When (x,y)∈C2(x,y)∈C2 is far from the origin, the value of the polynomial PdPd is "almost" the value of PP, since the other terms are smaller. Hence, the zeroes of PdPd are asymptotical lines of your curve (this is indeed true and can be checked with homogeneous coordinates for instance). Hence, the curve is not compact. The same argument shows that over RR, curves of odd degree are not compact and curves of even degree are compact if and only if the leading homogeneous polynomial has no real root. Go to this website for more ideas.
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