Proposing a New Theorem to Determine If an Algebraic Polynomial Is Nonnegative in an Interval
| Autor: | Andrew Yang, Ke-Pao Lin, Ruo-Yu Wang, Yi-Fan Wang |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
General Mathematics
Trigonometric polynomial 01 natural sciences Mathematics::Algebraic Geometry 0103 physical sciences Yau Number Theoretic Conjecture Computer Science (miscellaneous) 0101 mathematics Mathematics::Symplectic Geometry Engineering (miscellaneous) Sturm's theorem Mathematics Discrete mathematics Lemma (mathematics) Conjecture Degree (graph theory) 010308 nuclear & particles physics lcsh:Mathematics 010102 general mathematics integral points trigonometric polynomials lcsh:QA1-939 Yau Geometric Conjecture Face (geometry) Sturm’s Theorem Interval (graph theory) nonnegative Trigonometry Wang-Yau Lemma algebraic polynomial |
| Zdroj: | Mathematics, Vol 9, Iss 167, p 167 (2021) Mathematics Volume 9 Issue 2 |
| ISSN: | 2227-7390 |
| Popis: | 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&rsquo s Theorem. Many polynomials can use the new theorem but cannot use Sturm&rsquo s Theorem and Wang-Yau Lemma to judge whether they are nonnegative in an interval. New Theorem also performs better than Sturm&rsquo 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. |
| Databáze: | OpenAIRE |
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