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Norms of roots of trinomials. (English) Zbl 1379.12002

Summary: The behavior of norms of roots of univariate trinomials \(z^{s+t}+pz^t+q\in\mathbb C[z]\) for fixed support \(A=\{0,t,s+t\}\subset\mathbb N\) with respect to the choice of coefficients \(p,q\in\mathbb C\) is a classical late 19th and early 20th century problem. Although algebraically characterized by P. Bohl [Math. Ann. 65, 556–566 (1908; JFM 39.0134.01)], the geometry and topology of the corresponding parameter space of coefficients had yet to be revealed. Assuming \(s\) and \(t\) to be coprime we provide such a characterization for the space of trinomials by reinterpreting the problem in terms of amoeba theory. The roots of given norm are parameterized in terms of a hypotrochoid curve along a \(\mathbb C\)-slice of the space of trinomials, with multiple roots of this norm appearing exactly on the singularities. As a main result, we show that the set of all trinomials with support \(A\) and certain roots of identical norm, as well as its complement can be deformation retracted to the torus knot \(K(s+t,s)\), and thus are connected but not simply connected. An exception is the case where the \(t\)-th smallest norm coincides with the \((t+1)\)-st smallest norm. Here, the complement has a different topology since it has fundamental group \(\mathbb Z^2\).

MSC:

12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems)
14H10 Families, moduli of curves (algebraic)
14H50 Plane and space curves
26C10 Real polynomials: location of zeros
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
55R10 Fiber bundles in algebraic topology
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
55P15 Classification of homotopy type

Citations:

JFM 39.0134.01
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Full Text: DOI arXiv

References:

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