Complete classification scheme for the distribution of trinomial zeros with respect to their moduli. (English) Zbl 1513.26019
Consider the trinomial \(T_{a,b}^{k,m}(\lambda)=\lambda^k-a\lambda^{k-m}-b\) with \(a,b\in\mathbb{R}\) and let \(n_d^-\), \(n_d^0\), \(n_d^+\) be the number of its zeros whose modulus is less than, equal to, or larger than \(d\), respectively. The following problems are solved:
(P1) Given \(a\), \(b\), \(m\), \(k\), \(d\), find formulas for \(n_d^-\), \(n_d^0\), \(n_d^+\).
(P2) Given \(k\), \(m\), \(d\), \(r\), find the couples \((a,b)\in\mathbb{R}^2\) such that \(n_d^0=r\).
(P3) Given \(k\), \(m\), \(d\), \(r\), find the couples \((a,b)\in\mathbb{R}^2\) such that \(n_d^-=r\) and \(n_d^+=k-r\).
The problems are simplified to \((k,m)\) being coprime and \(d=1\). A survey of the existing literature is given extended with new results. Solving the three problems results in extensive tables that list the conditions on the parameters that result in the different distributions of \(k=n_d^-+n_d^0+n_d^+\).
(P1) Given \(a\), \(b\), \(m\), \(k\), \(d\), find formulas for \(n_d^-\), \(n_d^0\), \(n_d^+\).
(P2) Given \(k\), \(m\), \(d\), \(r\), find the couples \((a,b)\in\mathbb{R}^2\) such that \(n_d^0=r\).
(P3) Given \(k\), \(m\), \(d\), \(r\), find the couples \((a,b)\in\mathbb{R}^2\) such that \(n_d^-=r\) and \(n_d^+=k-r\).
The problems are simplified to \((k,m)\) being coprime and \(d=1\). A survey of the existing literature is given extended with new results. Solving the three problems results in extensive tables that list the conditions on the parameters that result in the different distributions of \(k=n_d^-+n_d^0+n_d^+\).
Reviewer: Adhemar Bultheel (Leuven)
MSC:
26C10 | Real polynomials: location of zeros |
12D10 | Polynomials in real and complex fields: location of zeros (algebraic theorems) |
30C15 | Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) |
39A30 | Stability theory for difference equations |