\documentclass[11pt]{report}
\input{header}
\pagestyle{empty}
\begin{document}
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    \begin{pspicture}(-4,-4)(4,4)
      \arraxes(0,0)(-4,-4)(4,4)
      \rput(1.5,-2.8){%
        \psframebox*{%
          \begin{minipage}{5cm}
            \textcolor{black}{$c_5: x=0$}\\
            \textcolor{blue}{$c_3: x^2-y^3=0$}\\
            \textcolor{olivegreen}{$c_4: y^2-x^2+1=0$}\\
            \textcolor{orange}{$c_1: x^4+y^3-1=0$}\\
            \textcolor{purple}{$c_2: 2x^2+5y^2-7=0$}
          \end{minipage}
        }
      }
      % C1: x^4+y^3-1
      \psplot[linecolor=orange,plotstyle=curve]{-2.82}{2.82}{
        /third 1 3 div def
        /a 1 x 4 exp sub def
        a 0 ge {a third exp} {a neg third exp neg} ifelse
      }
      % C2: 2*x^2+5*y^2-7
      \parametricplot[linecolor=purple,plotstyle=curve]{0}{180}{
        7 2 div sqrt t cos mul 7 5 div sqrt t sin mul
      }
      \parametricplot[linestyle=dashed,dash=2pt 2pt,
        linecolor=purple,plotstyle=curve]{180}{360}{
        7 2 div sqrt t cos mul 7 5 div sqrt t sin mul
      }
      % C3: x^2-y^3
      \psplot[linestyle=dashed,dash=2pt 2pt,
        linecolor=blue,plotstyle=curve]{-4}{-2}{
        /twothirds 2 3 div def
        x 0 ge {x twothirds exp} {x neg twothirds exp} ifelse
      }
      \psplot[linecolor=blue,plotstyle=curve]{-2}{2}{
        /twothirds 2 3 div def
        x 0 ge {x twothirds exp} {x neg twothirds exp} ifelse
      }
      \psplot[linestyle=dashed,dash=2pt 2pt,
        linecolor=blue,plotstyle=curve]{2}{3}{
        /twothirds 2 3 div def
        x 0 ge {x twothirds exp} {x neg twothirds exp} ifelse
      }
      \psplot[linecolor=blue,plotstyle=curve]{3}{4}{
        /twothirds 2 3 div def
        x 0 ge {x twothirds exp} {x neg twothirds exp} ifelse
      }
      % C4: y^2-x^2+1
      \parametricplot[linestyle=dashed,dash=2pt 2pt,
        linecolor=olivegreen,plotstyle=curve]{-75}{0}
                     {/cost t cos def 1 cost div t sin cost div}
      \parametricplot[linecolor=olivegreen,plotstyle=curve]{0}{60}
                     {/cost t cos def 1 cost div t sin cost div}
      \parametricplot[linestyle=dashed,dash=2pt 2pt,
        linecolor=olivegreen,plotstyle=curve]{60}{75}
                     {/cost t cos def 1 cost div t sin cost div}
      \parametricplot[linestyle=dashed,dash=2pt 2pt,
        linecolor=olivegreen,plotstyle=curve]{105}{255}
                     {/cost t cos def 1 cost div t sin cost div}
      % c5: vertical segment
      \psline(0,0)(0,1.18321)
      %
      \arrEmptyVertex(-1,0){p}             % degree 2
      \arrEmptyVertex(-1,1){p}             % degree 4
      \arrMainVertex(-1.87083,0){p}        % degree 1
      \arrMainVertex(-2,1.5874){p}         % degree 1
      \arrEmptyVertex(-0.786151,0.8518){p} % degree 4
      \arrEmptyVertex(0,0){p}              % degree 3
      \arrEmptyVertex(0,1){p}              % degree 4
      \arrEmptyVertex(0,1.18322){p}        % degree 3
      \arrEmptyVertex(0.786151,0.8518){p}  % degree 4
      \arrEmptyVertex(1,0){p}              % degree 3
      \arrEmptyVertex(1,1){p}              % degree 4
      \arrEmptyVertex(1.30931,0.845154){p} % degree 4
      \arrEmptyVertex(1.77423,1.46557){p}  % degree 4
      \arrMainVertex(1.87083,0){p}         % degree 1
      \arrMainVertex(2,1.5874){p}          % degree 1
      \arrMainVertex(2,1.73205){p}         % degree 1
      \arrMainVertex(3,2.08008){p}         % degree 1
      \arrp[45](-1,2){3}             % Isolated
      \arrp[45](1,2){2}              % Isolated
      \arrp[45](2,-1.73205){1}       % Isolated
      \psframe[linewidth=0.25pt](-4,-4)(4,4)
    \end{pspicture}
}
\end{document}
