We consider numerical certification of approximate solutions to a system of polynomial equations with more equations than unknowns by first certifying solutions to a square subsystem. We give several approaches that certifiably select which are solutions to the original overdetermined system. These approaches each use different additional information for this certification, such as liaison, Newton-Okounkov bodies, or intersection theory. They may be used to certify individual solutions, reject nonsolutions, or certify that we have found all solutions.

中文翻译：

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通过平方子系统对多项式系统进行认证

我们通过首先证明平方子系统的解来考虑对方程组多于未知数的多项式方程组的近似解的数值证明。我们给出了几种方法，可以确定地选择哪些是原始超定系统的解决方案。这些方法各自使用不同的附加信息来进行此认证，例如联络、Newton-Okounkov 实体或相交理论。它们可用于证明单个解决方案、拒绝非解决方案或证明我们已找到所有解决方案。