The Image of Critical Circle and Zero-free Curve for Quadrinomials

Abstract

The location of the zeros of a two-parameter family of complex-valued harmonic quadrinomials depends on the parameters. In this paper, we determine and demonstrate that the image of some critical circle under these two-parameter family of complex-valued harmonic quadrinomial is a hypocycloid. We also determine a zero-free curve for two-parameter family of quadratic qudrinomial.

11 pages, 5 figures. arXiv admin note: text overlap with arXiv:2106.15824, arXiv:2106.15162