Root Sets of Polynomials and Power Series with Finite Choices of Coefficients

摘要

AbstractGiven$$Hsubseteq mathbb {C}$$H⊆Ctwo natural objects to study are the set of zeros of polynomials with coefficients inH,$$begin{aligned} left{ zin mathbb {C}: exists k>0,, exists (a_n)in H^{k+1}, sum _{n=0}^{k}a_{n}z^n=0right} , end{aligned}$$and the set of zeros of a power series with coefficients inH,$$begin{aligned} left{ zin mathbb {C}: exists (a_n)in H^{mathbb {N}}, sum _{n=0}^{infty } a_nz^n=0right} . end{aligned}$$z∈C:∃(an)∈HN,∑n=0∞anzn=0.In this paper, we consider the case where each element ofHhas modulus 1. The main result of this paper states that for any$$rin (1/2,1),$$r∈(1/2,1),ifHis$$2cos ^{-1}(frac{5-4|r|^2}{4})$$2cos-1(5-4|r|24)-dense in$$S^1,$$S1,then the set of zeros of polynomials with coefficients inHis dense in$${zin {mathbb {C}}: |z|in [r,r^{-1}]},$${z∈C:|z|∈[r,r-1]},and the set of zeros of power series with coefficients inHcontains the annulus$${zin mathbb {C}: |z|in [r,1)}$${z∈C:|z|∈[r,1)}. These two statements demonstrate quantitatively how the set of polynomial zeros/power series zeros fill out the natural annulus containing them asHbecomes progressively more dense.