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The density of discriminants of quartic rings and fields
We determine, asymptotically, the number of quintic fields having bounded discriminant. Specifically, we prove that the asymptotic number of quintic fields having absolute discriminant at most X is a
Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves
We prove a theorem giving the asymptotic number of binary quartic forms having bounded invariants; this extends, to the quartic case, the classical results of Gauss and Davenport in the quadratic and
Higher composition laws III: The parametrization of quartic rings
In the first two articles of this series, we investigated various higher analogues of Gauss composition, and showed how several algebraic objects involving orders in quadratic and cubic fields could
Higher composition laws I: A new view on Gauss composition, and quadratic generalizations
Two centuries ago, in his celebrated work Disquisitiones Arithmeticae of 1801, Gauss laid down the beautiful law of composition of integral binary quadratic forms which would play such a critical
Higher composition laws
The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point
We prove that when all hyperelliptic curves of genus $n\geq 1$ having a rational Weierstrass point are ordered by height, the average size of the 2-Selmer group of their Jacobians is equal to 3. It
P-orderings and polynomial functions on arbitrary subsets of Dedekind rings.
We introduce the notion of a P-ordering of an arbitrary subset AOf a Dedekind ring R, and use it to investigate the functions from X to R which can be represented by polynomials. In the case when R
Higher composition laws IV: The parametrization of quintic rings
rst three parts of this series, we considered quadratic, cubic and quartic rings (i.e., rings free of ranks 2, 3, and 4 over Z) respectively, and found that various algebraic structures involving
The geometric sieve and the density of squarefree values of invariant polynomials
We develop a method for determining the density of squarefree values taken by certain multivariate integer polynomials that are invariants for the action of an algebraic group on a vector space. The