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In computer algebra, a triangular decomposition of a polynomial system is a set of simpler polynomial systems such that a point is a solution of if and only if it is a solution of one of the systems . When the purpose is to describe the solution set of in the algebraic closure of its coefficient field, those simpler systems are regular chains. If the coefficient of are real numbers, then the real solutions of can be obtained by a triangular decomposition into regular semi-algebraic systems. In both cases, each of these simpler systems has a triangular shape and remarkable properties, which justifies the terminology. == Formal definitions == Let be a field and be ordered variables. We denote by the corresponding polynomial ring. For , regarded as a system of polynomial equations, there are two notions of a triangular decomposition over the algebraic closure of . The first one is to decompose lazily, by representing only the generic points of the algebraic set in the so-called sense of Kalkbrener. : The second is to describe explicitly all the points of in the so-called sense of in Lazard and Wen-Tsun Wu. : In both cases are finitely many regular chains of and denotes the radical of the saturated ideal of while denotes the quasi-component of . Please refer to regular chain for definitions of these notions. Assume from now on that is a real closed field. Consider a semi-algebraic system with polynomials in . There exist〔Changbo Chen, James H. Davenport, John P. May, Marc Moreno-Maza, Bican Xia, Rong Xiao. Triangular decomposition of semi-algebraic systems. Proceedings of 2010 International Symposium on Symbolic and Algebraic Computation (ISSAC 2010), ACM Press, pp. 187--194, 2010.〕 finitely many regular semi-algebraic systems such that we have : where denotes the points of which solve . The regular semi-algebraic systems form a triangular decomposition of the semi-algebraic system . 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「triangular decomposition」の詳細全文を読む スポンサード リンク
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