A descent-set obstruction for the Casas-Alvero conjecture

[2608.14726] A descent-set obstruction for the Casas-Alvero conjecture Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Number Theory arXiv:2608.14726 (math) [Submitted on 12 Aug 2026] Title: A descent-set obstruction for the Casas-Alvero conjecture Authors: Mohammad F. Marashdeh View a PDF of the paper titled A descent-set obstruction for the Casas-Alvero conjecture, by Mohammad F. Marashdeh View PDF HTML (experimental) Abstract: The Casas--Alvero conjecture asserts that a monic polynomial $f$ of degree $d$ over a field of characteristic zero sharing a non-constant factor with each of $f',\dots,f^{(d-1)}$ is the $d$-th power of a linear polynomial. Two reductions are carried out. Once a root is translated to the origin, the condition at index $i$ is vacuous unless the coefficient $a_{d-i}$ is nonzero, so the Casas--Alvero locus stratifies by the support of the centred normal form; prescribing in addition which root realises each surviving condition makes the system unit lower triangular and eliminates the coefficients of $f$ entirely. When one root suffices, what remains is a single integer, and that integer is MacMahon's determinant: the number of permutations of $\{1,\dots,d\}$ with descent set the support of $f$. Since every subset is a descent set, no counterexample has fewer than three recycled roots; in characteristic $p$ a prime dividing one of these counts but none of the accompanying integers $z_{i}$ is a bad prime, the largest count in degree $d$ being the Euler zigzag number $A_{d}$, so that the irregular prime $691$ is bad for $d=11$. The stratification also yields short proofs of the two characteristic-$p$ propositions on which all known cases rest; supports of two elements are settled separately in every characteristic, whence a counterexample has at least four terms in centred normal form and a further Gröbner-free criterion for bad primes. Subjects: Number Theory (math.NT) Cite as: arXiv:2608.14726 [math.NT] (or arXiv:2608.14726v1 [math.NT] for this version) https://doi.org/10.48550/arXiv.2608.14726 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Mohammad Marashdeh [ view email ] [v1] Wed, 12 Aug 2026 17:33:19 UTC (27 KB) Full-text links: Access Paper: View a PDF of the paper titled A descent-set obstruction for the Casas-Alvero conjecture, by Mohammad F. Marashdeh View PDF HTML (experimental) TeX Source view license Current browse context: math.NT < prev | next > new | recent | 2026-08 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs 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 . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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