Smallest representatives of $\rm {SL}(2,\mathbb {Z})$-orbits of binary forms and endomorphisms of $\mathbb {P}^1$

Abstract

We develop an algorithm that determines, for a given squarefree binary form F with real coefficients, a smallest representative of its orbit under SL(2, Z), either with respect to the Euclidean norm or with respect to the maximum norm of the coefficient vector. This is based on earlier work of Cremona and Stoll [SC03]. We then generalize our approach so that it also applies to the problem of finding an integral representative of smallest height in the PGL(2, Q) conjugacy class of an endomorphism of the projective line. Having a small model of such an endomorphism is useful for various computations."Natural" means that this association is compatible with the action of PGL(2) = Aut(P 1 ) on both sides. In this way, we can set up a "reduction theory" for the objects Φ by selecting a suitable "reduced" representative F · γ in the SL(2, Z)-orbit of F and declaring Φ · γ to be the reduced representative in the SL(2, Z)-orbit of Φ. This is the approach taken in [Sto11] in the setting of P n for general n. This works reasonably well when we just want to have a way of selecting a canonical representative of moderate size. If we want to find a representative of smallest height, then more work is required: we have to relate the height of Φ to the size of F(Φ) and determine a bound on the distance we can move from the canonical representative without increasing the height. The application we have in mind is to endomorphisms of P 1 ; this is discussed in some detail in Section 2 below. Main ResultsOur main results are as follows.(1) We prove a result (Theorem 4.7) that bounds the size of a binary form F in terms of its Julia invariant θ(F) (see below) and the hyperbolic distance of its covariant z(F) (see below) to the "center" i of the hyperbolic plane H.(2) Based on this result, we construct an algorithm (Algorithm 5.1) that determines a representative of smallest size in the SL(2, Z)-orbit of a given binary form.(3) We generalize this algorithm so that it can be used with different notions of "size" and for more general objects associated to P 1 . We apply this specifically to automorphisms of P 1 ; see Section 6. (4) In the latter context, we also show how to find minimal representatives up to GL(2, Q)-conjugacy. This involves the determination of all GL(2, Z)-orbits of minimal models; see the algorithms in Section 7. In this context, we prove that an automorphism of even degree has only one orbit of minimal models (Proposition 7.2), but there can be an arbitrary number of orbits for odd degree (Proposition 7.4).

Minimization of hypersurfaces

“…(Since T is unimodular, acting on F 0 by T does not affect the minimality property.) This is known as reduction; algorithms that perform it are described in [SC03,HS19].…”

“…Minimization and reduction have been studied for 2-, 3-, 4and 5-coverings of elliptic curves in [CFS10] and [Fis13]. The reduction theory of binary forms is studied in [SC03] and [HS19] and that of point clusters in projective space in [Sto11]. The latter can be used to obtain a reduction method also for more general projective varieties; for example, we can reduce equations of plane curves by reducing their multiset of inflection points.…”

Algebraic Number Starscapes

“…One place this may arise is in trying to bound distances between cubic numbers. Thus, to control distances between cubic numbers in terms of their minimal polynomials, we do not need to use a full expression 28 for the unit tangent bundle metric on UTH 2 Coefs but rather just a means of measuring the hyperbolic distance between their projections to the base H 2 . While abstractly this is given exactly by the pull back of the metric on H 2…”

“…To compute this minimum, there is a recent algorithm due to Stoll-Cremona and Hutz-Stoll which generalizes the reduction theory of binary quadratic forms with respect to PSL(2; Z) to general binary forms [28,52]. It is not known how small a height one is guaranteed under their algorithm, nor where the minimum is attained.…”