A double Sylvester determinant

[1901.11109] A double Sylvester determinant Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Rings and Algebras arXiv:1901.11109 (math) [Submitted on 30 Jan 2019 ( v1 ), last revised 15 Apr 2026 (this version, v3)] Title: A double Sylvester determinant Authors: Darij Grinberg View a PDF of the paper titled A double Sylvester determinant, by Darij Grinberg View PDF HTML (experimental) Abstract: Given two $\left( n+1\right) \times\left( n+1\right)$-matrices $A$ and $B$ over a commutative ring, and some $k\in\left\{ 0,1,\ldots,n\right\}$, we consider the $\dbinom{n}{k}\times\dbinom{n}{k}$-matrix $W$ whose entries are $\left( k+1\right) \times\left( k+1\right)$-minors of $A$ multiplied by corresponding $\left( k+1\right) \times\left( k+1\right)$-minors of $B$. Here we require the minors to use the last row and the last column (which is why we obtain an $\dbinom{n}{k}\times\dbinom{n}{k}$-matrix, not an $\dbinom{n+1}{k+1}\times\dbinom{n+1}{k+1}$-matrix). We prove that the determinant $\det W$ is a multiple of $\det A$ if the $\left( n+1,n+1\right)$-th entry of $B$ is $0$. Furthermore, if the $\left( n+1,n+1\right)$-th entries of both $A$ and $B$ are $0$, then $\det W$ is a multiple of $\left( \det A\right) \left( \det B\right)$. This extends a previous result of Olver and the author ( arXiv:1802.02900 ). Comments: 16 pages. Slightly more detailed version available as ancillary file. Comments are welcome! v3 corrects some typos and updates references Subjects: Rings and Algebras (math.RA) ; Combinatorics (math.CO) MSC classes: 15A15, 11C20 Cite as: arXiv:1901.11109 [math.RA] (or arXiv:1901.11109v3 [math.RA] for this version) https://doi.org/10.48550/arXiv.1901.11109 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Ars Mathematica Contemporanea 20 No. 2 (2021), pp. 261--274 Related DOI : https://doi.org/10.26493/1855-3974.2248.d3f Focus to learn more DOI(s) linking to related resources Submission history From: Darij Grinberg [ view email ] [v1] Wed, 30 Jan 2019 21:36:47 UTC (15 KB) [v2] Tue, 25 Aug 2020 06:50:02 UTC (304 KB) [v3] Wed, 15 Apr 2026 08:13:54 UTC (314 KB) Full-text links: Access Paper: View a PDF of the paper titled A double Sylvester determinant, by Darij Grinberg View PDF HTML (experimental) TeX Source view license Ancillary-file links: Ancillary files ( details ) : bisyl-long.pdf Current browse context: math.RA < prev | next > new | recent | 2019-01 Change to browse by: math math.CO 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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