I have been studying algebraic curves from this book:
Rutter, J. (2000). Geometry of curves. United Kingdom: Taylor & Francis.
in which I found a few doubts
It should be remarked that I have also searched and read previous posts from this mathematics stack exchange. However, these did not help me with the questions to be presented here. Owing to the fact that algebraic curves and homogeneous polynomial equations are quite interesting topics in mathematics, I think these questions, and their possible answers, may be of great interest to those that subscribe to the mathematics stack exchange.
With that said, it is well-known that an algebraic curve is an analytical polynomial equation of the form:
$$P(x,y)=\displaystyle\sum^{n}_{i+j=0}a_{i j}x^{i}y^{j}=0$$
in which $n$ is the degree of the algebraic curve.
Based on the above, I ask:
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What is the best and most formal mathematical terminology for the points that satisfy the equation above?
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Let $(a,b)$ be a point where the above homogeneous polynomial equation is zero. Is that point necessarily a singular point?
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Are there any connections, results, or theorems that relate asymptotes and singular points of algebraic curves?
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Suppose that $x=k_{0}y+b_{0}$ is an asymptote of the above algebraic expression. Assume, further, that $P(x,y)$ is zero at
$$s_{1}\left(x^{*},y^{*}\right)=\left(b_{0},0\right) \quad\text{and}\quad s_{2}\left(x^{*},y^{*}\right)=\left(0, -b_{0}/k_{0}\right).$$
What is the most formal mathematical terminology to refer to $s_{1}$ and $s_{2}$? That is to say, should I refer to them as roots, zeros, equilibrium points or simply solutions of $P(x,y)$?
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What is the maximum number of asymptotes that an algebraic curve may exhibit?
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Based on your wide experience, could you suggest to me references in which such definitions are discussed?
Best Answer