This is a literature request for (hopefully) an English version to a rigorous proof that a complex algebraic curve cannot abruptly end. 


That is, if the algebraic curve enters a closed region it must also leave it. 


This has a historic significance because Gauss's proof in his Phd thesis assumed this property holds. From looking around it seems that A Ostrowski rigorously proved the result around the 1920's. Is this correct? I am unable to find the title of the paper. 


Is there also a proof that a real algebraic curve does not end abruptly?


I don't regard this property as obvious, but it doesn't seem to be well commented in the literature. Maybe, I'm wrong. 


Thanks in advance.


Abruptly end:
Given an irreducible polynomial $p$, we define $V(p)$ to be the complex algebraic curve associated to $p$. I say that $V(p)$ does not abruptly end at $(x,y)\in\mathbb{C}^2$ with $p(x,y)=0$ if there is a disc small enough so that the boundary contains exactly two points in $V(p)$. 


(This is a first attempt and maybe needs some corrections.)



Algebraic curve cannot suddenly end
Following the idea of Felipe Voloch, I try to give a simple proof based on Puiseux series expansion. Let $C$ be a real algebraic curve at the origin. Look at the Puiseux series expansion (say in terms of $x$) of $C$ near $O$. By assumption one of the branches (over $\mathbb{C}$), call it $C\_1$, has the form 
$$y = a\_1x^{r\_1} + a\_2x^{r\_2} + \cdots \quad\quad\quad (1)$$
for $a\_i \in \mathbb{R}$. Let $q$ be the least common multiple of the denominators of $r\_i$'s. If $q$ is odd, then the branch expands to  *both*  sides of the origin and therefore $C\_1$ does not end abruptly. So assume $q$ is even. Let $\zeta := e^{2\pi i/q}$. For each $j$, $1 \leq j \leq q$, the complex curve corresponding to $C\_1$ has a Puiseux expansion of the form $y = \sum\_i a\_i \zeta^{jp\_i}x^{r\_i}$, where $p\_i = qr\_i$. In particular, taking $j =q/2$ (so that $\zeta^j = -1$), we see that the complex curve corresponding to $C\_1$ has an expansion of the form 
$$y = \sum\_i a\_i (-1)^{p\_i}x^{r\_i}. \quad\quad\quad (2)$$
It follows by the minimality of assumption on $q$ that there is $i$ such that $a\_i\neq 0$ and $p\_i$ is  *odd* , and consequently, $(1)$ and $(2)$ give  *different*  real curves, and it follows that $C\_1$ does not end abruptly.


PS: The above arguments only show that $C\_1$ has at least two end points on the boundary of a small enough disk centered at $O$. But it can not have more than two, because for all $j \not\in \lbrace q/2, q\rbrace$, $\zeta^j$ is non-real, so the corresponding parametrization does not give any real points.


