Curves formed by Vanishing Discriminant and Roots of Complex-valued Harmonic Polynomials (Computer-Aided Case Study)

In this paper, we determine and specify the type of curves formed by the vanishing discriminant of some specified family of complex-valued harmonic polynomials with two parameters. We also classify the region formed by curves as bounded and unbounded connected components which in turn used to count the zeros of the complex-valued harmonic polynomials. Our study is a computer-aided case and the result shows that the curves formed are a teacup curve, a parabolic curve and a swallowtail catastrophe curve. These curves come up together to form a butterfly catastrophe.