division_algorithm_with_laurent_variables()

Nterm * PolyRing::division_algorithm_with_laurent_variables ( Nterm * f, Nterm * g, Nterm *& quot ) const

protected

Definition at line 1422 of file poly.cpp.

{
  // Step 1: replace f with f1, m1 (f = m1 * f1, f1 has only exponents >= 0, and no monomial factors)
  // Same: g = n1 * g1.
  // then if f1 = q * g1 + r, (q, r polynomials with only non-negative exponents)
  // then f = (q*m1*n1^(-1)) * g + m1*r
  //std::pair<Nterm*, int*> factor_out_inverse_variables(f);
  auto expf = setNegativeExponentMonomial(f);
  auto expg = setNegativeExponentMonomial(g);
  monomial m = getMonoid()->make_one();
  monomial n = getMonoid()->make_one();
  getMonoid()->from_expvector(expf.data(), m);
  getMonoid()->from_expvector(expg.data(), n);
 
  for (auto& a : expf) a = -a;
  for (auto& a : expg) a = -a;
  monomial minv = getMonoid()->make_one();
  monomial ninv = getMonoid()->make_one();
  getMonoid()->from_expvector(expf.data(), minv);
  getMonoid()->from_expvector(expg.data(), ninv);
 
  for (int i=0; i<n_vars(); ++i)
    expf[i] = -expf[i] + expg[i];
  monomial mninv = getMonoid()->make_one();
  getMonoid()->from_expvector(expf.data(), mninv);
  
  ring_elem c = getCoefficientRing()->from_long(1);
  Nterm* f1 = mult_by_term(f, c, minv); // f1 = m^-1 * f, m only involves the variables which allow negative exponents
  Nterm* g1 = mult_by_term(g, c, ninv); // g1 = n^-1 * g.
  Nterm* quot1 = nullptr;
  Nterm* rem1 = division_algorithm(f1, g1, quot1);
  Nterm* rem = mult_by_term(rem1, c, m);
  quot = mult_by_term(quot1, c, mninv);
  return rem;
}

References division_algorithm(), Monoid::from_expvector(), Ring::from_long(), PolynomialRing::getCoefficientRing(), PolynomialRing::getMonoid(), Monoid::make_one(), monomial, mult_by_term(), PolynomialRing::n_vars(), and setNegativeExponentMonomial().

Referenced by remainderAndQuotient().