Principal symmetric ideals in the coordinate rings of curves

Mathematics > Commutative Algebra [Submitted on 16 Jun 2026] Title:Principal symmetric ideals in the coordinate rings of curves View PDF HTML (experimental)Abstract:The study of principal symmetric ideals (PSIs) in ambient polynomial rings was complicated by the combinatorial instability of minimal generators for ideal powers. We resolve this instability in the two variable case by translating the problem into the arithmetic geometry of symmetric affine plane curves. By working topdown within the Dedekind domain of a symmetric coordinate ring, we establish a precise geometric dictionary for PSIs. We prove that the prime factorization of a PSI is strictly determined by the $S_2$-orbits of its symmetric intersection locus, and that ramification corresponds exactly to tangential intersections, which are detected globally by a novel Symmetric Discriminant ideal. Crucially, we demonstrate that the ideal class of any PSI is a $2$-torsion element in the Ideal Class Group. This establishes that the powers of a PSI exhibit strict periodicity, alternating between being principal and requiring exactly two generators. Finally, we localize this arithmetic obstruction to the axis of symmetry, culminating in a Parity Criterion that determines principality based on intersection multiplicities along the diagonal. Current browse context: math.AC References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.