In recent years, much work has been devoted to a systematic study of polynomial identities certifying strict or non-strict positivity of a polynomial f on a basic closed set K ⊂ R. The interest in such identities originates not least from their importance in polynomial optimization. The majority of the important results requires the archimedean condition, which implies that K has to be compact....

Cartan’s method is used to prove a several variable, non-Archimedean, Nevanlinna Second Main Theorem for hyperplanes in projective space. The corresponding defect relation is derived, but unlike in the complex case, we show that there can only be finitely many non-zero non-Archimedean defects. We then address the non-Archimedean Nevanlinna inverse problem, by showing that given a set of defects...

We prove that if X,X are closed subschemes of a torus T over a non-Archimedean field K, of complementary codimension and with finite intersection, then the stable tropical intersection along a (possibly positive-dimensional, possibly unbounded) connected component C of Trop(X) ∩ Trop(X) lifts to algebraic intersection points, with multiplicities. This theorem requires potentially passing to a s...

1.1. Motivation. This paper is largely concerned with constructing quotients by étale equivalence relations. We are inspired by questions in classical rigid geometry, but to give satisfactory answers in that category we have to first solve quotient problems within the framework of Berkovich’s k-analytic spaces. One source of motivation is the relationship between algebraic spaces and analytic s...

In recent years, much work has been devoted to a systematic study of polynomial identities certifying strict or non-strict positivity of a polynomial f on a basic closed set K ⊂ R. The interest in such identities originates not least from their importance in polynomial optimization. The majority of the important results requires the archimedean condition, which implies that K has to be compact....

We develop the theory of pinchings for non-archimedean analytic spaces. In particular, we show that although affinoid spaces do not have to be affinoid, Hausdorff always exist in category

In recent years, much work has been devoted to a systematic study of polynomial identities certifying strict or non-strict positivity of a polynomial f on a basic closed set K ⊂ R. The interest in such identities originates not least from their importance in polynomial optimization. The majority of the important results requires the archimedean condition, which implies that K has to be compact....

There is a natural analytification functor from the category of locally separated algebraic spaces locally of finite type over C to the category of complex-analytic spaces [Kn, Ch. I, 5.17ff]. (Recall that a map of algebraic spaces X → S is locally separated if the diagonal ∆X/S : X → X ×S X is an immersion. We require algebraic spaces to have quasi-compact diagonal over SpecZ.) It is natural t...

In this paper, we solve the additive ρ-functional inequalities ‖f(x+ y)− f(x)− f(y)‖ ≤ ∥∥∥∥ρ(2f (x+ y 2 ) − f(x)− f(y) )∥∥∥∥ , (1) ∥∥∥∥2f (x+ y 2 ) − f(x)− f(y) ∥∥∥∥ ≤ ‖ρ (f(x+ y)− f(x)− f(y))‖ , (2) where ρ is a fixed non-Archimedean number with |ρ| < 1 or ρ is a fixed complex number with |ρ| < 1. Using the direct method, we prove the Hyers-Ulam stability of the additive ρ-functional inequalit...