A geometric characterization of points of type $m$ on real submanifolds of $\mathbf{C}^n$

Complex vector fields and hypoelliptic partial differential operators

“…is the distribution of (0, 1)-vector fields of an almost CR manifold M , it follows from §2 below that the essential pseudoconcavity condition of [12] implies that E Z (M ) = Z(M ) + Z(M ) and that M is of finite type in the sense of [4]. Therefore, Corollary 1.15 also generalizes [12, Theorem 4.1].…”

“…There is only one positive real root, namely γ = α 1 + 2α 2 + 3α 3 + 2α 4 , that belongs to Q n Φ ∩Q n Φ . The corresponding Levi form L γ has rank one, and hence it is semidefinite.…”

“…Moreover, from (α +ᾱ) + (β +β) = 2(γ +γ), we deduce that R has rank 2 and contains roots of different lengths. Thus we can choose a basis {e j } in h * R such that α +ᾱ = 2e 1 , β +β = 2e 2 , γ +γ = e 1 + e 2 , and, furthermore, that α = e 1 + ae 3 , β = e 2 + ae 3 , γ = (1/2)e 1 + (1/2)e 2 + be 3 + ce 4 for some a, b, c ∈ R, and a > 0.…”

“…A motivation for [12] was to understand the structure of a large class of CR manifolds, of higher CR codimension, that are of finite type in the sense of [4], and are not pseudoconcave in the sense of [11], because of the vanishing of their scalar Levi forms in some characteristic codirections. To prove also in this case a subelliptic estimate for the tangential CauchyRiemann operator on functions, two of the authors invented in [12] the notion of essential pseudoconcavity.…”

Germs of holomorphic mappings between real algebraic hypersurfaces

“…This latter means that for any (3 e N^, there exists a positive integer k{f3) and holomorphic polynomials of their arguments R 3^ (with 0 ^ j ^ k( (3)) such that near 0, one has k{(3) (7) ^ ^ ((2,, (^/, r', ^)p=i,...,., A g^', rQ = 0, j=o with R^ ((2^ (^, r', ^))p=i,...,., z') ^ 0, and ((S,, (^ r', ^))p=i,...,., z') is a maximal set of algebraically independent elements as in the proof of Proposition 1. For (^w,^r) e M D U 1 x U°, putting z' = /($,r), = f{z,w), and r' = g(z,w) in the previous equation yields (8) fc(/3) E ^ ((=a, (7(^ w), p^, w), /($, r))^i,...,,, /($, r)) ^(^(^ w)) EE 0, J=0 / which can be rewritten in the following way:…”