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.


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)
Full Text: DOI Euclid


[1] 10.2140/pjm.2012.259.141 · Zbl 1268.12001 · doi:10.2140/pjm.2012.259.141
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.