{"title":"纳什条件独立曲线","authors":"Irem Portakal , Javier Sendra–Arranz","doi":"10.1016/j.jsc.2023.102255","DOIUrl":null,"url":null,"abstract":"<div><p><span>We study the Spohn conditional independence (CI) variety </span><span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span> of an <em>n</em>-player game <em>X</em><span> for undirected graphical models on </span><em>n</em><span> binary random variables consisting of one edge. For a generic game, we show that </span><span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span><span> is a smooth irreducible complete intersection curve (Nash conditional independence curve) in the Segre variety </span><span><math><msup><mrow><mo>(</mo><msup><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>)</mo></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msup><mo>×</mo><msup><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> and we give an explicit formula for its degree and genus. We prove two universality theorems for <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span><span>: The product of any affine real algebraic variety with the real line or any affine real algebraic variety in </span><span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span> defined by at most <span><math><mi>m</mi><mo>−</mo><mn>1</mn></math></span> polynomials is isomorphic to an affine open subset of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span> for some game <em>X</em>.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"122 ","pages":"Article 102255"},"PeriodicalIF":0.6000,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Nash conditional independence curve\",\"authors\":\"Irem Portakal , Javier Sendra–Arranz\",\"doi\":\"10.1016/j.jsc.2023.102255\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p><span>We study the Spohn conditional independence (CI) variety </span><span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span> of an <em>n</em>-player game <em>X</em><span> for undirected graphical models on </span><em>n</em><span> binary random variables consisting of one edge. For a generic game, we show that </span><span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span><span> is a smooth irreducible complete intersection curve (Nash conditional independence curve) in the Segre variety </span><span><math><msup><mrow><mo>(</mo><msup><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>)</mo></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msup><mo>×</mo><msup><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> and we give an explicit formula for its degree and genus. We prove two universality theorems for <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span><span>: The product of any affine real algebraic variety with the real line or any affine real algebraic variety in </span><span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span> defined by at most <span><math><mi>m</mi><mo>−</mo><mn>1</mn></math></span> polynomials is isomorphic to an affine open subset of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span> for some game <em>X</em>.</p></div>\",\"PeriodicalId\":50031,\"journal\":{\"name\":\"Journal of Symbolic Computation\",\"volume\":\"122 \",\"pages\":\"Article 102255\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2023-08-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Symbolic Computation\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S074771712300069X\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"COMPUTER SCIENCE, THEORY & METHODS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Symbolic Computation","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S074771712300069X","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
We study the Spohn conditional independence (CI) variety of an n-player game X for undirected graphical models on n binary random variables consisting of one edge. For a generic game, we show that is a smooth irreducible complete intersection curve (Nash conditional independence curve) in the Segre variety and we give an explicit formula for its degree and genus. We prove two universality theorems for : The product of any affine real algebraic variety with the real line or any affine real algebraic variety in defined by at most polynomials is isomorphic to an affine open subset of for some game X.
期刊介绍:
An international journal, the Journal of Symbolic Computation, founded by Bruno Buchberger in 1985, is directed to mathematicians and computer scientists who have a particular interest in symbolic computation. The journal provides a forum for research in the algorithmic treatment of all types of symbolic objects: objects in formal languages (terms, formulas, programs); algebraic objects (elements in basic number domains, polynomials, residue classes, etc.); and geometrical objects.
It is the explicit goal of the journal to promote the integration of symbolic computation by establishing one common avenue of communication for researchers working in the different subareas. It is also important that the algorithmic achievements of these areas should be made available to the human problem-solver in integrated software systems for symbolic computation. To help this integration, the journal publishes invited tutorial surveys as well as Applications Letters and System Descriptions.