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We pose and discuss several Hermitian analogues of Hilberts $17$-th problem. We survey what is known, offer many explicit examples and some proofs, and give applications to CR geometry. We prove one new algebraic theorem: a non-negative Hermitian symmetric polynomial divides a nonzero squared norm if and only if it is a quotient of squared norms. We also discuss a new example of Putinar-Scheiderer.
In this paper we reduce the generalized Hilberts third problem about Dehn invariants and scissors congruence classes to the injectivity of certain Chern--Simons invariants. We also establish a version of a conjecture of Goncharov relating scissors co
We develop the theory of resolvent degree, introduced by Brauer cite{Br} in order to study the complexity of formulas for roots of polynomials and to give a precise formulation of Hilberts 13th Problem. We extend the context of this theory to enumera
We formulate a property $P$ on a class of relations on the natural numbers, and formulate a general theorem on $P$, from which we get as corollaries the insolvability of Hilberts tenth problem, Godels incompleteness theorem, and Turings halting probl
The synchronization process of two mutually delayed coupled deterministic chaotic maps is demonstrated both analytically and numerically. The synchronization is preserved when the mutually transmitted signal is concealed by two commutative private fi
We study the signature pair for certain group-invariant Hermitian polynomials arising in CR geometry. In particular, we determine the signature pair for the finite subgroups of $SU(2)$. We introduce the asymptotic positivity ratio and compute it for