Super-ellipses are the equations that yield unit circles in different p-norms in $L^p$ space. I'm interested in the solution space in the unit square. My trouble is finding the solutions that are neither on the line $x=1/2$ nor on the line $y=1/2$. How do the following equations relate to $L^p$ space and how does one solve this system of super-ellipse equations for
$s,t=1,2,3.$ That is, what is the solution space for the intersections of these $6$ super-ellipse equations for,
$ x,y \in \Bbb R (0,1). $
The system is:
$ x^s+y^s=1 $
$ (1-x)^t+y^t=1. $
For $s=2$ and $t=3$ I first foiled, then added the equations and simplified to arrive at the following:
$-x^3+4x^2-3x+y^2+y^3=1.$
I know the solution must be algebraic.
For your $s=2,t=3$ example, which is the hardest, you can write $$(1-x)^3=1-y^3\\ 1-x=\sqrt[3]{1-y^3}\\ 1-\sqrt{1-y^2}=\sqrt[3]{1-y^3}$$ which will give you a sixth degree polynomial when you clear the roots. Alpha finds a root that is messy enough it gives it numerically as about $y=0.919311$