We provide a characterization of infinite algebraic Galois extensions of the rationals with uniformly bounded local degrees, giving a detailed proof of all the results announced in a paper by Checcoli and Zannier and obtaining relevant generalizations for them. In particular we show that that for an infinite Galois extension of the rationals the following three properties are equivalent: having uniformly bounded local degrees at every prime; having uniformly bounded local degrees at almost every prime; having Galois group of finite exponent. The proof of this result enlightens interesting connections with Zelmanov's work on the Restricted Burnside Problem. We give a formula to explicitly compute bounds for the local degrees of a infinite extension in some special cases. We relate the uniform boundedness of the local degrees to other properties: being a subfield of the compositum of all number fields of bounded degree; being generated by elements of bounded degree. We prove that the above properties are equivalent for abelian extensions, but not in general; we provide counterexamples based on group-theoretical constructions with extraspecial groups and their modules, for which we give explicit realizations.
We consider semigroup dynamical systems defined by several polynomials over a number field $\mathbb{K}$, and the orbit (tree) they generate at a given point. We obtain finiteness results for the set of preperiodic points of such systems that fall in the cyclotomic closure of $\mathbb{K}$. More generally, we consider the finiteness of initial points in the cyclotomic closure for which the orbit contains an algebraic integer of bounded house. This work extends previous results for classical obits generated by one polynomial over $\mathbb{K}$ obtained initially by Dvornicich and Zannier (for prep…
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…
The degree sequence of the algebraic numbers in an algebraic linear recurrence sequence is shown to be virtually periodic. This is proved using the Skolem-Mahler-Lech theorem. It has applications to the degree sequence and the minimal polynomial sequence for exponential sums over finite fields. The degree periodicity also holds for some more complicated non-linear recurrence sequences. We give one example from the iterations of a polynomial map. This depending on the dynamic Mordell-Lang conjecture which has been proved in some cases.
Given smooth, projective, geometrically integral algebraic curves $X$ and $Y$ defined over a number field $K$, assuming that there is a non-constant $K$-morphism $φ\colon X \to Y$, we give an upper bound on the minimum of the degrees of such morphisms. The proof is based on isogeny estimates between abelian varieties.
Given a number field $L$, we define the degree of an algebraic number $v \in L$ with respect to a choice of a primitive element of $L$. We propose the question of computing the minimal degrees of algebraic numbers in $L$, and examine these values in degree $4$ Galois extensions over $\mathbb{Q}$ and triquadratic number fields. We show that computing minimal degrees of non-rational elements in triquadratic number fields is closely related to solving classical Diophantine problems such as congruent number problem as well as understanding various arithmetic properties of elliptic curves.
In different areas of discrete mathematics, a certain type of polynomials, having coefficients in a field K of finite characteristic, has been considered. The form and the degree of these polynomials, here called projective, are simply linked to the characteristic p of K. Roots of these projective polynomials are particular algebraic elements over K, called hyperquadratic. For a general algebraic element of degree d over K, we discuss the possibility of being hyperquadratic. Using a method of differential algebra, we obtain, for particular fields K = Fp, projective polynomials only having poly…
Let $f(x)$ be an irreducible polynomial with integer coefficients of degree at least two. Hooley proved that the roots of the congruence equation $f(x)\equiv 0\mod n$ is uniformly distributed. as a parallel of Hooley's theorem under ideal theoretical setting, we prove the uniformity of the distribution of residues of an algebraic number modulo degree one ideals. Then using this result we show that the roots of a system of polynomial congruences are uniformly distributed. Finally, the distribution of digits of n-adic expansions of an algebraic number is discussed.
In the paper, we study the asymptotic distribution of real algebraic integers of fixed degree as their na\"ıve height tends to infinity. For an arbitrary interval $I \subset \mathbb{R}$ and sufficiently large $Q>0$, we obtain an asymptotic formula for the number of algebraic integers $α\in I$ of fixed degree $n$ and na\"ıve height $H(α)\le Q$. In particular, we show that the real algebraic integers of degree $n$, with their height growing, tend to be distributed like the real algebraic numbers of degree $n-1$. However, we reveal two symmetric "plateaux", where the distribution of real algebrai…
We prove a cohomological formula for non-critical residues of degree eight automorphic $L$-functions of $\mathrm{GSp}(4) \times \mathrm{GL}(2)$ in the spirit of Beilinson conjecture. We rely on the cohomological interpretation of an automorphic period integral and on the study of Novodvorsky's integral representation of the $L$-functions.
In the paper, we study the asymptotic distribution of real algebraic integers of fixed degree as their naive height tends to infinity. Let $I \subset \mathbb{R}$ be an arbitrary bounded interval, and $Q$ be a sufficiently large number. We obtain an asymptotic formula for the count of algebraic integers $α$ of fixed degree $n$ and naive height $H(α)\le Q$ lying in $I$. In this formula, we estimate the order of the error term from above and below. We show that algebraic integers of degree $n$ are distributed asymptotically like algebraic numbers of degree $(n-1)$ as the upper bound $Q$ of height…
Let $N_n(X)$ denote the number of degree $n$ number fields with discriminant bounded by $X$. In this note, we improve the best known upper bounds on $N_n(X)$, finding that $N_n(X) = O(X^{ c (\log n)^2})$ for an explicit constant $c$.
We study the algebraic Brauer classes on open del Pezzo surfaces of degree $4$. I.e., on the complements of geometrically irreducible hyperplane sections of del Pezzo surfaces of degree $4$. We show that the $2$-torsion part is generated by classes of two different types. Moreover, there are two types of $4$-torsion classes. For each type, we discuss methods for the evaluation of such a class at a rational point over a $p$-adic field.
We establish a generalized Ihara zeta function formula for simple graphs with bounded degree. This is a generalization of the formula obtained by G. Chinta, J. Jorgenson and A. Karlsson from a vertex-transitive graph.
Diagonals and algebraicity modulo $p$: a sharper degree bound
Univ. Claude Bernard Lyon 1（里昂第一大学）；CNRS UMR 5208, Institut Camille Jordan（法国国家科学研究中心联合研究单位5208，卡米尔·若尔当研究所）；Centre International de Rencontres Mathématiques (CIRM)（国际数学相遇中心）；Inria（法国国家信息与自动化研究所）；Sorbonne Université（索邦大学）；LIP6（计算机实验室6）；CNRS（法国国家科学研究中心）；IMB（布尔巴基数学研究所）；Université de Bordeaux（波尔多大学）
Let a and b be algebraic numbers such that exactly one of a and b is an algebraic integer, and let f_t(z):=z^2+t be a family of polynomials parametrized by t. We prove that the set of all algebraic numbers t for which there exist positive integers m and n such that f_t^m(a)=f_t^n(b) has bounded Weil height. This is a special case of a more general result supporting a new bounded height conjecture in dynamics. Our results fit into the general setting of the principle of unlikely intersections in arithmetic dynamics.
Let $D$ be a non-empty effective divisor on $\mathbb{P}^1$. We show that when ordered by height, any set of $(D,S)$-integral points on $\mathbb{P}^1$ of bounded degree has relative density zero. We then apply this to arithmetic dynamics: let $φ(z)\in \overline{\mathbb{Q}}(z)$ be a rational function of degree at least two whose second iterate $φ^2(z)$ is not a polynomial. We show that as we vary over points $P\in\mathbb{P}^1(\overline{\mathbb{Q}})$ of bounded degree, the number of algebraic integers in the forward orbit of $P$ is absolutely bounded and zero on average.
We prove a uniform version of non-Archimedean Yomdin-Gromov parametrizations in a definable context with algebraic Skolem functions in the residue field. The parametrization result allows us to bound the number of F_q[t]-points of bounded degrees of algebraic varieties, uniformly in the cardinality q of the finite field F_q and the degree, generalizing work by Sedunova for fixed q. We also deduce a uniform non-Archimedean Pila-Wilkie theorem, generalizing work by Cluckers-Comte-Loeser.
We give new constructions of two classes of algebraic code families which are efficiently list decodable with small output list size from a fraction $1-R-ε$ of adversarial errors where $R$ is the rate of the code, for any desired positive constant $ε$. The alphabet size depends only $ε$ and is nearly-optimal. The first class of codes are obtained by folding algebraic-geometric codes using automorphisms of the underlying function field. The list decoding algorithm is based on a linear-algebraic approach, which pins down the candidate messages to a subspace with a nice "periodic" structure. The …
