Polynomial complex equation with conjugates

Question

I am looking for bibliographical resources where I can read about polynomial equation with conjugates of the following form $$\sum_{k+l\leq n}a_{k,l}z^k\overline{z}^l=0$$

For example for $n=2$

$$a_{0,2}\overline{z}^2+a_{2,0}z^2+a_{1,1}z\overline{z}+a_{0,1}\overline{z}+a_{1,0}z+a_{0,0}=0$$

I have found it interesting because the set of solutions cannot be trivially described as in the case without conjugates.

Isn't it a good research topic for undergraduate dissertation?

Answer 1

1. "Polynomial and Rational Inequalities" by Peter Borwein and Tamas Erdelyi. This book has a chapter on complex conjugate polynomials, including examples similar to the one you provided.

2. "A Course in Complex Analysis and Riemann Surfaces" by Wilhelm Schlag. This book includes a chapter on polynomial equations, including equations with conjugates.

3. "Algebraic Numbers and Algebraic Functions" by Paul J. McCarthy. This book includes a section on complex conjugate polynomials.

4. "Topics in Algebraic and Analytic Geometry" by Maria Aparecida Soares Ruas and Jorge Vitório Pereira. This book includes a chapter on algebraic curves, which discusses polynomial equations with conjugates.