Where models of minimal height are hard to find: cubic rational maps with a multiple fixed point

Abstract

Given a rational point of the moduli space of cubic rational maps, we seek a map in its class with the smallest integer coefficients. A recent algorithm (Shaska, arXiv:2610.33990) combines an exact descent and minimisation of the resultant with a reduction and a local search; it is nearly exact where the minimal height is known. Beyond that range, a certified deeper search shows that the instances in which a smaller model lies beyond its reach are not random: a classifier predicts them from the six invariants of the point, and asking what it relies on leads to an exact algebraic surface, the maps with a triple fixed point. Invariant theory and the stability hypothesis of reduction theory (Hutz and Stoll, 2019) then identify the larger locus that contains it: the hypersurface $ξ_1^3=6ξ_2^2$ of maps with a multiple fixed point, on which the fixed-point quartic is not stable. It holds most of the hard instances. On it the two invariants of the fixed-point quartic are $(ξ_1,ξ_2)=(6m^2,6m^3)$ for an integer $m\ge0$; we parametrise the integer points of the surface, and show that on its generic part the sign of $ξ_5$ is the sign of the cubic coefficient at the parabolic fixed point. A covariant defined on the locus, added to the reduction, resolves only a minority of the hard instances: they appear to be mainly a matter of the search's reach. These findings lead to a certified structured search that never returns a larger model than the original algorithm and finds most of the smaller models within reach, also at heights not used to design it, at a small multiple of its running time.