Cubic and Quintic Approximants

Cubic one-line approximant Q(t2, t3)

The Bi–Tri slice admits a compact cubic approximant:

from geodepoly.geode import Q_cubic
Q = Q_cubic(t2=0.1, t3=-0.03)

This provides a fast low-order estimate on the t2,t3 plane.

Hybrid-cubic mode

method="hybrid-cubic" in solve_all uses Q_cubic as a warm-start heuristic:

1) Recenter and compute t=-a0/a1, map local a2,a3 to t2,t3.
2) Evaluate alpha = Q_cubic(t2,t3) and take y = t * alpha as a step.
3) Repeat briefly across a few centers; pass seeds to Aberth for finishing.

This can reduce time-to-first-root on many instances.

Quintic (Bring radical) note

The Bring radical/Eisenstein series is part of the theoretical framework. An explicit bring_radical_series API may be added in a future release.