When $\omega$ is a primitive $n$-th root of unity, the quadratic polynomial $F(z) = \omega z (1 -z)$ and the entire map $F(z) = \omega z \mathrm{e}^{-z}$ both have a parabolic fixed point at $0$. Their parabolic multiplicity is equal to $1$, that is, $F^{\circ n}(z) = z \bigl( 1 +c z^n +\mathcal{O}(z^{n+1}) \bigr)$ with $c \neq 0$. The classical proof of this fact is transcendental. We present an arithmetic proof which may be extracted from [Towards global models near homoclinic tangencies of dissipative diffeomorphisms; H. Broer, C. Sim\'{o}, J.C. Tatjer] in the transcendental case and requires working in $\mathbb{Z}/(n -1) \mathbb{Z}$, and which is new in the polynomial case and requires working in the $p$-adic field $\mathbb{Q}_p$ for a suitable prime $p$ such that the order of $2$ in $(\mathbb{Z}/p \mathbb{Z})^\times$ is exactly $n$.
We study first-order differential equations $f'=R(e^z,f)$, where $R\in\C(t,w)$. We prove that a meromorphic solution on the whole complex plane is algebraic over $\C(e^z)$ unless $R$ is a polynomial of degree at most two in its second variable. Such an algebraic solution necessarily has the form $S(e^{z/q})$, with $S$...
In this paper, we study isolated singularities of the following semilinear elliptic equation $-\Delta u+\frac12 x\cdot \nabla u+\frac{1}{q-1}u-u^q=0$ in $\Omega \setminus \{0\}$, where $n\ge 3$, $\Omega \subset \mathbb{R}^n$ is a domain, $0 \in \Omega$ and $q>1$. This equation arises in the study of blow-up profiles of...
Let $\Omega(n)$ denote the number of prime factors of $n$, counted with multiplicity. We prove that if $A\subset\mathbb{N}^3$ has positive upper Banach density, then there are $(x,y,z)\in\mathbb{N}^3$ and $d\in\mathbb{N}$ such that $$(x,y,z),(x+d,y,z),(x,y+d,z),(x,y,z+\Omega(d))\in A.$$ To establish the above result, w...
Let $\varrho\neq -1$ be a complex number with modulus $|\varrho|=1$ and $K(x,y)=P_1(x)+P_2(y)$ for two polynomials $P_1$ and $P_2$. We consider the non-oscillatory solutions such that $\lambda(f)<\infty$ of the generalized Hill equation $f''-K(e^z,e^{-\varrho z})f=0$ ($\sharp$). When $\varrho=1$, the Hill equation $f''...
We consider an elliptic differential operator $A_\varepsilon=- \frac{d}{dx} g(\frac x\varepsilon) \frac{d}{dx}$, $\varepsilon > 0$, with a periodic coefficient that acts in $L_2(\mathbb{R})$. We study the behaviour of solutions of the Cauchy problem for the hyperbolic equation $partial_\tau^2 w_\varepsilon (x,\tau)=- (...
For every $n \geq 1$ there is a distinguished $\mathfrak{sl}_2(\mathbb{R})$ action on the Poincar\'e disk, arising as the infinitesimal isometries of a hyperbolic metric with conical singularity of order $n-1$ at the origin. For $n=1$ this is the standard infinitesimal M\"obius action, and for $n>1$ these vector fields...
Rea Dalipi· 0 citations
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