How many times the parabola $y=x^2$ intersects the origin? The answer of this question is 1, right? but I'm about to study algebraic curves by this book
and I was surprised by this theorem:

If I'm right the immediate  corollary of that is $y=x^2$ intersects the origin two times.
Then he continues given an example:


I don't understand this, anyone can help me?
Thanks a lot
 A: Consider:
$$y = p(x) = x^2\\
g(x,y) = y-kx = 0$$
So $g(x,y)=0$ represents a straight line that passes through the origin. Notice $y-p(x) = y-x^2$ is never a factor of $y-kx$.
Now consider $g(x,p(x))$,
$$g(x,p(x)) = x^2-kx$$
And see that the lowest non-zero degree is $1$ unless $k = 0$. Otherwise the lowest non-zero degree is $2$. This shows the curve $y=x^2$ intersects almost every straight line through origin once, except the curve touches $y=0$.
For completeness, one can also show the curve intersects the vertical line $x=0$ once at the origin.
A: Here’s another discussion, more along the line of @peterwhy’s good answer, and not along the lines of your text.
Let’s look at a slightly more complicated curve than the parabola, the loop curve through the origin given by $y^2=x^3+x^2$. From a Calculus standpoint, it can also be given parametrically as $x=t^2-1$, $y=t^3-t$. Make sure you know what this curve looks like before you read any further.
Now look at the origin. Thinking of the parametrization, anyone will agree that the origin itself is a double point of the curve: after all it is passed through twice, when $t=-1$ and when $t=1$. Now consider lines $y=mx$ through the origin. Again, (almost) any such line has a double intersection with the curve. For instance the $x$-axis, given by $y=0$, does, since when you set $y=0$ into the equation $y^2=x^3+x^2$, you get the cubic $0=x^2(x+1)$: the double factor of $x$ counts the multiplicity at $x=0$. But what about a line of slope $1$ (or $-1$)? Then, we combine $y=x$ with our curve’s equation, and get $x^3=0$ for an intersection multiplicity of three, and when you look at the curve, you see that it’s tangent to the line $y=x$ at the origin. The “$3$” here counts two for the tangency, and one more for the intersection of the line with the descending “branch” of the curve.
A: You have to be a bit careful there. If we want to find the intersection of $y=x^2$ and the line $y=0$ within this framework, we would (or, at least, could) say
$$
y = p(x) = x^2\\
g(x,y) = y-x^2
$$
The theorem then tells us that as long as $\mathbf{y-p(x)}$ is not a factor of $\mathbf{g(x,y)}$, there are two intersections of the origin. Since $g(x,y)=y-p(x)$ (or, if you put things together a bit differently, $-(y-p(x))$) the theorem doesn't apply.
A: This is an instance where higher generality and scheme theory gives a more satisfactory description.
What is the intersection of two algebraic varieties? For simplicity, let both be affine subvarieties of the same affine space. For e.g., let the first curve be given in the affine plane by the vanishing ideal $I_1$ generated by the polynomial $y-x^2$. Let the second curve be given in the affine plane by the vanishing of ideal $I_2$  generated by the polynomial $y$.
Roughly, our algebraic varieties are zero sets of certain polynomials. So, the intersection of two of them should be the common zero set of all the polynomials in the ideals corresponding to either variety, i.e., in our case, it will be the zero set of the ideal $(I_1, I_2)$ generated by $I_1$ and $I_2$.
Which, unfortunately, is not an algebraic variety. This would have been an algebraic variety if the intersection were transversal, i.e., it were not tangential. The intersection would be a single point in that case, which is not quite the case for tangential intersection. There things are much more clear if we generalize to schemes. Consider the two relations $y=0$ and $y=x^2$. Taken together, this means the presence of an element $[\epsilon]$ in the structure sheaf such that $\epsilon ^2=0$ and this is an infinitesimal signifying in a "thick point" at the intersection. This POV is very clearly explained in Mumford's Red Book, which is a very good introductory  source for learning Varieties and Schemes at once. A book more at the level of the one you cite is Smith, Kahanpää, Kekäläinen and Traves, in which this is explained more informally.
