User:Youriens/sandbox

Consider a polynomial in two variables $$(z,w)\in\mathbb{C}$$ given by the expression

$$ F(z,w)=a_0(z)+a_1(z)w+a_2(z)w^2+\cdots+a_n(z)w^n $$ where $$a_i(z)$$ is a polynomial with rational coefficients. The zero set $$F(z,w)=0$$ defines an algebraic curve in $$\mathbb{C}^2$$. These curves are often quite complicated consisting of interleaved sheets twisting around singular points of $$F(z,w)$$ and although we cannot graph directly the curve in $$\mathbb{C}^2$$, we can obtain a very precise plot of the real or imaginary component of the curve. For example, consider the expression $$ F(z,w)= $$

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