The rational parametrisation of an algebraic curve in a projective space $P^n$ is a univariate function $\tau\rightarrow [P_1:\dots:P_{n+1}]$, where $P_k$ are coprime polynomials. If this parametrisation exists, it is generally claimed that a finite number of points of the curve can be "missed" by the parametrization. But it seems to me that the only point that might be missed is the one that corresponds to $\tau=\infty$. Are there other points?
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$\begingroup$ Over the reals, the curve $y^2 = x^3 - x^2$ (or rather, its projective completion $Y^2 Z = X^3 - X^2 Z$) has the rational parametrization $t \mapsto (t^2+1, t^3+t)$ which misses not only the point at infinity but also the point $(x,y) = (-1,0)$ which would correspond to $t = \pm\sqrt{-1}$. Your question is not precise enough to admit a clear answer (especially because you did not specify exactly what you meant by “algebraic curve” and over what field), but this should give an idea of what can go wrong. $\endgroup$– Gro-TsenCommented May 21, 2019 at 21:01
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$\begingroup$ Thanks for your remark. I should specify that the field is algebraically closed. I confess that I don't understand the question about the definition of algebraic curve. Is it not uniquely defined? $\endgroup$– AlmCommented May 21, 2019 at 21:44
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Eventually I found that my guess was right. In the book "rational algebraic curves, a computer algebra approach" there is the proof that only one point is missed by the parametrization.
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$\begingroup$ I suppose this is Rational Algebraic Curves, A Computer Algebra Approach, J. Rafael Sendra, Franz Winkler, Sonia Pérez-Diaz; Algorithms and Computation in Mathematics, Volume 22, 2008; Springer-Verlag Berlin Heidelberg; eBook ISBN 978-3-540-73725-4, DOI 10.1007/978-3-54073725-4, Hardcover ISBN 978-3-540-73724-7, Softcover ISBN 978-3-642-09291-6, Series ISSN 1431-1550, springer.com/gp/book/9783540737247 $\endgroup$ Commented Jun 20, 2019 at 22:52