For an algebraic number $α$ of degree $n$, let $\mathcal{M}_α$ be the $\mathbb{Z}$-module generated by $1,α,\ldots ,α^{n-1}$; then $\mathbb{Z}_α:=\{ξ\in\mathbb{Q} (α):\, ξ\mathcal{M}_α\subseteq\mathcal{M}_α\}$ is the ring of scalars of $\mathcal{M}_α$. We call an order of the shape $\mathbb{Z}_α$ \emph{rationally monogenic}. If $α$ is an algebraic integer, then $\mathbb{Z}_α=\mathbb{Z}[α]$ is monogenic. Rationally monogenic orders are special types of invariant orders of binary forms, which have been studied intensively. If $α,β$ are two $\text{GL}_2(\mathbb{Z})$-equivalent algebraic numbers, i.e., $β=(aα+b)/(cα+d)$ for some $\big(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\big)\in\text{GL}_2(\mathbb{Z})$, then $\mathbb{Z}_α=\mathbb{Z}_β$. Given an order $\mathcal{O}$ of a number field, we call a $\text{GL}_2(\mathbb{Z})$-equivalence class of $α$ with $\mathbb{Z}_α=\mathcal{O}$ a \emph{rational monogenization} of $\mathcal{O}$.
We prove the following. If $K$ is a quartic number field, then $K$ has only finitely many orders with more than two rational monogenizations. This is best possible. Further, if $K$ is a number field of degree $\geq 5$, the Galois group of whose normal closure is $5$-transitive, then $K$ has only finitely many orders with more than one rational monogenization. The proof uses finiteness results for unit equations, which in turn were derived from Schmidt's Subspace Theorem. We generalize the above results to rationally monogenic orders over rings of $S$-integers of number fields. Our results extend work of Bérczes, Győry and the author from 2013 on multiply monogenic orders.
We show that an order in a quartic field has fewer than $3000$ essentially different generators as a $\mathbb Z$-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 $2^{80}$.
Jones conjectures the arboreal representation of a degree two rational map will have finite index in the full automorphism group of a binary rooted tree except under certain conditions. We prove a version of Jones' Conjecture for quadratic and cubic polynomials assuming the $abc$-Conjecture and Vojta's Conjecture. We also exhibit a family of degree $2$ rational maps and give examples of degree $3$ polynomial maps whose arboreal representations have finite index in the appropriate group of tree automorphisms.
We exploit some properties of the Hurwitz zeta function $ζ(n,x)$ in order to study sums of the form $\frac{1}{π^{n}}\sum_{j=-\infty}^{\infty}1/(jk+l)^{n}$ and $\frac{1}{π^{n}}\sum_{j=-\infty}^{\infty}(-1)^{j}/(jk+l)^{n}$ for $% 2\leq n,k\in \mathbb{N},$ and integer $l\leq k/2$. We show that these sums are algebraic numbers. We also show that $1<n\in \mathbb{N}$ and $p\in \mathbb{Q\cap (}0,1\mathbb{)}$ $:$ the numbers $(ζ(n,p)+(-1)^{n}ζ(n,1-p))/π^{n}$ are algebraic. On the way we find polynomials $s_{m}$ and $c_{m}$ of order respectively $2m+1$ and $2m+2$ such that their $n-$th coefficients of …
We study the problem of counting the number of varieties in families which have a rational point. We give conditions on the singular fibres that force very few of the varieties in the family to contain a rational point, in a precise quantitative sense. This generalises and unifies existing results in the literature by Serre, Browning-Dietmann, Bright-Browning-Loughran, Graber-Harris-Mazur-Starr, et al.
We compute new polynomials with Galois group $M_{11}$ over $\mathbb{Q}(t)$. These polynomials stem from various families of covers of $\mathbb{P}^1\mathbb{C}$ ramified over at least 4 points. Each of these families has features that make a detailed study interesting. Some of the polynomials lead, via specialization, to number fields with very small discriminant or few ramified primes.
In this paper we give a general upper bound for the irrationality exponent of algebraic Laurent series with coefficients in a finite field. Our proof is based on a method introduced in a different framework by Adamczewski and Cassaigne. It makes use of automata theory and, in our context, of a classical theorem due to Christol. We then introduce a new approach which allows us to strongly improve this general bound in many cases. As an illustration, we give few examples of algebraic Laurent series for which we are able to compute the exact value of the irrationality exponent.
We revisit the abstract framework underlying the fibration method for producing rational points on the total space of fibrations over the projective line. By fine-tuning its dependence on external arithmetic conjectures, we render the method unconditional when the degree of the non-split locus is $\leq 2$, as well as in various instances where it is $3$. We are also able to obtain improved results in the regime that is conditionally accessible under Schinzel's hypothesis, by incorporating into it, for the first time, a technique due to Harari for controlling the Brauer--Manin obstruction in fa…
We give an explicit upper bound for the number of equivalence classes of binary forms with rational integral coefficients of given degree and given discriminant, and with given splitting field. Further, we give an explicit upper bound for the number of irreducible binary forms with rational integral coefficients with given invariant order. Our bounds depend on as few parameters as possible. For instance, we show that the number of equivalence classes of irreducible binary forms with rational integral coefficients of degree r with given invariant order has an upper bound depending only on r. We…
It is well known that sin(aπ/b), cos(aπ/b), etc., are only rational numbers for a few select integers a and b. We show that this is equivalent to the fact that only for d = 1,2,3,4, and 6 is the primitive dth root of unity of degree 2 over Q.
In this paper we study a family of curves obtained by fibre products of hyperelliptic curves. We then exploit this family to construct examples of curves of given genus g over a finite field Fq with many rational points. The results obtained improve the known bounds for a few pairs (q, g).
We show that an iterative method for computing the center of mass (CM) of q units of mass, placed on a unit interval [0, 1] along the x- axis, give rise to a simple procedure for expanding rational numbers less than unity in powers of r/s < 1, with r, s, integers larger than 0. The method is then extended to all numbers, real or complex, though the procedure for none rational numbers is more time consuming. We also show how our method provides a natural way to generalize Jacobsthal numbers. Moreover, the method provides a way to generate infinitely many sequences of numbers, of which many play…
In this paper we obtain three undecidable results for exponential diophantine equations over the field $\mathbb Q$ of rational numbers. For example, we prove that there is no algorithm to decide the solvability of a general exponential diophantine equation $F(x_1,\ldots,x_{8})=0$ over $\mathbb Q$ with eight unknowns.
Suppose L is any finite algebraic extension of either the ordinary rational numbers or the p-adic rational numbers. Also let g_1,...,g_k be polynomials in n variables, with coefficients in L, such that the total number of monomial terms appearing in at least one g_i is exactly m. We prove that the maximum number of isolated roots of G:=(g_1,...,g_k) in L^n is finite and depends solely on (m,n,L), i.e., is independent of the degrees of the g_i. We thus obtain an arithmetic analogue of Khovanski's Theorem on Fewnomials, extending earlier work of Denef, Van den Dries, Lipshitz, and Lenstra.
Let $C$ be a curve of genus at least three defined over a number field, and let $r$ be the rank of the rational points of its Jacobian. Under mild hypotheses on $r$, recent results by Katz, Rabinoff, Zureick-Brown, and Stoll bound the number of rational points on $C$ by a constant that depends only on its genus. Yet one expects an even stronger bound that depends favorably on $r$: when $r$ is small, there should be fewer points on $C$. In a 2013 paper, Stoll established such a "rank-favorable" bound for hyperelliptic curves using Chabauty's method. In the present work we extend Stoll's results…
We investigate the density of rational points on Cayley's cubic surface whose coordinates have few prime factors. The key tools used are the circle method and universal torsors.
This paper presents expressions for gamma values at rational points with the denominator dividing 24 or 60. These gamma values are expressed in terms of 10 distinct gamma values and rational powers of $π$ and a few real algebraic numbers. Our elementary list of formulas can be conveniently used to evaluate, for example, algebraic Gauss hypergeometric functions by the Gauss identity. Also, algebraic independence of gamma values and their relation to the elliptic K-function are briefly discussed.
The multiple zeta values (MZV) are a set of real numbers with a beautiful structure as an algebra over the rational numbers. They are related to maybe the most important conjecture on mathematics today, the Riemann hypothesis. In this paper we will show partial solutions to five well known conjectures about the MZV which we partially proved using just linear systems and shuffles products. We saw that to partially solve four of this conjectures we only need linear algebra.
1973 Schinzel proved that the standard logarithmic height h on the maximal totally real field extension of the rationals is either zero or bounded from below by a positive constant. In this paper we study this property for canonical heights associated to rational functions and the corresponding dynamical system on the affine line. At the end, we will give a few remarks on the behavior of h on finite extensions of the maximal totally real field.
We investigate the density of rational points on the Fermat cubic surface and the Cayley cubic surface whose coordinates have few prime factors. The key tools used are the weighted sieve, the circle method and universal torsors.
Let k be an algebraically closed field complete with respect to a non-Archimedean absolute value of arbitrary characteristic. Let D_1,...,D_n be effective nef divisors intersecting transversally in an n-dimensional nonsingular projective variety X. We study the degeneracy of non-Archimedean analytic maps from k into $X\setminus \cup_{i=1}^nD_i$ under various geometric conditions. When X is a rational ruled surface and D_1 and D_2 are ample, we obtain a necessary and sufficient condition such that there is no non-Archimedean analytic map from k into $X\setminus D_1 \cup D_2$. Using a dictionary…
Locally repairable codes, or locally recoverable codes (LRC for short) are designed for application in distributed and cloud storage systems. Similar to classical block codes, there is an important bound called the Singleton-type bound for locally repairable codes. In this paper, an optimal locally repairable code refers to a block code achieving this Singleton-type bound. Like classical MDS codes, optimal locally repairable codes carry some very nice combinatorial structures. Since introduction of the Singleton-type bound for locally repairable codes, people have put tremendous effort on cons…
