On the number of monogenizations of a quartic order

Abstract

We show that an order in a quartic field has fewer than 3000 essentially different generators as a Z-algebra (and fewer than 200 if the discriminant of the order is sufficiently large). This significantly improves the previously best known bound of 2 72 .Analogously, we show that an order in a quartic field is isomorphic to the invariant order of at most 10 classes of integral binary quartic forms (and at most 7 if the discriminant is sufficiently large). This significantly improves the previously best known bound of 2 80 .

Quartic Index Form Equations and Monogenizations of Quartic Orders

“…The following is Theorem A. 1 of [10], where results from [2,3,9] are combined to obtain upper bounds for the number of integral solutions to quartic Thue equations. The following is part of the main theorem in [3].…”

“…The identity ( 22) is not used in our proofs, but it is crucial in confirming that the ternary quadratic forms Q 1 and Q 2 form a pair that parametrizes a quartic ring in the sense of Bhargava's work [11]. Such a parametrization is used in Bhargava's proof of Theorem 1.1 in [10]. Another ingredient in [10] is a beautiful parametrization due to Wood in [28] for quartic rings.…”

“…Such a parametrization is used in Bhargava's proof of Theorem 1.1 in [10]. Another ingredient in [10] is a beautiful parametrization due to Wood in [28] for quartic rings. We do not use any of these two parametrizations.…”

“…In the case n = 4 , Evertse's result shows that an order in a quartic field can have at most 2 72 monogenizations. Recently Bhargava gave an improved bound in [10], showing that an order in a quartic number field can have at most 2760 monogenizations (and even fewer when the discriminant of the order is large enough). We give another proof for Theorem 1.1 of [10].…”

“…Using this method, we will be able to associate explicit polynomials and binary and ternary forms to a monogenized order and eventually reduce our problem to the resolution of a number of Thue equations of degree 3 and 4. The proof in [10] uses a more abstract viewpoints by utilizing two ways of parametrizing quartic rings, one established by Bhargava in [11] and another one established by Wood in [28] .…”