We extend the notion of polynomial integration over an arbitrary circle C in the Euclidean geometry over general fields $\mathbb {F}$ of characteristic zero as a normalised $\mathbb {F}$-linear functional on $\mathbb {F}[\alpha _1, \alpha _2]$ that maps polynomials that evaluate to zero on C to zero and is $\mathrm {SO}(2,\mathbb {F})$-invariant. This allows us to not only build a purely algebraic integration theory in an elementary way, but also give the super Catalan numbers

$$ \begin{align*} S(m,n) = \frac{(2m)!(2n)!}{m!n!(m+n)!} \end{align*} $$

an algebraic interpretation in terms of values of this algebraic integral over some circle applied to the monomials $\alpha _1^{2m}\alpha _2^{2n}$.

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. Monthly 112(3) (2005), 269–270] gave a generalisation of Ram Murty’s irreducibility criterion. We further generalise these criteria.

We introduce and study a natural class of fields in which certain first-order definable sets are existentially definable, and characterise this class by a number of equivalent conditions. We show that global fields belong to this class, and in particular obtain a number of new existential (or diophantine) predicates over global fields.

We answer a question posed by Mordell in 1953, in the case of repeated radical extensions, and find necessary and sufficient conditions for $[F[\sqrt [m_1]{N_1},\dots ,\sqrt [m_\ell ]{N_\ell }]:F]=m_1\cdots m_\ell $ , where F is an arbitrary field of characteristic not dividing any $m_i$ .

A cyclotomic polynomial $\unicode[STIX]{x1D6F7}_{k}(x)$ is an essential cyclotomic factor of $f(x)\in \mathbb{Z}[x]$ if $\unicode[STIX]{x1D6F7}_{k}(x)\mid f(x)$ and every prime divisor of $k$ is less than or equal to the number of terms of $f.$ We show that if a monic polynomial with coefficients from $\{-1,0,1\}$ has a cyclotomic factor, then it has an essential cyclotomic factor. We use this result to prove a conjecture posed by Mercer [‘Newman polynomials, reducibility, and roots on the unit circle’, Integers12(4) (2012), 503–519].

Given $f\in \mathbb{Z}[t]$ of positive degree, we investigate the existence of auxiliary polynomials $g\in \mathbb{Z}[t]$ for which $f(g(t))$ factors as a product of polynomials of small relative degree. One consequence of this work shows that for any quadratic polynomial $f\in \mathbb{Z}[t]$ and any $\unicode[STIX]{x1D700}>0$, there are infinitely many $n\in \mathbb{N}$ for which the largest prime factor of $f(n)$ is no larger than $n^{\unicode[STIX]{x1D700}}$.

Let $v$ be a henselian valuation of any rank of a field $K$ and let $\bar{v}$ be the unique extension of $v$ to a fixed algebraic closure $\overline{K}$ of $K$. In 2005, we studied properties of those pairs $\left( \theta ,\,\alpha \right)$ of elements of $\overline{K}$ with $\left[ K\left( \theta \right):K \right]\,>\,\left[ K\left( \alpha \right):K \right]$ where $\alpha $ is an element of smallest degree over $K$ such that

$$\bar{v}\left( \theta \,-\,\alpha \right)\,=\,\sup \left\{ \bar{v}\left( \theta \,-\,\beta \right)\,|\,\beta \,\in \,\bar{K},\,\left[ K\left( \beta \right):K \right]\,<\,\left[ K\left( \theta \right):K \right] \right\}\,.$$

Such pairs are referred to as distinguished pairs. We use the concept of liftings of irreducible polynomials to give a different characterization of distinguished pairs.

This article explores recursive and integral equations for ruin probabilities of generalised risk processes, under rates of interest with homogenous Markov chain claims and homogenous Markov chain premiums. We assume that claim and premium take a countable number of non-negative values. Generalised Lundberg inequalities for the ruin probabilities of these processes are derived via a recursive technique. Recursive equations for finite time ruin probabilities and an integral equation for the ultimate ruin probability are presented, from which corresponding probability inequalities and upper bounds are obtained. An illustrative numerical example is discussed.

We give a survey of old and new results concerning the expressibility of the real roots of a solvable polynomial over a real number field by real radicals. A characterization of Fermat primes is obtained in terms of solvability by real radicals for certain ploynomials.

We consider a discrete-time risk model which describes the evolution of the reserves of an insurance company at periodic dates fixed in advance. The amount of loss per unit of time corresponds to independent and identically distributed random variables with arithmetic distribution, and the process of the receipt of premiums is assumed to be deterministic, nonnegative but not uniform (instead of being constant and equal to 1 as in the standard, compound binomial model). For this model, we determine the probability of ruin (or of non-ruin), as well as the distribution of the severity of the eventual ruin, with some finite horizon. A compact and efficient exact expression is found by bringing up-to-date a generalised family of Appell polynomials. The method used is illustrated with some numerical examples.

We continue the study of the discrete-time risk model introduced by Picard et al. (2003). The cumulative loss process (St)t∊ℕ has independent and stationary increments, the increments per unit of time having nonnegative integer values with distribution {ai, i ∊ ℕ and mean ā. The premium receipt process (ck)k∊ℕ is deterministic, nonnegative and nonuniform; in addition, we assume it to be regular in order for there to exist a constant c > ā such that the deviation is bounded as the time t varies. We are interested in whether or not ruin occurs within a finite time. If T is the time of ruin, we obtain P(T = ∞) as the limit of P(T > t) as t → ∞, firstly in the particular case where c = 1/d for some positive d ∊ ℕ, and then in the general case for positive c under the condition that a0 > ½.

Let $G$ be an algebraic group and let $X$ be a generically free $G$-variety. We show that $X$ can be transformed, by a sequence of blowups with smooth $G$-equivariant centers, into a $G$-variety ${{X}^{'}}$ with the following property: the stabilizer of every point of ${{X}^{'}}$ is isomorphic to a semidirect product $U\rtimes A$ of a unipotent group $U$ and a diagonalizable group $A$.

As an application of this result, we prove new lower bounds on essential dimensions of some algebraic groups. We also show that certain polynomials in one variable cannot be simplified by a Tschirnhaus transformation.

A classical theorem of Hermite and Joubert asserts that any field extension of degree $n\,=\,5\,\text{or}\,\text{6}$ is generated by an element whose minimal polynomial is of the form ${{\lambda }^{n}}\,+\,{{c}_{1}}{{\lambda }^{n-1}}\,+\,\cdot \cdot \cdot +\,{{c}_{n-1}}\lambda \,+\,{{c}_{n}}$ with ${{c}_{1\,}}\,=\,\,{{c}_{3}}\,=\,0$. We show that this theorem fails for $n\,=\,{{3}^{m}}$ or ${{3}^{m}}+{{3}^{l}}$ (and more generally, for $n={{p}^{m}}$ or ${{p}^{m}}+{{p}^{l}}$, if 3 is replaced by another prime $p$), where $m\,>\,1\,\ge \,0$. We also prove a similar result for division algebras and use it to study the structure of the universal division algebra $\text{UD}\left( n \right)$.

We also prove a similar result for division algebras and use it to study the structure of the universal division algebra $\text{UD}\left( n \right)$.

L'objet de cet article est d'une part la comparaison de la mesure de Mahler et de la longueur d'un polynôme à coefficients entiers dont toutes les racines sont réelles positives. Nous comparons ensuite la mesure de Mahler d'un polynôme à coefïîcients entiers ayant toutes ses racines réelles à une mesure généralisant la longueur.

We study polynomials over an integral domain R which, for infinitely many prime ideals P, induce a permutation of R/P. In many cases, every polynomial with this property must be a composition of Dickson polynomials and of linear polynomials with coefficients in the quotient field of R. In order to find out which of these compositions have the required property we investigate some number theoretic aspects of composition of polynomials. The paper includes a rather elementary proof of ‘Schur's Conjecture’ and contains a quantitative version for polynomials of prime degree.

Let H(x) be a homogeneous polynomial in n indeterminates over an algebraically closed field K. A necesssary and sufficient condition is given for H(x) to admit a factorization of the form

a, b∈ Kn, and “∘” is the usual inner product. This condition involves the linear derivatives of H(x).