= Parity and upper bound for transverse ellipse intersections
{title2=$|E_1\cap E_2|\in\{0,2,4\}$}
Parametrize one <ellipse> as $\gamma(t)=c+A(\cos t,\sin t)$ and substitute it into the quadratic equation $Q=0$ of the other. At a <transverse intersection>, $g(t)=Q(\gamma(t))$ has a simple zero, so its sign changes. Periodicity makes the number of zeros even. The function is a <trigonometric polynomial> of degree at most two, so multiplication by $z^2$, with $z=e^{it}$, gives a nonzero polynomial of degree at most four. Hence there are at most four distinct intersection points. Coincident ellipses are excluded by transversality.
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