Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2504.10722

On GL-domains and the ascent of the IDF property

Victor Gonzalez·Ishan Panpaliya
View original source ↗Read source document ↗Export metadata ↓

Abstract

Following the terminology introduced by Arnold and Sheldon back in 1975, we say that an integral domain $D$ is a GL-domain if the product of any two primitive polynomials over $D$ is again a primitive polynomial. In this paper, we study the class of GL-domains. First, we propose a characterization of GL-domain in terms of certain elements we call prime-like. Then we identify a new class of GL-domains. An integral domain $D$ is also said to have the IDF property provided that each nonzero element of $D$ is divisible by only finitely many non-associate irreducible divisors. It was proved by Malcolmson and Okoh in 2009 that the IDF property ascends to polynomial extensions when restricted to the class of GCD-domains. This result was recently strengthened by Gotti and Zafrullah to the class of PSP-domains. We conclude this paper by proving that the IDF property does not ascend to polynomial extensions in the class of GL-domains, answering an open question posed by Gotti and Zafrullah.

Explore related subjects

math.AC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Victor Gonzalez, Ishan Panpaliya. 2025-04-19. On GL-domains and the ascent of the IDF property. https://arxiv.org/abs/2504.10722

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Unified Common-Root and Interpolation Bounds Based on Leading Monomial Data

For general fields the footprint bound from Gröbner basis theory estimates the number of common affine roots of any set of multivariate polynomials using information on their leading monomials. In this paper we develop an interpolation bound with a similar flavor extending a previously known result for only a single polynomial to any prescribed number of polynomials. Surprisingly, our interpolation theorem and the footprint bound can be shown to be two sides of the same coin, solving similar problems, but for dual spaces. As discussed the footprint bound compares well with the improved Alon-Füredi bound and for finite fields the presented interpolation theorem is sharp. Our work can be viewed as a comment to a question raised by Tao in [Tao, 2014]

math.AC↗
arXiv2026

Poincaré Duality and Quadratic Refinements over Laurent Rings

We develop a Poincaré duality theory for defects of nondegenerate sesquilinear pairings over Laurent polynomial rings. A key ingredient is a novel flat resolution of the character module, constructed from a triangulation of the sphere at infinity associated with a fan. The cup product on this resolution turns Poincaré duality on the sphere into canonical pairings between the resulting defect modules. In middle degrees, we construct distinguished quadratic refinements using equivariant cohomology of the sphere with the antipodal action. The effective replacement of the sphere with a projective space provides a geometric substitute for division by two. Applied to translation-invariant Pauli stabilizer codes, our results establish the nondegeneracy of higher-dimensional braiding pairings. They extend the two-dimensional T-junction formula for topological spin to higher dimensions, while giving it a geometric interpretation.

math.AC↗