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摘要翻译：
我们提供了一个实代数符号-数值算法来计算理想$I$的实变数$V_R(I)$,假设它是有限的，而$V_C(I)$可能不是。我们的方法使用$R[X]$上的线性泛函集，它消失在生成$I$的给定多项式集上，并且消失在由适当定义的半正定最大秩矩矩阵的核得到的$I$的实根理想的多项式上。我们对这些线性泛函集的投影维数提出了一个条件，作为我们算法的停止准则。该算法基于标准的数值线性代数程序和半定优化，结合了作者以前工作中的技术和现有的复杂变量算法。这就为真实和复杂的案例提供了一个统一的方法。
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英文标题：
《A prolongation-projection algorithm for computing the finite real
variety of an ideal》
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作者：
J.B. Lasserre (LAAS-CNRS, Toulouse), M. Laurent (CWI, Amsterdam), P.
Rostalski (ETH, Zurich)
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最新提交年份：
2009
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分类信息：
一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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一级分类：Mathematics 数学
二级分类：Optimization and Control 优化与控制
分类描述：Operations research, linear programming, control theory, systems theory, optimal control, game theory
运筹学，线性规划，控制论，系统论，最优控制，博弈论
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英文摘要：
We provide a real algebraic symbolic-numeric algorithm for computing the real variety $V_R(I)$ of an ideal $I$, assuming it is finite while $V_C(I)$ may not be. Our approach uses sets of linear functionals on $R[X]$, vanishing on a given set of polynomials generating $I$ and their prolongations up to a given degree, as well as on polynomials of the real radical ideal of $I$, obtained from the kernel of a suitably defined moment matrix assumed to be positive semidefinite and of maximum rank. We formulate a condition on the dimensions of projections of these sets of linear functionals, which serves as stopping criterion for our algorithm. This algorithm, based on standard numerical linear algebra routines and semidefinite optimization, combines techniques from previous work of the authors together with an existing algorithm for the complex variety. This results in a unified methodology for the real and complex cases.
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PDF链接：
https://arxiv.org/pdf/0806.3874
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