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IMAGINARY PROJECTIONS OF POLYNOMIALS

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 n >0 = ∅, the notion offers a novel geometric view underlying stability questions of polynomials. We show that the connected components of the closure of the complement 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.
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https://hal.inria.fr/hal-01829918
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Submitted on : Wednesday, July 4, 2018 - 1:59:51 PM
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Thorsten Jörgens, Thorsten Theobald, Timo de Wolff. IMAGINARY PROJECTIONS OF POLYNOMIALS. MEGA 2017 - International Conference on Effective Methods in Algebraic Geometry, Jun 2017, Nice, France. ⟨hal-01829918⟩

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