The basic facts about $\mathbb{R}[x,y].$

Question

There's all this basic stuff about $\mathbb{R}[x,y]$ that I don't know, and this impacts negatively on my ability to teach e.g. Year 10 and 11 mathematics effectively (for example).

So let $R$ denote a ring. A well-known result says that if $R$ is a UFD, then $R[x]$ is, too. This tells us that $\mathbb{R}[x,y]$ is a UFD. So prime elements are the same as irreducible elements, and every element of $\mathbb{R}[x,y]$ factors uniquely as a product of such elements. That's about all I know. So, here's a few questions designed to help me patch up my knowledge.

• Are the irreducible elements in $\mathbb{R}[x,y]$ well-known and easy to describe, and if so, what are they? When one variable is concerned, the discriminant tells us everything we need to know, of course.

• It would seem that both the "circle polynomial" $x^2+y^2-1$ and the "hyperbola polynomial" $x^2-y^2-1$ are both irreducible in $\mathbb{R}[x,y]$. Is that correct? If so, how do we know?

• At the risk of asking a meaningless question:) assuming they're both irreducible, why is zero-locus of $x^2+y^2-1$ connected, but the zero-locus of $x^2-y^2-1$ is not?

• More generally, can we say anything about the zero-loci of irreducible polynomials that, say, a high school student can understand?

• Does homogenization somehow help us? If so, how?

Answer 1

(1) The polynomial ring $K[x]$ over a field $K$ is a Euclidean ring, that is one can perform divisions with remainder. Therefore irreducible and prime elements are one and the same. But observe that even in this case irreducibles are in general not easy to describe: consider the case $K=\mathbb{Q}$. The structure of the field $K$ plays a role. For $K=\mathbb{R}$ things are simple because of the fundamental theorem of algebra and the fact that the complex numbers are an extension of degree 2 of the reals.

The polynomial ring $\mathbb{R}[x,y]$ can be considered as a subring of the polynomial ring $\mathbb{R}(x)[y]$, where $\mathbb{R}(x)$ is the field of rational functions in $x$ with real coefficients, a field that is in a sense as complex as the rational numbers. Irreducibles of $\mathbb{R}(x)[y]$ are therefore not easy to describe.

On the other side a result of Gauß says: a polynomial $f=a_ny^n+a_{n-1}y^{n-1}+\ldots +a_0\in\mathbb{R}[x,y]$, $a_k\in\mathbb{R}[x]$ relatively prime, is irreducible if and only if it is irreducible in $\mathbb{R}(x)[y]$.

(2) A nice check for irreducibility is Eisenstein's criterion: let $A[y]$ be a polynomial ring in $y$ over the UFD $A$. If the coefficients $a_k$ of the polynomial $f=y^n+a_{n-1}y^{n-1}+\ldots +a_0\in A[y]$ are divisible by a prime $p\in A$ and $a_0$ is not divisible by $p^2$, then $f$ is irreducible.

Apply this for $A=\mathbb{R}[x]$ and $p=x-1$ and $p=x^2+1$ to obtain the irreducibility of $y^2+x^2-1$ and $y^2-x^2-1$.

(3) Suppose you draw an irreducible $f$ randomly from $\mathbb{R}[x]$. Then it may or may not have a real root resp. $n$ real roots, $n$ the degree of $f$. Typically some roots are non-real complex numbers. Similar for the zero locus of an irreducible $f\in\mathbb{R}[x,y]$: the zero locus $C(f)$ of $f$ in $\mathbb{C}\times\mathbb{C}$ is always an infinite set - a so-called plane algebraic curve. We have the maps

$ \mathbb{R}\times\mathbb{R}\rightarrow\mathbb{C}\times\mathbb{C},\; (a_1,a_2)\mapsto (a_1+0i,a_2+0i) $

and

$ \mathbb{C}\times\mathbb{C}\rightarrow\mathbb{R}^2\times\mathbb{R}^2,\; (a_1+ib_1,a_2+ib_2)\mapsto (a_1,b_1,a_2,b_2). $

Using them we can consider the zero locus $C(f)$ as a subset of $\mathbb{R}^4$. The intersection

$ C(f)\cap\{(a_1,0,a_2,0) : a_1,a_2\in\mathbb{R}\} $

consists of the points in $C(f)$ having real coordinates ("real zeros"). Depending on the geometric position of $C(f)$ in the space $\mathbb{R}^4$ this intersection may or may not be empty resp. connected.

(4) An interesting fact for high school students: let $f,g\in\mathbb{R}[x,y]$ be two irreducible polynomials such that $f\neq cg$ for all $c\in\mathbb{R}$, then the intersection of their zero loci is finite and bounded from above by $\deg(f)\deg(g)$. Or in other words: the system of polynomials equations

$f(x,y)=0, g(x,y)=0$

has at most $\deg(f)\deg(g)$ solutions.

Answer 2

What you are asking about is essentially the basics of classical algebraic geometry. Instead of answering your question directly, I would like to discuss two things that your question made me think of.

First, there are a lot of strange phenomena that happen because $\mathbb R$ is not algebraically closed. Things are substantially simpler when you work over the complex numbers. Suddenly, the zero locus of $y^2-x^2-1$ isn't two branches, but one giant sheet. It just turned out that you had a slice of the sheet that contained two separate parts. Moreover, the map $(a,b)\mapsto (ia,b)$ gives a bijection between the points in your hyperbola and circle examples. Two objects that seemed to be different are actually the same if you change your perspective, which has interesting consequences.

Second, homogenization is very useful. I don't know what your background is, but likely you have studied compactness and seen the wonderful things that you can do with it. If you have seen how you can add a point at infinity to the euclidean plane to make a sphere (an instance of the one point compactification), you know that we can sometimes better understand things by replacing them with compact versions, where things are slightly better behaved.

It turns out that homogenization gives an algebraic analogue of compactification. Given a field $k$, define $n$ dimensional projective space over $k$ \mathbb P^n(k) to be the quotient of $k^{n+1}-0$ by the equivalence relation $(a_0,a_1,\ldots, a_n)\sim (\lambda a_0, \lambda a_1,\ldots, \lambda a_n)$ for $\lambda \in k-\{0\}$. When $k=\mathbb R$ this is the same as taking the $n$-dimensional sphere and identifying antipodal points.

Every point of projective space falls into two sets, those where $a_0=0$, and those where $a_0\neq 0$. For points of the latter type, every point is equivalent to a unique point of the form $(1,a_1,a_2,\ldots, a_n)$. Thus, projective space may be thought of as taking $k^n$ adjoining the points of the form $[0:a_1:a_2:\ldots:a_n]$ "at infinity" (where the notation stands for the equivalence class of the point). This is an algebraic way of compactifying $k^n$.

What does this have to do with homogenization? Suppose that $f$ is a homogenous polynomial of degree $d$ in $n+1$ variables. Then $f(\lambda x)=\lambda^d f(x)$, and so if $x$ is a root of $f$, then so is $\lambda x$. This means the zero locus of $f$ can be viewed as living in $\mathbb P^n(k)$ instead of just living in $k^{n+1}$.

Moreover, given a polynomial $g$ in $n$ variables, we can form its homogenizations $\widetilde{g}$ in $n+1$ variables, and (letting $x_0$ be the new variable), $g(x_1,x_2,\ldots,x_n)=0 \Leftrightarrow \widetilde{g}(x_0,x_1,\ldots, x_n)=0$. Therefore, the zero locus of $\widetilde{g}$ in projective space is the zero locus of $g$ with added points at infinity.

So why do we care? One reason is that adding back these points at infinity can simplify matters, just as adding in complex points can. Things that looked different can be seen to be different views of the same object. But other things can happen too. For an example, I give Bezout's theorem.

Theorem: Let $f,g \in \mathbb C[x,y]$ of degrees $m$ and $n$, and suppose that $f$ and $g$ have no nontrivial factors in common. Then, considering the intersections in projective space and properly counting the multiplicities of the intersection (e.g., the curve $y=x^3$ intersects the curve $y=0$ with multiplicity $3$ at $(0,0)$), the zero loci of $f$ and $g$ intersect in exactly $mn$ points.

Compare this to the result given by Hagen. You would like the number of points of intersection to be more than just bounded by the product of the degrees. Unfortunately, some points of intersection are complex, but even taking care of these, sometimes two curves can intersect at infinity (e.g., in the projective plane every pair of lines has a unique point of intersection, but if you take two parallel lines, they intersect at infinity). Thus, by adding in complex points and points at infinity, we can strengthen the link between the algebraic side and the geometric side.