Imaginary projections of polynomials. (March 2019)

• Record Type: Journal Article
• Authors: Jörgens, Thorsten; Theobald, Thorsten; de Wolff, Timo
• Is Part Of: Journal of symbolic computation. Volume 91(2019)
• Page Start: 181
• Page End: 199
• Publication Date: 2019-03
• Subjects: 14P10 -- 12D10; Imaginary projection -- Stable polynomial -- Convex algebraic geometry -- Amoeba -- Component of the complement; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2018.06.020
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: We introduce the imaginary projection of a multivariate polynomial f ∈ C [ z ] as the projection of the variety of f onto its imaginary part, I ( f ) = { Im ( z ) : z ∈ V ( f ) } . Since a polynomial f is stable if and only if I ( f ) ∩ R > 0 n = ∅, the notion offers a novel geometric view underlying stability questions of polynomials. We show that the connected components of the complement of the closure of the imaginary projections are convex, thus opening a central connection to the theory of amoebas and coamoebas. Building upon this, the paper establishes structural properties of the components of the complement, such as lower bounds on their maximal number, proves a complete classification of the imaginary projections of quadratic polynomials and characterizes the limit directions for polynomials of arbitrary degree.