On some algebraic and geometric extensions of Goldbach's conjecture

Abstract

The goal of this paper is to study Goldbach's conjecture for rings of regular functions of affine algebraic varieties over a field. Among our main results, we define the notion of Goldbach condition for Newton polytopes, and we prove in a constructive way that any polynomial in at least two variables over a field can be expressed as sum of at most $2r$ absolutely irreducible polynomials, where $r$ is the number of its non--zero monomials. We also study other weak forms of Goldbach's conjecture for localizations of these rings. Moreover, we prove the validity of Goldbach's conjecture for a particular instance of the so--called forcing algebras introduced by Hochster. Finally, we prove that, for a proper multiplicative closed set $S$ of $\mathbb{Z}$, the collection of elements of $S^{-1}\mathbb{Z}$ that can be written as finite sum of primes forms a dense subset of the real numbers, among other results.