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Keywords = Wang-Yau Lemma

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12 pages, 262 KiB  
Article
Proposing a New Theorem to Determine If an Algebraic Polynomial Is Nonnegative in an Interval
by Ke-Pao Lin, Yi-Fan Wang, Ruo-Yu Wang and Andrew Yang
Mathematics 2021, 9(2), 167; https://doi.org/10.3390/math9020167 - 15 Jan 2021
Viewed by 1762
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Section A: Algebra and Logic)
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