Rational curves of degree four with two inner Galois points

TL;DR

This paper characterizes degree four rational plane curves with multiple Galois points, verifies the existence of a curve with three such points, and confirms the sharpness of Miura's bound for these curves.

Contribution

It provides a complete characterization of degree four rational curves with two or more Galois points and demonstrates the existence of a curve with exactly three Galois points.

Findings

Degree four rational curves with two or more Galois points are characterized.

Existence of a degree four rational curve with three Galois points is verified computationally.

Miura's bound is shown to be sharp for rational curves.

Abstract

We characterize plane rational curves of degree four with two or more inner Galois points. A computer verifies the existence of plane rational curves of degree four with three inner Galois points. This would be the first example of a curve with exactly three them. Our result implies that Miura's bound is sharp for rational curves.