uint64_t root = polynomial_detail::normalize_mod_u64(-factor.get_coeff(0), p_u64);
            uint64_t modulus = p_u64;
            bool simple = true;
 
            for (size_t step = 1; step < target_steps; ++step)
              {
                const uint64_t next_mod = checked_mul_u64(modulus, p_u64, "next_mod");
                const uint64_t f_val = eval_mod_u64(f, root, next_mod);
                const uint64_t deriv_modp = eval_mod_u64(derivative, root % p_u64, p_u64);
 
                if (f_val % modulus != 0 or std::gcd(deriv_modp, p_u64) != 1)
                  {
                    simple = false;
                    break;
                  }
 
                const uint64_t e = (f_val / modulus) % p_u64;
                const uint64_t inv = mod_inv(deriv_modp, p_u64);
                const uint64_t t = (p_u64 - mod_mul(e, inv, p_u64)) % p_u64;
 
                const uint64_t increment = checked_mul_u64(modulus, t, "root increment");
#if defined(_MSC_VER) or defined(_WIN32)
                // MSVC has no __int128. clang-cl on Windows has __int128 but
                // lacks the __umodti3 runtime helper for 128-bit modulo.
                // Compute (root+increment) % next_mod without 128-bit arithmetic.
                {
                  const uint64_t a = root % next_mod;
                  const uint64_t b = increment % next_mod;
                  root = (a >= next_mod - b) ? (a - (next_mod - b)) : (a + b);
                }
#else
                root = static_cast<uint64_t>((static_cast<unsigned __int128>(root) + increment) %
                                             next_mod);
#endif
                modulus = next_mod;
              }
 
            if (simple)
              {
                Gen_Polynomial lifted;
                lifted.set_coeff(0,
                                 polynomial_detail::centered_from_mod_u64<Coefficient>(
                                   (target_u64 - (root % target_u64)) % target_u64, target_u64));
                lifted.set_coeff(1, Coefficient(1));
                result.append(lifted);
                lifted_linear_factor = true;
              }
          }
 
        if (lifted_linear_factor)
          continue;
 
        Gen_Polynomial normalized;
        factor.for_each_term([&normalized, p_power](size_t exp, const Coefficient &coeff)
          {
            Coefficient canonical = coeff % p_power;
            if (canonical < Coefficient{})
              canonical += p_power;
            if (canonical != Coefficient{})
              normalized.set_coeff(exp, canonical);
          });
        result.append(normalized);
      }
 
    return result;
  }
 
  [[nodiscard]] DynList<SfdTerm> factorize() const
    requires std::is_integral_v<Coefficient>
  {
    DynList<SfdTerm> result;
 
    if (is_zero() or is_constant())
      return result;
 
    const Coefficient c = content();
    if (c != Coefficient(1))
      result.append(SfdTerm{Gen_Polynomial(c), 1});
 
    // Step 1: Get primitive part
    Gen_Polynomial prim = primitive_part();
 
    // Step 2: Compute square-free decomposition
    DynList<SfdTerm> sfd_terms = prim.yun_sfd();
 
    if (sfd_terms.is_empty())
      return result;
 
    // Step 3: For each square-free group, attempt factorization
    for (auto it = sfd_terms.get_it(); it.has_curr(); it.next_ne())
      {
        auto &sfd_term = it.get_curr();
        Gen_Polynomial factor = sfd_term.factor;
        size_t mult = sfd_term.multiplicity;
        Gen_Polynomial remaining = factor.primitive_part();
 
        while (remaining.degree() > 0)
          {
            Gen_Polynomial exact_linear, quotient;
            if (not try_extract_rational_linear_factor(remaining, exact_linear, quotient))
              break;
 
            result.append(SfdTerm{exact_linear.primitive_part(), mult});
            remaining = quotient.is_zero() ? quotient : quotient.primitive_part();
          }
 
        while (remaining.degree() > 1)
          {
            Gen_Polynomial exact_quadratic, quotient;
            if (not try_extract_primitive_quadratic_factor(remaining, exact_quadratic, quotient))
              break;
 
            result.append(SfdTerm{exact_quadratic.primitive_part(), mult});
            remaining = quotient.is_zero() ? quotient : quotient.primitive_part();
          }
 
        while (remaining.degree() > 2)
          {
            Gen_Polynomial exact_cubic, quotient;
            if (not try_extract_primitive_cubic_factor(remaining, exact_cubic, quotient))
              break;
 
            result.append(SfdTerm{exact_cubic.primitive_part(), mult});
            remaining = quotient.is_zero() ? quotient : quotient.primitive_part();
          }
 
        if (remaining.is_constant() or remaining.is_zero())
          continue;
 
        // Try modular factorization with a few small primes and exact redivision.
        for (Coefficient p : {Coefficient(3), Coefficient(5), Coefficient(7), Coefficient(11)})
          {
            if (remaining.leading_coeff() % p == Coefficient{})
              continue;
 
            auto mod_factors = factor_mod_p(remaining, p);
            if (mod_factors.size() <= 1)
              continue;
 
            auto lifted = hensel_lift(remaining, mod_factors, p, 2);
 
            bool changed = false;
            for (auto jt = lifted.get_it(); jt.has_curr(); jt.next_ne())
              {
                Gen_Polynomial cand = jt.get_curr().primitive_part();
                if (cand.is_zero() or cand.is_constant() or cand.degree() != 1)
                  continue;
 
                Gen_Polynomial quotient;
                if (not try_divide_by_linear_factor(remaining, cand.get_coeff(1), cand.get_coeff(0),
                                                    quotient))
                  continue;
 
                result.append(SfdTerm{cand, mult});
                remaining = quotient.is_zero() ? quotient : quotient.primitive_part();
                changed = true;
 
                if (remaining.is_constant())
                  break;
              }
 
            if (changed)
              {
                if (remaining.is_constant())
                  break;
              }
          }
 
        if (not remaining.is_constant() and not remaining.is_zero())
          result.append(SfdTerm{remaining.primitive_part(), mult});
      }
 
    return result;
  }
};
 
// =================================================================
//  Free functions
// =================================================================
 
template <typename C>
[[nodiscard]] Gen_Polynomial<C> operator * (const C &s, const Gen_Polynomial<C> &p)
{
  return p * s;
}
 
template <typename C>
[[nodiscard]] Gen_Polynomial<C> operator + (const C &s, const Gen_Polynomial<C> &p)
{
  return p + s;
}
 
template <typename C>
[[nodiscard]] Gen_Polynomial<C> operator - (const C &s, const Gen_Polynomial<C> &p)
{
  return Gen_Polynomial<C>(s) - p;
}
 
template <typename C>
std::ostream &operator << (std::ostream &out, const Gen_Polynomial<C> &p)
{
  return out << p.to_str();
}
 
// =================================================================
//  Convenience type aliases
// =================================================================
 
using Polynomial = Gen_Polynomial<double>;
 
using PolynomialF = Gen_Polynomial<float>;
 
using PolynomialLD = Gen_Polynomial<long double>;
 
using IntPolynomial = Gen_Polynomial<long long>;
 
}  // end namespace Aleph
 
#endif  // TPL_POLYNOMIAL_H