On some algebraic and geometric extensions of Goldbach's conjecture

• Published in: arXiv.org 2023-12
• Main Authors: Boix, Alberto F, Gómez-Ramírez, Danny A J
• Format: Article
• Language: English
• Subjects: Algebra; Fields (mathematics); Mathematical analysis; Polynomials; Polytopes; Real numbers; Rings (mathematics); Set theory; Sums
• ISSN: 2331-8422

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.