2017/06/16 by Trevor Richards, Richards, Trevor
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #Mathematical Dynamics and Fractals #Polynomial and algebraic computation #math.CV #msc:30C15
paper · pdf · doi:10.48550/arxiv.1706.05410
9 pages
arxiv created 2017/06/16 · arxiv updated 2017/06/20
The Gauss--Lucas theorem states that any convex set K⊂ℂ which contains all n zeros of a degree n polynomial p∈ℂ[z] must also contain all n-1 critical points of p. In this paper we explore the following question: for which choices of positive integers n and k, and positive real number ε, will it follow that for every degree n polynomial p with at least k zeros lying in K, p will have at least k-1 critical points lying in the ε-neighborhood of K. We supply an inequality relating n, k, and ε which, when satisfied, guarantees a positive answer to the above question.