We develop a recursive method for computing the $r$-removed $P$-orderings and $P$-orderings of order $h,$ the characteristic sequences associated to these and limits associated to these sequences for subsets $S$ of a Dedekind domain $D.$ This method is applied to compute these objects for $S=\mathbb{Z}$ and $S=\mathbb{Z}\setminus p\mathbb{Z}$.

In this paper we study an action $D$ of the absolute Galois group $\Gamma =Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ on bicolored plane trees. In distinction with the similar action provided by the Grothendieck theory of “Dessins d’enfants” the action $D$ is induced by the action of $\Gamma$ on equivalence classes of conservative polynomials which are the simplest examples of postcritically finite rational functions. We establish some basic properties of the action $D$ and compare it with the Grothendieck action.

For n ∈ ℕ, L > 0, and p ≥ 1 let ${\kappa }_p(n,L)$ be the largest possible value of k for which there is a polynomial P ≢ 0 of the form $P\left(x\right)={\sum }_{j=0}^n{a}_j{x}^j$, $|a_0\left|\ge L\right({\sum }_{j=1}^n|a_j{{|}^p)}^{1/p}$, $a_j\in ℂ$, such that ${(x-1)}^k$ divides P(x). For n ∈ ℕ, L > 0, and q ≥ 1 let ${\mu }_q(n,L)$ be the smallest value of k for which there is a polynomial Q of degree k with complex coefficients such that $\left|Q\left(0\right)\right|>1/L({\sum }_{j=1}^n{\left|Q\left(j\right)\right|}^q{)}^{1/q}$. We find the size of ${\kappa }_p(n,L)$ and ${\mu }_q(n,L)$ for all n ∈ ℕ, L > 0, and 1 ≤ p,q ≤ ∞. The result about ${\mu }_{\infty }(n,L)$ is due to Coppersmith and Rivlin, but our proof is completely different and much shorter even in that special...

On page 211, line 9, and on page 213, line -6, the assumption should be added that F is not the product of generalized cyclotomic polynomials.

Consider the group ${SL}_2\left({\mathbf{O}}_K\right)$ over the ring of algebraic integers of a number field $K$. Define the height of a matrix to be the maximum over all the conjugates of its entries in absolute value. Let ${SL}_2({\mathbf{O}}_K,t)$ be the number of matrices in ${SL}_2\left({\mathbf{O}}_K\right)$ with height bounded by $t$. We determine the asymptotic behaviour of ${SL}_2({\mathbf{O}}_K,t)$ as $t$ goes to infinity including an error term,$${SL}_2({\mathbf{O}}_K,t)=C{t}^{2n}+O\left(t^{2n-\eta }\right)$$with $n$ being the degree of $K$. The constant $C$ involves the discriminant of $K$, an integral depending only on the signature of $K$, and the value of the Dedekind zeta function...

This is a survey of recent results concerning (integer) matrices whose leading principal minors are well-known sequences such as Fibonacci, Lucas, Jacobsthal and Pell (sub)sequences. There are different ways for constructing such matrices. Some of these matrices are constructed by homogeneous or nonhomogeneous recurrence relations, and others are constructed by convolution of two sequences. In this article, we will illustrate the idea of these methods by constructing some integer matrices of this...

Let $S=\{x_1,\cdots ,x_n\}$ be a finite subset of a partially ordered set $P$. Let $f$ be an incidence function of $P$. Let $\left[f(x_i\wedge x_j)\right]$ denote the $n\times n$ matrix having $f$ evaluated at the meet $x_i\wedge x_j$ of $x_i$ and $x_j$ as its $i,j$-entry and $\left[f(x_i\vee x_j)\right]$ denote the $n\times n$ matrix having $f$ evaluated at the join $x_i\vee x_j$ of $x_i$ and $x_j$ as its $i,j$-entry. The set $S$ is said to be meet-closed if $x_i\wedge x_j\in S$ for all $1\le i,j\le n$. In this paper we get explicit combinatorial formulas for the determinants of matrices $\left[f(x_i\wedge x_j)\right]$ and $\left[f(x_i\vee x_j)\right]$ on any meet-closed set $S$. We also obtain necessary and sufficient conditions for the matrices...