Let $F=\left(\omega \right)$ be a codim 1 local foliation generated by a germ $\omega$ of the form $\omega =f_1...f_p{\sum }_{i=1}^p{\lambda }_i\frac{d{f}_i}{f_i}$ for some complex numbers ${\lambda }_i$ and germs $f_i$ of holomorphic functions at the origin in ${\mathbf{C}}^n$. We determine, under some conditions, the set of equivalence classes of first order unfoldings and construct explicitly a universal unfolding of $F$. Special cases of this include foliations with holomorphic or meromorphic first integrals. We also show that the unfolding theory for $F$ is equivalent to the unfolding theory for the multiform...

Let $\Gamma$ be a non-pluripolar set in ${ℂ}^N$. Let $f$ be a function holomorphic in a connected open neighborhood $G$ of $\Gamma$. Let $\left\{P_n\right\}$ be a sequence of polynomials with $deg{P}_n\le d_n(d_n\<d_{n+1})$ such that $$\underset{n\to \infty }{lim\; sup}{|f\left(z\right)-P_n\left(z\right)|}^{1/d_n}\<1,z\in \Gamma .$$ We show that if $$\underset{n\to \infty }{lim\; sup}{\left|P_n\left(z\right)\right|}^{1/d_n}\le 1,z\in E,$$ where $E$ is a set in ${ℂ}^N$ such that the global extremal function $V_E\equiv 0$ in ${ℂ}^N$, then the maximal domain of existence $G_f$ of $f$ is one-sheeted, and $$\underset{n\to \infty }{lim\; sup}{\parallel f-P_n\parallel }_K^{\frac{1}{d_n}}\<1$$ for every compact set $K\subset G_f$. If, moreover, the sequence $\{d_{n+1}/d_n\}$ is bounded then $G_f={ℂ}^N$. If $E$ is a closed...

To a plurisubharmonic function $u$ on ${\mathbf{C}}^n$ with logarithmic growth at infinity, we may associate the Robin function $${\displaystyle {\rho }_u\left(z\right)=\underset{\lambda \to \infty }{lim\; sup}u\left(\lambda z\right)-log\left(\lambda z\right)}$$ defined on ${\mathbf{P}}^{n-1}$, the hyperplane at infinity. We study the classes $L_{+}$, and (respectively) $L_p$ of plurisubharmonic functions which have the form $u=log(1+|z\left|\right)+O\left(1\right)$ and (respectively) for which the function ${\rho }_u$ is not identically $-\infty$. We obtain an integral formula which connects the Monge-Ampère measure on the space ${\mathbf{C}}^n$ with the Robin function on ${\mathbf{P}}^{n-1}$. As an application we obtain a criterion...

Given an irreducible algebraic curves $A$ in ${ℂ}^N$, let $m_d$ be the dimension of the complex vector space of all holomorphic polynomials of degree at most $d$ restricted to $A$. Let $K$ be a nonpolar compact subset of $A$, and for each $d=1,2,...,$ choose $m_d$ points ${\left\{A_{dj}\right\}}_{j=1,...,m_d}$ in $K$. Finally, let ${\Lambda }_d$ be the $d$-th Lebesgue constant of the array $\left\{A_{dj}\right\}$; i.e., ${\Lambda }_d$ is the operator norm of the Lagrange interpolation operator $L_d$ acting on $C\left(K\right)$, where $L_d\left(f\right)$ is the Lagrange interpolating polynomial for $f$ of degree $d$ at the points ${\left\{A_{dj}\right\}}_{j=1,...,m_d}$. Using techniques...

We consider a singularity perturbed nonlinear differential equation $\epsilon u^{\text{'}}=f\left(x\right)u++\epsilon P(x,u,\epsilon )$ which we suppose real analytic for $x$ near some interval $[a,b]$ and small $\left|u\right|$, $\left|\epsilon \right|$. We furthermore suppose that 0 is a turning point, namely that $xf\left(x\right)$ is positive if $x\ne 0$. We prove that the existence of nicely behaved (as $ϵ\to 0$) local (at $x=0$) or global, real analytic or $C^{\infty }$ solutions is equivalent to the existence of a formal series solution $\sum u_n\left(x\right){\epsilon }^n$ with $u_n$ analytic at $x=0$. The main tool of a proof is a new “principle of analytic continuation” for...