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.

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 ${}_{R,n}$.

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 x be an indeterminate over ℂ. We investigate solutions $$\begin{array}{c}\hfill \alpha (s,x)=\sum _{n\ge 0}{\alpha }_n\left(s\right)x^n,\end{array}$$αn : ℂ → ℂ, n ≥ 0, of the first cocycle equation $$\begin{array}{c}\hfill \alpha (s+t,x)=\alpha (s,x)\alpha \left(t,F(s,x)\right),s,t\in ,\left(\mathrm{Co}1\right)\end{array}$$in ℂ [[x]], the ring of formal power series over ℂ, where (F(s,x))s ∈ ℂ is an iteration group of type II, i.e. it is a solution of the translation equation $$\begin{array}{c}\hfill F(s+t,x)=F(s,F(t,x\left)\right),s,t\in ,\left(\mathrm{T}\right)\end{array}$$of the form F(s,x) ≡ x + ck(s)xk mod xk+1, where k ≥ 2 and ck ≠ 0 is necessarily an additive function. It is easy to prove that the coefficient functions αn(s) of $$\begin{array}{c}\hfill \alpha (s,x)=1+\sum _{n\ge 1}{\alpha }_n\left(s\right)x^n\end{array}$$are polynomials in ck(s).It is possible to replace...

Let $\overline{X}$ be a germ of normal surface with local ring $\overline{R}$ covering a germ of regular surface $X$ with local ring $R$ of characteristic $p\>0$. Given an extension of valuation rings $W/V$ birationally dominating $\overline{R}/R$, we study the existence of a new such pair of local rings ${\overline{R}}^{\text{'}}/R^{\text{'}}$ birationally dominating $\overline{R}/R$, such that $R^{\text{'}}$ is regular and ${\overline{R}}^{\text{'}}$ has only toric singularities. This is achieved when $W/V$ is defectless or when $[W:V]$ is equal to $p$

Let k[[x,y]] be the formal power series ring in two variables over a field k of characteristic zero and let d be a nonzero derivation of k[[x,y]]. We prove that if Ker(d) ≠ k then Ker(d) = Ker(δ), where δ is a jacobian derivation of k[[x,y]]. Moreover, Ker(d) is of the form k[[h]] for some h ∈ k[[x,y]].