Elliptic tales 1

I’ve been reading Elliptic Tales, a fantastic book on elliptic curves by Avner Ash and Robert Gross, and I’d like to summarize what I’ve learned from the book in three long-ish posts corresponding to the three parts of the book. This post is the first one in this series.

We are going to consider polynomial equations and their degrees. There are two different ways to define the degree of a polynomial equation and of the curve it describes. The first definition is purely algebraic: the algebraic degree of a polynomial is the largest degree of any monomial appearing in the polynomial: e.g. the algebraic degree of $x^3y^2z + x^2y^2z+xz^3$ is 6, because the monomial with the largest degree, i.e. $x^3y^2z$, has degree 6. The second notion of degree is purely geometric: informally, we want to say that the geometric degree of a curve described by a polynomial equation $f(x,y,\ldots)=0$ is the number of intersection points it has with an arbitrary line $L$. Of course, the problem is that this is not well-defined, because the number of intersection points depends on the line $L$ we pick! Consider the figure below (adapted from Figure 1.6 in the book):

The line $K$ intersects the parabola $G$ at two points, the lines $M$ and $L$ intersect $G$ at one point only and the line $N$ doesn’t intersect $G$ at all!

The uncreative and uninteresting way to deal with this nuisance is to define the geometric degree to be the maximum number of intersection points we could get by varying the line $L$. But instead we’re going to be bold and demand that the geometric degree of a curve to be the same regardless of the line $L$ we choose! It turns out that this can be done and is the source of a lot of rich, beautiful and exciting mathematics. But, enforcing this demand will require us to reconsider which number system to adopt, and to carefully redefine what we mean by an “intersection point” and how to count them.

It turns out that we need to do three things: 1) define our polynomials over an algebraically closed field of numbers; 2) switch from the Euclidean plane to the projective plane; 3) count intersection points by their “multiplicities”. I will now explain what each of these means.

1) $F$ is an algebraically closed field means that any non-constant polynomial whose coefficients are in $F$ has a root in $F$. We can see immediately that the reals $\mathbb{R}$ is not algebraically closed, since e.g. the polynomial $x^2 + 1$ has no roots in  $\mathbb{R}$. However, the fundamental theorem of algebra tells us that the complex numbers $\mathbb{C}$ is algebraically closed. It is intuitively clear why we need to work with an algebraically closed field, hence why e.g. $\mathbb{R}$ won’t do for us: we want to say that algebraically $x^2 + 1$ has degree 2, but geometrically $x^2+1=0$ does not even make sense in $\mathbb{R}$.

Switching to an algebraically closed field helps us deal with cases such as the line $N$ in the above figure. Although this line does not intersect the parabola in the real plane, it does intersect the parabola in the complex plane at exactly two points.

2) In the projective plane, we describe points by three coordinates instead of the usual two: $(x:y:z)$. Setting $z=1$ gives us the usual (“finite”) plane. Points with $z=0$ correspond to “points at infinity”. The coordinates $(x:y:z)$ and $(\lambda x: \lambda y: \lambda z)$ with $\lambda\neq 0$ describe the same point. So, the points at infinity form a single line at infinity that can be parametrized by $(1:t:0)$ or by $(t:1:0)$ where $t$ can be any real number.

Polynomial curves in the usual “finite” plane might get points at infinity attached to them in the projective plane. To find out what specific points at infinity get attached to a particular curve, we need to define the homogenization of a polynomial: for a polynomial of degree $d \geq 1$ in two variables, $f(x,y)$, this means multiplying each monomial in $f(x,y)$ by an appropriate power of $z$ so that all monomials have the same degree $d$. For example, the homogenization of $f(x,y)=y-x^2$ is $F(x,y,z)=yz - x^2$. Now, to find out which points at infinity get attached to the curve described by the polynomial equation $F(x,y,z)=0$, we simply set $z=0$: $y \cdot 0 - x^2 =0$ to which the solutions are $(0:y:0)$ with $y \neq 0$. By the property above, these describe a single point which we might as well denote by $(0:1:0)$.

Why do we need this homogenization of polynomials and these points at infinity? Intuitively, it is to deal with cases such as the line $L$ in the figure above, which intersects the parabola at a single point, whereas we desire every line to intersect the parabola at exactly two points. The second intersection point turns out to be a point at infinity: namely $(0:1:0)$. This point at infinity belongs to both $L$ and the parabola, as you can easily check.

3) Finally, we have to count intersection points by their multiplicity. Suppose a given curve $C$ and a line $L$ intersect at a point $P$. Suppose further that the line is parametrized by the parameter $t$ such that $P$ corresponds to $t=0$. Then $t=0$ is a root of the polynomial $g(t)$ obtained by plugging the parametric form of $L$ in the polynomial describing $C$. If the degree of this root is $k$, we say that the intersection point has multiplicity $k$ and express it by $I(C,L,P)=k$. This definition was a mouthful, so let’s do an example: suppose our curve is defined by the polynomial $f(x,y)=x^2 + y + 4$ and the line $L$ is parametrized by $x = 2t+1$ and $y=t-5$. If we plug these in $f(x,y)$, we get $g(t)=5t+4t^2=(t-0)(4t+5)$. The root $t=0$ appears with multiplicity $1$ in $g(t)$, so we say that the intersection point corresponding to $t=0$, i.e. $P=(1,-5)$ has multiplicity $1$.

When can the multiplicity of an intersection point be larger than $1$? This can happen either when the line is tangent to the curve at the intersection point, or when the intersection point is singular, i.e. the tangent is not well-defined at the intersection point, i.e. the partial derivatives all vanish at the intersection point. Incidentally, because we are dealing with polynomials only, we can define the derivative of a polynomial purely procedurally (i.e. the derivative of $f(x)=a_0+a_1 x\ldots+a_n x^n$ is defined to be $f^\prime (x) = a_1 + \ldots + n x^{n-1}$), without having to worry about whether the standard analytic definition in terms of limits even makes sense in a given field. Ditto for partial derivatives.

It is again clear why we need to count intersection points by their multiplicity: it is to deal with cases such as the line $M$ in the above figure: this line is tangent to the parabola, hence intersects it at a single point only, but the intersection has multiplicity $2$, as you can easily check.

We are now ready to state Bézout’s theorem which is the culmination of our desire to make the algebraic degree and the geometric degree of polynomial equations or curves to always match each other. Suppose $C$ and $D$ are two projective curves that intersect at a finite number of points: $P_1$, $P_2$, $\ldots$, $P_n$. Then Bézout’s theorem states that the product of the degrees of the homogeneous polynomials defining $C$ and $D$ exactly matches what we call the global intersection multiplicity $\mathcal{I}(C,D)=\sum_k I(C,D,P_k)$, i.e. the sum of the multiplicities of all intersection points. The fact that any line intersects a curve $C$ at the same number of points and that this number is equal to the algebraic degree of the polynomial describing $C$ are immediate consequences of Bézout’s theorem.