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In mathematics, a thin set in the sense of Serre, named after Jean-Pierre Serre, is a certain kind of subset constructed in algebraic geometry over a given field ''K'', by allowed operations that are in a definite sense 'unlikely'. The two fundamental ones are: solving a polynomial equation that may or may not be the case; solving within ''K'' a polynomial that does not always factorise. One is also allowed to take finite unions. ==Formulation== More precisely, let ''V'' be an algebraic variety over ''K'' (assumptions here are: ''V'' is an irreducible set, a quasi-projective variety, and ''K'' has characteristic zero). A type I thin set is a subset of ''V''(''K'') that is not Zariski-dense. That means it lies in an algebraic set that is a finite union of algebraic varieties of dimension lower than ''d'', the dimension of ''V''. A type II thin set is an image of an algebraic morphism (essentially a polynomial mapping) φ, applied to the ''K''-points of some other ''d''-dimensional algebraic variety ''V''′, that maps essentially onto ''V'' as a ramified covering with degree ''e'' > 1. Saying this more technically, a thin set of type II is any subset of :φ(''V''′(''K'')) where ''V''′ satisfies the same assumptions as ''V'' and φ is generically surjective from the geometer's point of view. At the level of function fields we therefore have :(''K''(''V''′) ) = ''e'' > 1. While a typical point ''v'' of ''V'' is φ(''u'') with ''u'' in ''V''′, from ''v'' lying in ''K''(''V'') we can conclude typically only that the coordinates of ''u'' come from solving a degree ''e'' equation over ''K''. The whole object of the theory of thin sets is then to understand that the solubility in question is a rare event. This reformulates in more geometric terms the classical Hilbert irreducibility theorem. A thin set, in general, is a subset of a finite union of thin sets of types I and II . The terminology ''thin'' may be justified by the fact that if ''A'' is a thin subset of the line over Q then the number of points of ''A'' of height at most ''H'' is ≪ ''H'': the number of integral points of height at most ''H'' is 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Thin set (Serre)」の詳細全文を読む スポンサード リンク
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