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Necessary and sufficient conditions for a polytope of real polynomials to contain a Hurwitz polynomial

L.R. Pujara

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Abstract

Necessary and sufficient conditions for a polytope of real polynomials to contain a Hurwitz polynomial are established. More specifically, it is proved that a polytope of real polynomials contains a stable polynomial if and only if a certain semipolytope of the polytope contains a stable polynomial. The existence theorem is followed by an algorithm to determine a stable polynomial in the polytope, if one exists. The polytope problem for discrete polynomials is also solved. Two illustrative examples are included.

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What this paper is about

Necessary and sufficient conditions for a polytope of real polynomials to contain a Hurwitz polynomial are established. More specifically, it is proved that a polytope of real polynomials contains a stable polynomial if and only if a certain semipolytope of the polytope contains a stable polynomial. The existence theorem is followed by an algorithm to determine a stable polynomial in the polytope, if one exists. The polytope problem for discrete polynomials is also solved. Two illustrative examples are included.

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Available abstract

Necessary and sufficient conditions for a polytope of real polynomials to contain a Hurwitz polynomial are established. More specifically, it is proved that a polytope of real polynomials contains a stable polynomial if and only if a certain semipolytope of the polytope contains a stable polynomial. The existence theorem is followed by an algorithm to determine a stable polynomial in the polytope, if one exists. The polytope problem for discrete polynomials is also solved. Two illustrative examples are included.

Key concepts: Polytope, Birkhoff polytope, Hurwitz polynomial, Mathematics, Combinatorics, Polynomial, Discrete mathematics, Mathematical analysis

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