We show that the number of derivatives of a non negative 2-order symbol needed to establish the classical Fefferman-Phong inequality is bounded by $\frac{n}{2}+4+ϵ$ improving thus the bound $2n+4+ϵ$ obtained recently by N. Lerner and Y. Morimoto. In the case of symbols of type $S_{0,0}^0$, we show that this number is bounded by $n+4+ϵ$; more precisely, for a non negative symbol $a$, the Fefferman-Phong inequality holds if ${\partial }_x^{\alpha }{\partial }_{\xi }^{\beta }a(x,\xi )$ are bounded for, roughly, $4\le \left|\alpha \right|+\left|\beta \right|\le n+4+ϵ$. To obtain such results and others, we first prove an abstract result which...

Let $M^n$ be a germ at $0\in {ℂ}^m$ of an irreducible analytic set of dimension $n$, where $n\ge 2$ and $0$ is a singular point of $M$. We study the question: when does there exist a germ of holomorphic map $\phi :({ℂ}^n,0)\to (M,0)$ such that ${\phi }^{-1}\left(0\right)=\left\{0\right\}$ ? We prove essentialy three results. In Theorem 1 we consider the case where $M$ is a quasi-homogeneous complete intersection of $k$ polynomials $F=(F_1,...,F_k)$, that is there exists a linear holomorphic vector field $X$ on ${ℂ}^m$, with eigenvalues ${\lambda }_1,...,{\lambda }_m\in {\mathbb{Q}}_{+}$ such that $X\left(F^T\right)=U·F^T$, where $U$ is a $k\times k$ matrix with entries in ${𝒪}_m$. We prove that if...

Let $A,B$ denote generic binary forms, and let ${𝔲}_r={(A,B)}_r$ denote their $r$-th transvectant in the sense of classical invariant theory. In this paper we classify all the quadratic syzygies between the $\left\{{𝔲}_r\right\}$. As a consequence, we show that each of the higher transvectants $\{{𝔲}_r:r\ge 2\}$ is redundant in the sense that it can be completely recovered from ${𝔲}_0$ and ${𝔲}_1$. This result can be geometrically interpreted in terms of the incomplete Segre imbedding. The calculations rely upon the Cauchy exact sequence of $S{L}_2$-representations,...

A continuous linear extension operator, different from Whitney’s, for ${𝒞}^p$-Whitney fields (p finite) on a closed o-minimal subset of ${ℝ}^n$ is constructed. The construction is based on special geometrical properties of o-minimal sets earlier studied by K. Kurdyka with the author.