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By analogy with the trace of an algebraic integer
$\alpha $
with conjugates
$\alpha _1=\alpha , \ldots , \alpha _d$
, we define the G-measure
$ {\mathrm {G}} (\alpha )= \sum _{i=1}^d ( |\alpha _i| + 1/ | \alpha _i | )$
and the absolute
${\mathrm G}$
-measure
${\mathrm {g}}(\alpha )={\mathrm {G}}(\alpha )/d$
. We establish an analogue of the Schur–Siegel–Smyth trace problem for totally positive algebraic integers. Then we consider the case where
$\alpha $
has all its conjugates in a sector
$| \arg z | \leq \theta $
,
$0 < \theta < 90^{\circ }$
. We compute the greatest lower bound
$c(\theta )$
of the absolute G-measure of
$\alpha $
, for
$\alpha $
belonging to
$11$
consecutive subintervals of
$]0, 90 [$
. This phenomenon appears here for the first time, conforming to a conjecture of Rhin and Smyth on the nature of the function
$c(\theta )$
. All computations are done by the method of explicit auxiliary functions.
Let
$\alpha $
be a totally positive algebraic integer of degree d, with conjugates
$\alpha _1=\alpha , \alpha _2, \ldots , \alpha _d$
. The absolute
$S_k$
-measure of
$\alpha $
is defined by
$s_k(\alpha )= d^{-1} \sum _{i=1}^{d}\alpha _i^k$
. We compute the lower bounds
$\upsilon _k$
of
$s_k(\alpha )$
for each integer in the range
$2\leq k \leq 15$
and give a conjecture on the results for integers
$k>15$
. Then we derive the lower bounds of
$s_k(\alpha )$
for all real numbers
$k>2$
. Our computation is based on an improvement in the application of the LLL algorithm and analysis of the polynomials in the explicit auxiliary functions.
Let
$a,b$
and n be positive integers and let
$S=\{x_1, \ldots , x_n\}$
be a set of n distinct positive integers. For
${x\in S}$
, define
$G_{S}(x)=\{d\in S: d<x, \,d\mid x \ \mathrm {and} \ (d\mid y\mid x, y\in S)\Rightarrow y\in \{d,x\}\}$
. Denote by
$[S^a]$
the
$n\times n$
matrix having the ath power of the least common multiple of
$x_i$
and
$x_j$
as its
$(i,j)$
-entry. We show that the bth power matrix
$[S^b]$
is divisible by the ath power matrix
$[S^a]$
if
$a\mid b$
and S is gcd closed (that is,
$\gcd (x_i, x_j)\in S$
for all integers i and j with
$1\le i, j\le n$
) and
$\max _{x\in S} \{|G_S (x)|\}=1$
. This confirms a conjecture of Shaofang Hong [‘Divisibility properties of power GCD matrices and power LCM matrices’, Linear Algebra Appl.428 (2008), 1001–1008].
In this paper, we prove the assertion that the number of monic cubic polynomials $F(x) = x^3 + a_2 x^2 + a_1 x + a_0$ with integer coefficients and irreducible, Galois over ${\mathbb {Q}}$ satisfying $\max \{|a_2|, |a_1|, |a_0|\} \leq X$ is bounded from above by $O(X (\log X)^2)$. We also count the number of abelian monic binary cubic forms with integer coefficients up to a natural equivalence relation ordered by the so-called Bhargava–Shankar height. Finally, we prove an assertion characterizing the splitting field of 2-torsion points of semi-stable abelian elliptic curves.
We use circulant matrices and hyperelliptic curves over finite fields to study some arithmetic properties of certain determinants involving Legendre symbols and kth power residues.
A finite set of integers A tiles the integers by translations if $\mathbb {Z}$ can be covered by pairwise disjoint translated copies of A. Restricting attention to one tiling period, we have $A\oplus B=\mathbb {Z}_M$ for some $M\in \mathbb {N}$ and $B\subset \mathbb {Z}$. This can also be stated in terms of cyclotomic divisibility of the mask polynomials $A(X)$ and $B(X)$ associated with A and B.
In this article, we introduce a new approach to a systematic study of such tilings. Our main new tools are the box product, multiscale cuboids and saturating sets, developed through a combination of harmonic-analytic and combinatorial methods. We provide new criteria for tiling and cyclotomic divisibility in terms of these concepts. As an application, we can determine whether a set A containing certain configurations can tile a cyclic group $\mathbb {Z}_M$, or recover a tiling set based on partial information about it. We also develop tiling reductions where a given tiling can be replaced by one or more tilings with a simpler structure. The tools introduced here are crucial in our proof in [24] that all tilings of period $(pqr)^2$, where $p,q,r$ are distinct odd primes, satisfy a tiling condition proposed by Coven and Meyerowitz [2].
We obtain an effective analytic formula, with explicit constants, for the number of distinct irreducible factors of a polynomial $f \in \mathbb {Z}[x]$. We use an explicit version of Mertens’ theorem for number fields to estimate a related sum over rational primes. For a given $f \in \mathbb {Z}[x]$, our result yields a finite list of primes that certifies the number of distinct irreducible factors of f.
Girstmair [‘On an irreducibility criterion of M. Ram Murty’, Amer. Math. Monthly112(3) (2005), 269–270] gave a generalisation of Ram Murty’s irreducibility criterion. We further generalise these criteria.
By making use of the Cauchy double alternant and the Laplace expansion formula, we establish two closed formulae for the determinants of factorial fractions that are then utilised to evaluate several determinants of binomial coefficients and Catalan numbers, including those obtained recently by Chammam [‘Generalized harmonic numbers, Jacobi numbers and a Hankel determinant evaluation’, Integral Transforms Spec. Funct.30(7) (2019), 581–593].
We study a class of two-variable polynomials called exact polynomials which contains $A$-polynomials of knot complements. The Mahler measure of these polynomials can be computed in terms of a volume function defined on the vanishing set of the polynomial. We prove that the local extrema of the volume function are on the two-dimensional torus and give a closed formula for the Mahler measure in terms of these extremal values. This formula shows that the Mahler measure of an irreducible and exact polynomial divided by $\pi$ is greater than the amplitude of the volume function. We also prove a K-theoretic criterion for a polynomial to be a factor of an $A$-polynomial and give a topological interpretation of its Mahler measure.
We study intersections of orbits in polynomial semigroup dynamics with lines on the affine plane over a number field, extending previous work of D. Ghioca, T. Tucker, and M. Zieve (2008).
where
$\chi $
is a primitive Dirichlet character and F belongs to a class of L-functions. The class we consider includes L-functions associated with automorphic representations of
$GL(n)$
over
${\mathbb {Q}}$
.
Given $d\in \mathbb{N}$, we establish sum-product estimates for finite, nonempty subsets of $\mathbb{R}^{d}$. This is equivalent to a sum-product result for sets of diagonal matrices. In particular, let $A$ be a finite, nonempty set of $d\times d$ diagonal matrices with real entries. Then, for all $\unicode[STIX]{x1D6FF}_{1}<1/3+5/5277$,
which strengthens a result of Chang [‘Additive and multiplicative structure in matrix spaces’, Combin. Probab. Comput.16(2) (2007), 219–238] in this setting.
We prove that for every sufficiently large integer $n$, the polynomial $1+x+x^{2}/11+x^{3}/111+\cdots +x^{n}/111\ldots 1$ is irreducible over the rationals, where the coefficient of $x^{k}$ for $1\leqslant k\leqslant n$ is the reciprocal of the decimal number consisting of $k$ digits which are each $1$. Similar results following from the same techniques are discussed.
Lapkova [‘On the average number of divisors of reducible quadratic polynomials’, J. Number Theory180 (2017), 710–729] uses a Tauberian theorem to derive an asymptotic formula for the divisor sum $\sum _{n\leq x}d(n(n+v))$ where $v$ is a fixed integer and $d(n)$ denotes the number of divisors of $n$. We reprove this result with additional terms in the asymptotic formula, by investigating the relationship between this divisor sum and the well-known sum $\sum _{n\leq x}d(n)d(n+v)$.
The aim of this note is to give a simple topological proof of the well-known
result concerning continuity of roots of polynomials. We also consider a
more general case with polynomials of a higher degree approaching a given
polynomial. We then examine the continuous dependence of solutions of linear
differential equations with constant coefficients.
Let $n\geq 1$ be an integer and $f$ be an arithmetical function. Let $S=\{x_{1},\ldots ,x_{n}\}$ be a set of $n$ distinct positive integers with the property that $d\in S$ if $x\in S$ and $d|x$. Then $\min (S)=1$. Let $(f(S))=(f(\gcd (x_{i},x_{j})))$ and $(f[S])=(f(\text{lcm}(x_{i},x_{j})))$ denote the $n\times n$ matrices whose $(i,j)$-entries are $f$ evaluated at the greatest common divisor of $x_{i}$ and $x_{j}$ and the least common multiple of $x_{i}$ and $x_{j}$, respectively. In 1875, Smith [‘On the value of a certain arithmetical determinant’, Proc. Lond. Math. Soc.7 (1875–76), 208–212] showed that $\det (f(S))=\prod _{l=1}^{n}(f\ast \unicode[STIX]{x1D707})(x_{l})$, where $f\ast \unicode[STIX]{x1D707}$ is the Dirichlet convolution of $f$ and the Möbius function $\unicode[STIX]{x1D707}$. Bourque and Ligh [‘Matrices associated with classes of multiplicative functions’, Linear Algebra Appl.216 (1995), 267–275] computed the determinant $\det (f[S])$ if $f$ is multiplicative and, Hong, Hu and Lin [‘On a certain arithmetical determinant’, Acta Math. Hungar.150 (2016), 372–382] gave formulae for the determinants $\det (f(S\setminus \{1\}))$ and $\det (f[S\setminus \{1\}])$. In this paper, we evaluate the determinant $\det (f(S\setminus \{x_{t}\}))$ for any integer $t$ with $1\leq t\leq n$ and also the determinant $\det (f[S\setminus \{x_{t}\}])$ if $f$ is multiplicative.
We classify all polynomials $P(X)\in \mathbb{Q}[X]$ with rational coefficients having the property that the quotient $(\unicode[STIX]{x1D706}_{i}-\unicode[STIX]{x1D706}_{j})/(\unicode[STIX]{x1D706}_{k}-\unicode[STIX]{x1D706}_{\ell })$ is a rational number for all quadruples of roots $(\unicode[STIX]{x1D706}_{i},\unicode[STIX]{x1D706}_{j},\unicode[STIX]{x1D706}_{k},\unicode[STIX]{x1D706}_{\ell })$ with $\unicode[STIX]{x1D706}_{k}\neq \unicode[STIX]{x1D706}_{\ell }$.
For a finite field of odd cardinality $q$, we show that the sequence of iterates of $aX^{2}+c$, starting at $0$, always recurs after $O(q/\text{log}\log q)$ steps. For $X^{2}+1$, the same is true for any starting value. We suggest that the traditional “birthday paradox” model is inappropriate for iterates of $X^{3}+c$, when $q$ is 2 mod 3.