On some algebraic and geometric extensions of Goldbach's conjecture

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 var...

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Bibliographic Details
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
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Summary: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.
ISSN:2331-8422