多项式方程组实解集完整性的数值验证
Numerically validating the completeness of the real solution set of a system of polynomial equations
AI总结：
本文结合数值代数几何与求和平方编程，提出验证给定集合是否为实解集的完整集合的方法，通过 Zariski 封闭性与实根理想的关系进行测试，包含有限、无限解及多项式不等式实例。
摘要：
计算多项式方程组的实解是一个具有挑战性的问题，特别是验证是否已找到所有解。本文描述了一种结合数值代数几何和求和平方编程的方法，用于测试给定集合是否在实解集上是完整的。具体而言，我们测试给定集合的 Zariski 封闭性是否确实等于由给定多项式生成的理想的真实根理想。提供了有限和无限实解的例子，以及包含多项式不等式的例子。
英文摘要：
Computing the real solutions to a system of polynomial equations is a challenging problem, particularly verifying that all solutions have been computed. We describe an approach that combines numerical algebraic geometry and sums of squares programming to test whether a given set is "complete" with respect to the real solution set. Specifically, we test whether the Zariski closure of that given set is indeed equal to the solution set of the real radical of the ideal generated by the given polynomials. Examples with finitely and infinitely many real solutions are provided, along with an example having polynomial inequalities.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
