_gr_poly_divrem, gr_poly_divrem, _gr_poly_divrem_basecase_preinv1, _gr_poly_divrem_basecase_noinv, _gr_poly_divrem_basecase, gr_poly_divrem_basecase, _gr_poly_divrem_divconquer_preinv1, _gr_poly_divrem_divconquer_noinv, _gr_poly_divrem_divconquer, gr_poly_divrem_divconquer, _gr_poly_divrem_newton, gr_poly_divrem_newton

The basecase versions perform long division. The divconquer versions divide blocks of $2 \mathit{lenB} - 1$ coefficients of the dividend by $B$ recursively, with two divisions by the upper half of $B$ and products by its lower half, and use long division for divisors shorter than cutoff. The newton versions use Newton division. The preinv1 versions take a precomputed inverse invB of the leading coefficient, the noinv versions divide by the leading coefficient with gr_div at each step, and the plain versions try to invert the leading coefficient and fall back to the noinv algorithm.

The default method _gr_poly_divrem_generic uses long division when $\mathit{lenB} = 1$ or $\mathit{lenA} - \mathit{lenB} = 1$, and over finite rings also when the divisor or the quotient has at most 10 coefficients; otherwise it uses Newton division, falling back to long or divide-and-conquer division without inversion when the leading coefficient is not invertible. Rings with fast arithmetic override the method: nmod, fmpz_mod, mpn_mod, fq, fq_nmod, fq_zech, arb and acb call the division of their own polynomial modules.

The underscore functions assume:

• $\mathit{lenA} \ge \mathit{lenB} \ge 1$ and a nonzero leading coefficient of $B$;
• $Q$ has room for $\mathit{lenA} - \mathit{lenB} + 1$ and $R$ for $\mathit{lenB} - 1$ coefficients;
• $Q$ is not aliased with $A$ or $B$, and $R$ is not aliased with $B$;
• $R$ may be aliased with $A$, in which case all $\mathit{lenA}$ coefficients are used as scratch space and only the low $\mathit{lenB} - 1$ are meaningful on output.

@misc{flint-doc-gr-poly-gr-poly-divrem,
  author       = {{The FLINT team}},
  title        = {{\_gr\_poly\_divrem, gr\_poly\_divrem, \_gr\_poly\_divrem\_basecase\_preinv1, \_gr\_poly\_divrem\_basecase\_noinv, \_gr\_poly\_divrem\_basecase, gr\_poly\_divrem\_basecase, \_gr\_poly\_divrem\_divconquer\_preinv1, \_gr\_poly\_divrem\_divconquer\_noinv, \_gr\_poly\_divrem\_divconquer, gr\_poly\_divrem\_divconquer, \_gr\_poly\_divrem\_newton, gr\_poly\_divrem\_newton (gr\_poly.h)}},
  howpublished = {FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev},
  year         = {2026},
  url          = {https://flintlib.org/doc/gr_poly/_gr_poly_divrem.html},
  note         = {Generated 2026-11-02}
}