I know Ellipses can be represented by rational parametric functions, ie. as the ratio of two polynomials.
For example, an arc of an ellipse with minor axis $1$ and major axis $2$ can be represented by the rational parametric function:
$$ C(u) = \left(\frac{1-u^2}{1+u^2}, \frac{4u}{1+u^2}\right) $$
with $u \in [0,1]$
I also know the same ellipse can be represented by:
$$ C'(u) = \left(sin(u), 2cos(u) \right) $$
with $u \in [0,\frac{\pi}{2}]$
But this second way isn't a rational parametrization.
I wanna know if there is a way to proof that an Ellipse can't be represented by a polynomial parametric function and where to find it (or how to build this proof).
My attempt until now (by contradiction):
Suppose $ x = P(u)$ and $y = Q(u)$ are polynomials such that
$$E(u) = \left(P(u), Q(u) \right) \text{ with } u \in [0,1] \text{ such that } E(0) = (a,0) \text{ and } E(1) = (0,b)$$
is an ellipse arch centered at the origin.
But I couldn't find any contradiction until now.
I would be very thankful with any help to this problem. :)
The implicit equation of this ellipse is
$$x^2+\frac{y^2}4=1.$$
No polynomials of degree $>0$ are such that
$$p^2(t)+\frac{q^2(t)}4=1.$$