We give a deepened version of a lemma of Gabrielov and then use it to prove the following fact: if h ∈ 𝕂[[X]] (𝕂 = ℝ or ℂ) is a root of a non-zero polynomial with convergent power series coefficients, then h is convergent.

We consider two issues concerning polynomial cycles. Namely, for a discrete valuation domain $R$ of positive characteristic (for $N\ge 1$) or for any Dedekind domain $R$ of positive characteristic (but only for $N\ge 2$), we give a closed formula for a set $𝒞YCL(R,N)$ of all possible cycle-lengths for polynomial mappings in $R^N$. Then we give a new property of sets $𝒞YCL(R,1)$, which refutes a kind of conjecture posed by W. Narkiewicz.

Let R be a real closed field, and denote by ${}_{R,n}$ the ring of germs, at the origin of Rⁿ, of $C^{\infty }$ functions in a neighborhood of 0 ∈ Rⁿ. For each n ∈ ℕ, we construct a quasianalytic subring ${}_{R,n}{\subset }_{R,n}$ with some natural properties. We prove that, for each n ∈ ℕ, ${}_{R,n}$ is a noetherian ring and if R = ℝ (the field of real numbers), then ${}_{ℝ,n}=ₙ$, where ₙ is the ring of germs, at the origin of ℝⁿ, of real analytic functions. Finally, we prove the Real Nullstellensatz and solve Hilbert’s 17th Problem for the ring ${}_{R,n}$.

Let $R$ be a commutative ring with an identity different from zero and $n$ be a positive integer. Anderson and Badawi, in their paper on $n$-absorbing ideals, define a proper ideal $I$ of a commutative ring $R$ to be an $n$-absorbing ideal of $R$, if whenever $x_1\cdots x_{n+1}\in I$ for $x_1,...,x_{n+1}\in R$, then there are $n$ of the $x_i$’s whose product is in $I$ and conjecture that ${\omega }_{R\left[X\right]}\left(I\left[X\right]\right)={\omega }_R\left(I\right)$ for any ideal $I$ of an arbitrary ring $R$, where ${\omega }_R\left(I\right)=min\{n:I\text{is}\text{an}n\text{-absorbing}\text{ideal}\text{of}R\}$. In the present paper, we use content formula techniques to prove that their conjecture is true, if one of the following conditions...

We give a simplified approach to the Abhyankar-Moh theory of approximate roots. Our considerations are based on properties of the intersection multiplicity of local curves.

Let ℕ represent the positive integers and ℕ₀ the non-negative integers. If b ∈ ℕ and Γ is a multiplicatively closed subset of ${\mathbb{Z}}_b=\mathbb{Z}/b\mathbb{Z}$, then the set $H_{\Gamma }=x\in \mathbb{N}|x+b\mathbb{Z}\in \Gamma \cup 1$ is a multiplicative submonoid of ℕ known as a congruence monoid. An arithmetical congruence monoid (or ACM) is a congruence monoid where Γ = ā consists of a single element. If $H_{\Gamma }$ is an ACM, then we represent it with the notation M(a,b) = (a + bℕ₀) ∪ 1, where a, b ∈ ℕ and a² ≡ a (mod b). A classical 1954 result of James and Niven implies that the only ACM...