Masser and Vaaler have given an asymptotic formula for the number of algebraic numbers of given degree $d$ and increasing height. This problem was solved by counting lattice points (which correspond to minimal polynomials over $\mathbb{Z}$) in a homogeneously expanding star body in $\mathbb{R}^{d+1}$. The volume of this star body was computed by Chern and Vaaler, who also computed the volume of the codimension-one "slice" corresponding to monic polynomials -- this led to results of Barroero on counting algebraic integers. We show how to estimate the volume of higher-codimension slices, which allows us to count units, algebraic integers of given norm, trace, norm and trace, and more. We also refine the lattice point-counting arguments of Chern-Vaaler to obtain explicit error terms with better power savings, which lead to explicit versions of some results of Masser-Vaaler and Barroero.
Let $k$ be a number field. For $\mathcal{H}\rightarrow \infty$, we give an asymptotic formula for the number of algebraic integers of absolute Weil height bounded by $\mathcal{H}$ and fixed degree over $k$.
We consider the set of points in projective $n$-space that generate an extension of degree $e$ over given number field $k$, and deduce an asymptotic formula for the number of such points of absolute height at most $X$, as $X$ tends to infinity. We deduce a similar such formula with instead of the absolute height, a so-called adelic-Lipschitz height.
Let $k$ be a finite field extension of the function field $\bfF_p(T)$ and $\bar{k}$ its algebraic closure. We count points in projective space $\Bbb P ^{n-1}(\bar{k})$ with given height and of fixed degree $d$ over the field $k$. If $n>2d+3$ we derive an asymptotic estimate for their number as the height tends to infinity. As an application we also deduce asymptotic estimates for certain decomposable forms.
In this paper, we give sharp upper and lower bounds for the number of degenerate monic (and arbitrary, not necessarily monic) polynomials with integer coefficients of fixed degree $n \ge 2$ and height bounded by $H \ge 2$. The polynomial is called degenerate if it has two distinct roots whose quotient is a root of unity. In particular, our bounds imply that non-degenerate linear recurrence sequences can be generated randomly.
Let $k$ be a number field and $K$ a finite extension of $k$. We count points of bounded height in projective space over the field $K$ generating the extension $K/k$. As the height gets large we derive asymptotic estimates with a particularly good error term respecting the extension $K/k$. In a future paper we will use these results to get asymptotic estimates for the number of points of fixed degree over $k$. We also introduce the notion of an adelic Lipschitz height generalizing that of Masser and Vaaler. This will lead to further applications involving points of fixed degree on linear variet…
We give an asymptotic formula for the number of elliptic curves over $\mathbb{Q}$ with bounded Faltings height. Silverman has shown that the Faltings height for elliptic curves over number fields can be expressed in terms of modular functions and the minimal discriminant of the elliptic curve. We use this to recast the problem as one of counting lattice points in a particular region in $\mathbb{R}^2$.
An important problem in analytic and geometric combinatorics is estimating the number of lattice points in a compact convex set in a Euclidean space. Such estimates have numerous applications throughout mathematics. In this note, we exhibit applications of a particular estimate of this sort to several counting problems in number theory: counting integral points and units of bounded height over number fields, counting points of bounded height over positive definite quaternion algebras, and counting points of bounded height with a fixed support over global function fields. Our arguments use a co…
Given a set of endomorphisms on $\mathbb{P}^N$, we establish an upper bound on the number of points of bounded height in the associated monoid orbits. Moreover, we give a more refined estimate with an associated lower bound when the monoid is free. Finally, we show that most sets of rational functions in one variable satisfy these more refined bounds.
The present note is devoted to an amendment to a recent paper of Ellenberg, Lawrence and Venkatesh. Roughly speaking, the main result here states the subpolynomial growth of the number of integral points with bounded height of a variety over a number field whose fundamental group is large. Such an improvement, i.e. requiring large fundamental group as opposed to the existence of a geometric variation of pure Hodge structures, was already asked in op.cit..
A polynomial over a ring is called decomposable if it is a composition of two nonlinear polynomials. In this paper, we obtain sharp lower and upper bounds for the number of decomposable polynomials with integer coefficients of fixed degree and bounded height. Moreover, we obtain asymptotic formulas for the number of decomposable monic polynomials of even degree. For example, the number of monic sextic integer polynomials which are decomposable and of height at most $H$ is asymptotic to $(16ζ(3)-5/4)H^3$ as $H \to \infty$.
A condition is given, under which a general lattice point counting function is asymptotic to the corresponding ball volume growth function. This is then used to give height asymptotics in the style of the Batyrev-Manin Conjecture for certain intrinsically defined heights on semisimple groups.
We provide an asymptotic estimate for the number of rational points of bounded height on a non-singular conic over the rationals. The estimate is uniform in the coefficients of the underlying quadratic form.
An asymptotic formula is established for the number of rational points of bounded anticanonical height which lie on a certain Zariski open subset of an arbitrary smooth biquadratic hypersurface in sufficiently many variables. The proof uses the Hardy--Littlewood circle method.
We count the number of rational elliptic curves of bounded naive height that have a rational $N$-isogeny, for $N \in \{2,3,4,5,6,8,9,12,16,18\}$. For some $N$, this is done by generalizing a method of Harron and Snowden. For the remaining cases, we use the framework of Ellenberg, Satriano and Zureick-Brown, in which the naive height of an elliptic curve is the height of the corresponding point on a moduli stack.
We count rational points of bounded height on the Cayley ruled cubic surface and interpret the result in the context of general conjectures due to Batyrev and Tschinkel.
For any number field k, upper bounds are established for the number of k-rational points of bounded height on non-singular del Pezzo surfaces defined over k, which are equipped with suitable conic bundle structures over k.
We fix a counting function of multiplicities of algebraic points in a projective hypersurface over a number field, and take the sum over all algebraic points of bounded height and fixed degree. An upper bound for the sum with respect to this counting function will be given in terms of the degree of the hypersurface, the dimension of the singular locus, the upper bounds of height, and the degree of the field of definition.
For a given conic bundle X over a curve C defined over F_q, we count irreducible branch covers of C in X of degree d and height e>>1. As a special case, we get the number of algebraic numbers of degree d and height e over the function field F_q (C).
We give an upper bound for the number of rational points of height at most $B$, lying on a surface defined by a quadratic form $Q$. The bound shows an explicit dependence on $Q$. It is optimal with respect to $B$, and is also optimal for typical forms $Q$.
We prove an estimate on the number of rational points on the Grassmannian variety of bounded twisted height, refining the classical results of Schmidt ([12]) and Thunder ([20]) over the rational field: most importantly, our formula counts all points. Among the consequences are a couple of new implications on the classical subject of counting rational points on flag varieties.
In this paper, we prove a general result computing the number of rational points of bounded height on a projective variety $V$ which is covered by lines. The main technical result used to achieve this is an upper bound on the number of rational points of bounded height on a line. This upper bound is such that it can be easily controlled as the line varies, and hence is used to sum the counting functions of the lines which cover the original variety $V$.
Let $S$ be a Shimura variety and let $h$ be a Weil height function on $S$. We conjecture that the heights of special points in $S$ are discriminant negligible. Assuming this conjecture to be true, we prove that the sizes of the Galois orbits of special points grow as a fixed power of their discriminant (an invariant we will define in the text). In the case of Shimura varieties of abelian type, the height bound holds by the recently proved averaged Colmez formula, and our theorem gives a new proof of Tsimerman's Galois lower bound in this case. The main novelty is that our approach avoids the u…
Let $F(x_1,...,x_n)$ be a form of degree $d\geq 2$, which produces a geometrically irreducible hypersurface in $\mathbb{P}^{n-1}$. This paper is concerned with the number of rational points on F=0 which have height at most $B$. Whenever $n<6$, or whenever the hypersurface is not a union of lines, we obtain estimates that are essentially best possible and that are uniform in $d$ and $n$.
Deng (arXiv:math/9812082) gave an asymptotic formula for the number of rational points on a weighted projective space over a number field with respect to a certain height function. We prove a generalization of Deng's result involving a morphism between weighted projective spaces, allowing us to count rational points whose image under this morphism has bounded height. This method provides a more general and simpler proof for a result of the first-named author and Najman on counting elliptic curves with prescribed level structures over number fields. We further include some examples of applicati…
We obtain upper bounds for the number of monic irreducible polynomials over $\mathbb Z$ of a fixed degree $n$ and a growing height $H$ for which the field generated by one of its roots has a given discriminant. We approach it via counting square-free parts of polynomial discriminants via two complementing approaches. In turn, this leads to a lower bound on the number of distinct discriminants of fields generated by roots of polynomials of degree $n$ and height at most $H$. We also give an upper bound for the number of trinomials of bounded height with given square-free part of the discriminant…
In this paper we consider the problem of counting algebraic numbers $α$ of fixed degree $n$ and bounded height $Q$ such that the derivative of the minimal polynomial $P_α(x)$ of $α$ is bounded, $|P_α'(α)| < Q^{1-v}$. This problem has many applications to the problems of the metric theory of Diophantine approximation. We prove that the number of $α$ defined above on the interval $\left(-\frac12, \frac12\right)$ doesn't exceed $c_1(n)Q^{n+1-\frac{1}{7}v}$ for $Q>Q_0(n)$ and $1.4 \le v \le \frac{7}{16}(n+1)$. Our result is based on an improvement to the lemma on the order of zero approximation by…
Let $k$ be a number field, let $θ$ be a nonzero algebraic number, and let $H(\cdot)$ be the Weil height on the algebraic numbers. In response to a question by T. Loher and D. W. Masser, we prove an asymptotic formula for the number of $α\in k$ with $H(αθ)\leq X$. We also prove an asymptotic counting result for a new class of height functions defined via extension fields of $k$. This provides a conceptual framework for Loher and Masser's problem and generalizations thereof. Moreover, we analyze the leading constant in our asymptotic formula for Loher and Masser's problem. In particular, we prov…
A conjecture of Serre concerns the number of rational points of bounded height on a finite cover of projective space P^{n-1}. In this paper, we achieve Serre's conjecture in the special case of smooth cyclic covers of any degree when n is at least 10, and surpass it for covers of degree 3 or higher when n > 10. This is achieved by a new bound for the number of perfect r-th power values of a polynomial with nonsingular leading form, obtained via a combination of an r-th power sieve and the q-analogue of van der Corput's method.
Let $K$ be a global field of positive characteristic. We give an asymptotic formula for the number of $K$-points of bounded height on the Hilbert scheme $\text{Hilb}^2\mathbb{P}^2$ and show that by eliminating an exceptional thin set, the constant in front of the main term agrees with the prediction of Peyre in the function field setting. Moreover, we extend the analogy between the integers and $0$-cycles on a variety $V$ over a finite field to $0$-cycles on a variety $V$ over $K$ and establish a version of the prime number theorem in the case when $V = \mathbb{P}^2$.
In this paper, we concentrate on counting and testing dominant polynomials with integer coefficients. A polynomial is called dominant if it has a simple root whose modulus is strictly greater than the moduli of its remaining roots. In particular, our results imply that the probability that the dominant root assumption holds for a random monic polynomial with integer coefficients tends to 1 in some setting. However, for arbitrary integer polynomials it does not tend to 1. For instance, the proportion of dominant quadratic integer polynomials of height $H$ among all quadratic integer polynomials…
Given a number field $K$ of degree $n_K$ and with absolute discriminant $d_K$, we obtain an explicit bound for the number $N_K(T)$ of non-trivial zeros (counted with multiplicity), with height at most $T$, of the Dedekind zeta function $ζ_K(s)$ of $K$. More precisely, we show that for $T \geq 1$, $$ \Big| N_K (T) - \frac{T}π \log \Big( d_K \Big( \frac{T}{2πe}\Big)^{n_K}\Big)\Big| \le 0.228 (\log d_K + n_K \log T) + 23.108 n_K + 4.520, $$ which improves previous results of Kadiri and Ng, and Trudgian. The improvement is based on ideas from the recent work of Bennett $et$ $al.$ on counting zeros…
Let $Z$ be a projective geometrically integral algebraic variety. This paper is concerned with estimating the number of rational points on $Z$ which have height at most $B$. The bounds obtained are uniform in varieties of fixed degree and fixed dimension, and are essentially best possible for varieties of degree at least six.
Manin's conjecture for the asymptotic behavior of the number of rational points of bounded height on del Pezzo surfaces can be approached through universal torsors. We prove several auxiliary results for the estimation of the number of integral points in certain regions on universal torsors. As an application, we prove Manin's conjecture for a singular quartic del Pezzo surface.
