Let $S$ be an abelian semigroup, and $A$ a finite subset of $S$. The sumset $hA$ consists of all sums of $h$ elements of $A$, with repetitions allowed. Let $\left|hA\right|$ denote the cardinality of $hA$. Elementary lattice point arguments are used to prove that an arbitrary abelian semigroup has polynomial growth, that is, there exists a polynomial $p\left(t\right)$ such that $\left|hA\right|=p\left(h\right)$ for all sufficiently large $h$. Lattice point counting is also used to prove that sumsets of the form $h_1{A}_1+\cdots +h_r{A}_r$ have multivariate polynomial growth.

Let $R$ be a finite commutative ring with unity. We determine the set of all possible cycle lengths in the ring of polynomials with rational integral coefficients.

For an odd prime p and an integer w ≥ 1, polynomial quotients $q_{p,w}\left(u\right)$ are defined by $q_{p,w}\left(u\right)\equiv (u^w-u^{wp})/pmodp$ with $0\le q_{p,w}\left(u\right)\le p-1$, u ≥ 0, which are generalizations of Fermat quotients $q_{p,p-1}\left(u\right)$. First, we estimate the number of elements $1\le u<N\le p$ for which $f\left(u\right)\equiv q_{p,w}\left(u\right)modp$ for a given polynomial f(x) over the finite field ${}_p$. In particular, for the case f(x)=x we get bounds on the number of fixed points of polynomial quotients. Second, before we study the problem of estimating the smallest number (called the Waring number) of summands needed to express each element of...

For an algebraic number field and a subset ${\alpha }_1,...,{\alpha }_r{\subseteq }_{}$, we establish a lower bound for the average of the logarithmic heights that depends on the ideal of polynomials in $\mathbb{Q}[x_1,...,x_r]$ vanishing at the point $({\alpha }_1,...,{\alpha }_r)$.

Let $𝒜$ be the algebra of quaternions $ℍ$ or octonions $𝕆$. In this manuscript an elementary proof is given, based on ideas of Cauchy and D’Alembert, of the fact that an ordinary polynomial $f\left(t\right)\in 𝒜\left[t\right]$ has a root in $𝒜$. As a consequence, the Jacobian determinant $\left|J\right(f\left)\right|$ is always non-negative in $𝒜$. Moreover, using the idea of the topological degree we show that a regular polynomial $g\left(t\right)$ over $𝒜$ has also a root in $𝒜$. Finally, utilizing multiplication ($*$) in $𝒜$, we prove various results on the topological degree of products...