A Geometric Modulus Principle for Polynomials

作者：Kalantari Bahman^*

来源：American Mathematical Monthly, 2011, 118(10): 931-935.

DOI：10.4169/amer.math.monthly.118.10.931

We characterize the ascent and descent directions for the modulus of a complex polynomial p(z) at an arbitrary point z(0) in the complex plane. We prove that when p(z(0)) not equal 0, the cones of ascent and descent directions partition the unit disc centered at z(0) into alternating sectors of ascent and descent, each having angle pi/k, where k >= 1 is the smallest index with p((k)) (Z(0)) not equal 0. Applying this geometric modulus principle, we give new proofs for the maximum modulus principle, the fundamental theorem of algebra, and the Gauss-Lucas theorem.

• 出版日期2011-12