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Locating trinomial zeros. (English) Zbl 1393.30009

Let \(p(z) = z^n + z^k - 1\), with \(n\geq 2\), \(1 \leq k \leq n-1\), and \(g=\gcd(n,k)\). The authors study the location of the zeros of \(p\).
Their main result states that the trinomial \(p\) has \(2g\lceil (n+k)/(6g) \rceil-g \) interior zeros (zeros of modulus \(<1\)), \(n-2g\lceil (n+k)/(6g) \rceil-g\) exterior zeros (zeros of modulus \(>1\)), and when \(6g\) divides \(n+k,\) it has \(2g \) unimodular zeros.
This result extends that of M. Brilleslyper and B. Schaubroeck [“Locating unimodular roots”, Coll. Math. J. 45, No. 3, 162–168 (2014; doi:10.4169/college.math.j.45.3.162)], who proved the statement concerning unimodular zeros.

MSC:

30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems)
30C10 Polynomials and rational functions of one complex variable
97I80 Complex analysis (educational aspects)
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References:

[1] 10.2140/pjm.2012.259.141 · Zbl 1268.12001 · doi:10.2140/pjm.2012.259.141
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