{"title":"除法多项式的常见估值","authors":"Bartosz Naskręcki, Matteo Verzobio","doi":"10.1017/prm.2024.7","DOIUrl":null,"url":null,"abstract":"<p>In this note, we prove a formula for the cancellation exponent <span><span><span data-mathjax-type=\"texmath\"><span>$k_{v,n}$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240223151238591-0447:S0308210524000076:S0308210524000076_inline1.png\"/></span></span> between division polynomials <span><span><span data-mathjax-type=\"texmath\"><span>$\\psi _n$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240223151238591-0447:S0308210524000076:S0308210524000076_inline2.png\"/></span></span> and <span><span><span data-mathjax-type=\"texmath\"><span>$\\phi _n$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240223151238591-0447:S0308210524000076:S0308210524000076_inline3.png\"/></span></span> associated with a sequence <span><span><span data-mathjax-type=\"texmath\"><span>$\\{nP\\}_{n\\in \\mathbb {N}}$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240223151238591-0447:S0308210524000076:S0308210524000076_inline4.png\"/></span></span> of points on an elliptic curve <span><span><span data-mathjax-type=\"texmath\"><span>$E$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240223151238591-0447:S0308210524000076:S0308210524000076_inline5.png\"/></span></span> defined over a discrete valuation field <span><span><span data-mathjax-type=\"texmath\"><span>$K$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240223151238591-0447:S0308210524000076:S0308210524000076_inline6.png\"/></span></span>. The formula greatly generalizes the previously known special cases and treats also the case of non-standard Kodaira types for non-perfect residue fields.</p>","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"14 1","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Common valuations of division polynomials\",\"authors\":\"Bartosz Naskręcki, Matteo Verzobio\",\"doi\":\"10.1017/prm.2024.7\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>In this note, we prove a formula for the cancellation exponent <span><span><span data-mathjax-type=\\\"texmath\\\"><span>$k_{v,n}$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240223151238591-0447:S0308210524000076:S0308210524000076_inline1.png\\\"/></span></span> between division polynomials <span><span><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\psi _n$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240223151238591-0447:S0308210524000076:S0308210524000076_inline2.png\\\"/></span></span> and <span><span><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\phi _n$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240223151238591-0447:S0308210524000076:S0308210524000076_inline3.png\\\"/></span></span> associated with a sequence <span><span><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\{nP\\\\}_{n\\\\in \\\\mathbb {N}}$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240223151238591-0447:S0308210524000076:S0308210524000076_inline4.png\\\"/></span></span> of points on an elliptic curve <span><span><span data-mathjax-type=\\\"texmath\\\"><span>$E$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240223151238591-0447:S0308210524000076:S0308210524000076_inline5.png\\\"/></span></span> defined over a discrete valuation field <span><span><span data-mathjax-type=\\\"texmath\\\"><span>$K$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240223151238591-0447:S0308210524000076:S0308210524000076_inline6.png\\\"/></span></span>. The formula greatly generalizes the previously known special cases and treats also the case of non-standard Kodaira types for non-perfect residue fields.</p>\",\"PeriodicalId\":54560,\"journal\":{\"name\":\"Proceedings of the Royal Society of Edinburgh Section A-Mathematics\",\"volume\":\"14 1\",\"pages\":\"\"},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2024-02-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the Royal Society of Edinburgh Section A-Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1017/prm.2024.7\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/prm.2024.7","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
In this note, we prove a formula for the cancellation exponent $k_{v,n}$ between division polynomials $\psi _n$ and $\phi _n$ associated with a sequence $\{nP\}_{n\in \mathbb {N}}$ of points on an elliptic curve $E$ defined over a discrete valuation field $K$. The formula greatly generalizes the previously known special cases and treats also the case of non-standard Kodaira types for non-perfect residue fields.
期刊介绍：
A flagship publication of The Royal Society of Edinburgh, Proceedings A is a prestigious, general mathematics journal publishing peer-reviewed papers of international standard across the whole spectrum of mathematics, but with the emphasis on applied analysis and differential equations.
An international journal, publishing six issues per year, Proceedings A has been publishing the highest-quality mathematical research since 1884. Recent issues have included a wealth of key contributors and considered research papers.
between division polynomials $\psi _n$
and $\phi _n$
associated with a sequence $\{nP\}_{n\in \mathbb {N}}$
of points on an elliptic curve $E$
defined over a discrete valuation field $K$
. The formula greatly generalizes the previously known special cases and treats also the case of non-standard Kodaira types for non-perfect residue fields.
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