The concepts of tropical semiring and tropical hypersurface, are extended to the case of an arbitrary ordered group. Then, we define the tropicalization of a polynomial with coefficients in a Krull-valued field. After a close study of the properties of the operator “tropicalization” we conclude with an extension of Kapranov’s theorem to algebraically closed fields together with a valuation over...

1480 NOTICES OF THE AMS VOLUME 42, NUMBER 12 T he theory of functions (what we now call the theory of functions of a complex variable) was one of the great achievements of nineteenth century mathematics. Its beauty and range of applications were immense and immediate. The desire to generalize to higher dimensions must have been correspondingly irresistible. In this desire to generalize, there w...

We study singular real analytic Levi-flat subsets invariant by holomorphic foliations in complex projective spaces. give sufficient conditions for a subset to be the pull-back of semianalytic hypersurface surface under rational map or algebraic curve meromorphic function. In particular, we an application case hypersurface. Our results improve previous ones due Lebl and Bretas -- Fernandez-Perez...

Given any real-analytic CR manifold M, we provide general conditions on M guaranteeing that the group of all its global real-analytic CR automorphisms AutCR (M) is a Lie group (in an appropriate topology). In particular, we obtain a Lie group structure for AutCR (M) when M is an arbitrary compact real-analytic hypersurface embedded in some Stein manifold.

The Ricci curvature at an isolated singularity of an immersed hypersurface exhibits a local behaviour echoing a global property of the Ricci curvature on a complete hypersurface in euclidean space (i.e., the Efimov theorem [12], that sup Ric ≥ 0). This local behaviour takes the form of a near universal bound on the decay of the Ricci curvature at a simple singularity (eg. a cone singularity) in...

being the number of coefficients in a generic homogenous polynomial of degree d in n+1 variables. One associates with it the universal hypersurface Γ ⊂ P × P = {(p, x) | p(x) = 0}, which is a nonsingular variety, and the projection pr : Γ → P . The critical locus D ⊂ P of pr is the so-called discriminant hypersurface: p ∈ D if and only if the hypersurface p = 0 in P is singular, and the induced...