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2020•arXiv (Cornell University)Open access

Positivity Conditions for Quintic Polynomials

Liqun Qi, Yisheng Song, Xinzhen Zhang

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Abstract

In this paper, we present a necessary and sufficient conditions for a quintic polynomial to be positive for all positive reals, and a necessary and sufficient conditions for a quintic polynomial to be nonnegative for all positive reals. Only polynomials of the coefficients of the quintic polynomial are used in these conditions. A practical method to determine such positivity and nonnegativity is presented. An application for determining a fifth order two-dimensional symmetric tensor to be copositive and strictly copositive is also presented.

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What this paper is about

In this paper, we present a necessary and sufficient conditions for a quintic polynomial to be positive for all positive reals, and a necessary and sufficient conditions for a quintic polynomial to be nonnegative for all positive reals. Only polynomials of the coefficients of the quintic polynomial are used in these conditions. A practical method to determine such positivity and nonnegativity is presented. An application for determining a fifth order two-dimensional symmetric tensor to be copositive and strictly copositive is also presented.

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Available abstract

In this paper, we present a necessary and sufficient conditions for a quintic polynomial to be positive for all positive reals, and a necessary and sufficient conditions for a quintic polynomial to be nonnegative for all positive reals. Only polynomials of the coefficients of the quintic polynomial are used in these conditions. A practical method to determine such positivity and nonnegativity is presented. An application for determining a fifth order two-dimensional symmetric tensor to be copositive and strictly copositive is also presented.

Key concepts: Quintic function, Mathematics, Polynomial, Order (exchange), Pure mathematics, Tensor (intrinsic definition), Combinatorics, Applied mathematics

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