{"title":"多项式映射的热带非完备性集合","authors":"Boulos El Hilany","doi":"10.1007/s00454-024-00684-4","DOIUrl":null,"url":null,"abstract":"<p>We study some discrete invariants of Newton non-degenerate polynomial maps <span>\\(f: {\\mathbb {K}}^n \\rightarrow {\\mathbb {K}}^n\\)</span> defined over an algebraically closed field of Puiseux series <span>\\({\\mathbb {K}}\\)</span>, equipped with a non-trivial valuation. It is known that the set <span>\\({\\mathcal {S}}(f)\\)</span> of points at which <i>f</i> is not finite forms an algebraic hypersurface in <span>\\({\\mathbb {K}}^n\\)</span>. The coordinate-wise valuation of <span>\\({\\mathcal {S}}(f)\\cap ({\\mathbb {K}}^*)^n\\)</span> is a piecewise-linear object in <span>\\({\\mathbb {R}}^n\\)</span>, which we call the tropical non-properness set of <i>f</i>. We show that the tropical polynomial map corresponding to <i>f</i> has fibers satisfying a particular combinatorial degeneracy condition exactly over points in the tropical non-properness set of <i>f</i>. We then use this description to outline a polyhedral method for computing this set, and to recover the fan dual to the Newton polytope of the set at which a complex polynomial map is not finite. The proofs rely on classical correspondence and structural results from tropical geometry, combined with a new description of <span>\\({\\mathcal {S}}(f)\\)</span> in terms of multivariate resultants.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The Tropical Non-Properness Set of a Polynomial Map\",\"authors\":\"Boulos El Hilany\",\"doi\":\"10.1007/s00454-024-00684-4\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We study some discrete invariants of Newton non-degenerate polynomial maps <span>\\\\(f: {\\\\mathbb {K}}^n \\\\rightarrow {\\\\mathbb {K}}^n\\\\)</span> defined over an algebraically closed field of Puiseux series <span>\\\\({\\\\mathbb {K}}\\\\)</span>, equipped with a non-trivial valuation. It is known that the set <span>\\\\({\\\\mathcal {S}}(f)\\\\)</span> of points at which <i>f</i> is not finite forms an algebraic hypersurface in <span>\\\\({\\\\mathbb {K}}^n\\\\)</span>. The coordinate-wise valuation of <span>\\\\({\\\\mathcal {S}}(f)\\\\cap ({\\\\mathbb {K}}^*)^n\\\\)</span> is a piecewise-linear object in <span>\\\\({\\\\mathbb {R}}^n\\\\)</span>, which we call the tropical non-properness set of <i>f</i>. We show that the tropical polynomial map corresponding to <i>f</i> has fibers satisfying a particular combinatorial degeneracy condition exactly over points in the tropical non-properness set of <i>f</i>. We then use this description to outline a polyhedral method for computing this set, and to recover the fan dual to the Newton polytope of the set at which a complex polynomial map is not finite. The proofs rely on classical correspondence and structural results from tropical geometry, combined with a new description of <span>\\\\({\\\\mathcal {S}}(f)\\\\)</span> in terms of multivariate resultants.</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-08-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00454-024-00684-4\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00454-024-00684-4","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
The Tropical Non-Properness Set of a Polynomial Map
We study some discrete invariants of Newton non-degenerate polynomial maps \(f: {\mathbb {K}}^n \rightarrow {\mathbb {K}}^n\) defined over an algebraically closed field of Puiseux series \({\mathbb {K}}\), equipped with a non-trivial valuation. It is known that the set \({\mathcal {S}}(f)\) of points at which f is not finite forms an algebraic hypersurface in \({\mathbb {K}}^n\). The coordinate-wise valuation of \({\mathcal {S}}(f)\cap ({\mathbb {K}}^*)^n\) is a piecewise-linear object in \({\mathbb {R}}^n\), which we call the tropical non-properness set of f. We show that the tropical polynomial map corresponding to f has fibers satisfying a particular combinatorial degeneracy condition exactly over points in the tropical non-properness set of f. We then use this description to outline a polyhedral method for computing this set, and to recover the fan dual to the Newton polytope of the set at which a complex polynomial map is not finite. The proofs rely on classical correspondence and structural results from tropical geometry, combined with a new description of \({\mathcal {S}}(f)\) in terms of multivariate resultants.