We face the problem to determine whether an algebraic polynomial is nonnegative in an interval the Yau Number Theoretic Conjecture and Yau Geometric Conjecture is proved. In this paper, we propose a new theorem to determine if an algebraic polynomial is nonnegative in an interval. It improves Wang-Yau Lemma for wider applications in light of Sturm’s Theorem. Many polynomials can use the new theorem but cannot use Sturm’s Theorem and Wang-Yau Lemma to judge whether they are nonnegative in an interval. New Theorem also performs better than Sturm’s Theorem when the number of terms and degree of polynomials increase. Main Theorem can be used for polynomials whose coefficients are parameters and to any interval we use. It helps us to find the roots of complicated polynomials. The problem of constructing nonnegative trigonometric polynomials in an interval is a classical, important problem and crucial to many research areas. We can convert a given trigonometric polynomial to an algebraic polynomial. Hence, our proposed new theorem affords a new way to solve this classical, important problem.

中文翻译：

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提出一个新的定理，确定区间内的代数多项式是否为负

我们面临的问题是确定在证明了Yau数理论猜想和Yau几何猜想的区间内代数多项式是否为非负数。在本文中，我们提出了一个新的定理，以确定代数多项式在区间内是否为非负。根据Sturm定理，它改进了Wang-Yau Lemma的应用范围。许多多项式可以使用新定理，但不能使用Sturm定理和Wang-Yau Lemma来判断它们在一个区间内是否为负。当多项式的项数和次数增加时，新定理的性能也比Sturm定理好。主定理可以用于系数为参数的多项式，并且可以使用任何区间。它帮助我们找到复杂多项式的根。在区间中构造非负三角多项式的问题是一个经典的重要问题，对许多研究领域都至关重要。我们可以将给定的三角多项式转换为代数多项式。因此，我们提出的新定理提供了解决这一经典重要问题的新方法。