Given the circunference centered in the origin of a cartesian reference frame, its equation is: $x^2+y^2=r^2$, Assuming $r=1$, we have: $x^2+y^2=1$. The intersections of this curve with the curve described by the equation: $y=ax^k$ with $k\in\mathbb{N}$ can be found solving the equation: $$x^2+a^2x^{2k}=1$$ Is there some analytical method to solve this equation? Thanks.
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Hint: if $y=x^2$ then $y^k=$? – Sergio Parreiras Jul 28 '14 at 13:57
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It simplifies matters slightly to solve for $z=x^2$ instead. Even with this, though, this is the $k$th degree polynomial $z+a^2 z^k=1$ in $z$ which in general is not analytically solvable.
Semiclassical
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