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Real root polynomials and real root preserving transformations. (English) Zbl 1486.26024

Summary: Polynomials can be used to represent real-world situations, and their roots have real-world meanings when they are real numbers. The fundamental theorem of algebra tells us that every nonconstant polynomial \(p\) with complex coefficients has a complex root. However, no analogous result holds for guaranteeing that a real root exists to \(p\) if we restrict the coefficients to be real. Let \(n\geq 1\) and \(P_n\) be the vector space of all polynomials of degree \(n\) or less with real coefficients. In this article, we give explicit forms of polynomials in \(P_n\) such that all of their roots are real. Furthermore, we present explicit forms of linear transformations on \(P_n\) which preserve real roots of polynomials in a certain subset of \(P_n\).

MSC:

26C10 Real polynomials: location of zeros
65H05 Numerical computation of solutions to single equations

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References:

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