Proposing a New Theorem to Determine If an Algebraic Polynomial Is Nonnegative in an Interval

Ke-Pao Lin, Yi-Fan Wang, Ruo-Yu Wang and Andrew Yang
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Ke-Pao Lin: Department of Liberal Arts, Chang Gung University of Science and Technology, Chang Gung Memorial Hospital, Tao-Yuan 33303, Taiwan
Yi-Fan Wang: Institute of Information and Decision Sciences, National Taipei University of Business, Taipei City 100, Taiwan
Ruo-Yu Wang: School of Mathematics Sciences, Beihang University, Beijing 102206, China
Andrew Yang: Department of Mathematics, Swansea University, Swansea SA2 8PP, UK

Mathematics, 2021, vol. 9, issue 2, 1-12

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 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.

Keywords: algebraic polynomial; trigonometric polynomials; nonnegative; Wang-Yau Lemma; Sturm’s Theorem; Yau Number Theoretic Conjecture; Yau Geometric Conjecture; integral points (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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