Let $K$ be a finite extension of ${\mathbf{Q}}_p$. The field of norms of a $p$-adic Lie extension $K_{\infty }/K$ is a local field of characteristic $p$ which comes equipped with an action of $\mathrm{Gal}(K_{\infty }/K)$. When can we lift this action to characteristic $0$, along with a compatible Frobenius map? In this note, we formulate precisely this question, explain its relevance to the theory of $(\varphi ,\Gamma )$-modules, and give a condition for the existence of certain types of lifts.

We give a description of all local derivations (in the Kadison sense) in the polynomial ring in one variable in characteristic two. Moreover, we describe all local derivations in the power series ring in one variable in any characteristic.

H. Sharif et C. Woodcock donnent dans [26] une caractérisation des séries formelles à coefficients dans un corps $K$ de caractéristique non nulle et algébriques sur $K\left(X\right)$ ; ils en déduisent simplement l’algébricité du produit de Hadamard ou des diagonales de séries algébriques. (Ces résultats ont aussi été obtenus par T. Harase [14]). Nous donnons ici une démonstration légèrement différente de leur théorème et montrons comment on peut en déduire une généralisation intéressante de la notion de $p^k$-substitution...

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...