参系数半代数系统的实解完全分类算法研究

半代数系统的求解是计算实代数几何的基本问题，而参系数半代数系统的实解完全分类是半代数系统求解中最重要也最具挑战性的问题。实解完全分类方法不仅能够更精确地刻画半代数系统解集的性质, 还在实际问题中具有广泛的应用。目前参系数半代数系统的求解方法都无法提供实解重数信息，因而在应用中受到局限。本项目旨在研发参系数半代数系统实解完全分类的高效算法。项目拟围绕参数多项式系统保零点重数的完全三角分解、三角形式半代数系统的实解完全分类以及参系数多项式实根完全分类的高效算法等三个研究内容，发展保零点重数的多项式系统三角分解理论, 给出蕴含参系数三角形式半代数系统实解重数信息的边界/判别多项式的构造方法, 建立一个融合判别式序列和广义子结式的完全判别系统, 在此基础上设计包含解的重数信息的参系数半代数系统实解完全分类算法。本项目的研发成果预计将丰富半代数系统的求解理论和方法，拓广半代数系统求解方法的应用领域。

Solving semi-algebraic systems is a fundamental problem in computational algebraic geometry, where complete real solution classification of parametric semi-algebraic systems is the most important and challenging subproblem. Methods for complete real solution classification can help to depict the properties of solution sets more accurately and thus have a broad range of applications. However, all the methods developed so far cannot provide the multiplicity information of real solutions for parametric semi-algebraic systems, which limits its application and popularization. This project aims to develop an efficient algorithm for complete real solution classification of parametric semi-algebraic systems. We focus on the following three aspects, i.e., comprehensive triangular decomposition of parametric polynomial systems with multiplicities of zeros preserved, complete real solution classification of triangular semi-algebraic systems, and efficient algorithms of real root classification for a parametric polynomial. We will develop the theory of multiplicity-preserved triangular decomposition and provide a method for constructing the border polynomial and discrimination polynomial with multiplicity information embedded. Furthermore, we will establish a complete discrimination system integrating the discriminant sequence and generalized subresultant polynomials. Combining the above ingredients will yield a complete real solution classification algorithm for parametric semi-algebraic systems. The project's achievements are expected to enrich the theory and methodology for solving semi-algebraic systems and expand the application field of methods for solving semi-algebraic systems.