  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.

Number Theory;Commutative Algebra;Algebraic Geometry