Enumerating places of $\mathbf P^1$ up to automorphisms of $\mathbf P^1$ in quasilinear time (2407.05534v1)

Abstract: We present an algorithm that, for every fixed degree $n\ge 3$, will enumerate all degree-$n$ places of the projective line over a finite field $k$ up to the natural action of $\operatorname{PGL}_2(k)$ using $O(\log q)$ space and $\widetilde{O}(q^{n-3})$ time, where $q=#k$. Since there are $\Theta(q^{n-3})$orbits of $\operatorname{PGL}_2(k)$ acting on the set of degree-$n$ places, the algorithm is quasilinear in the size of its output. The algorithm is probabilistic unless we assume the extended Riemann hypothesis, because its complexity depends on that of polynomial factorization: for odd $n$, it involves factoring $\Theta(q^{n-3})$ polynomials over $k$ of degree up to ${1 + ((n-1)/2)^n}$. For composite degrees $n$, earlier work of the author gives an algorithm for enumerating $\operatorname{PGL}_2(k)$ orbit representatives for degree-$n$ places of $\mathbf{P}^1$ over $k$ that runs in time $\widetilde{O}(q^{n-3})$ independent of the extended Riemann hypothesis, but that uses $O(q^{n-3})$ space. We also present an algorithm for enumerating orbit representatives for the action of $\operatorname{PGL}_2(k)$ on the degree-$n$ effective divisors of $\mathbf{P}^1$ over finite fields $k$. The two algorithms depend on one another; our method of enumerating orbits of places of odd degree $n$ depends on enumerating orbits of effective divisors of degree $(n+1)/2$.