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Notes on Speyer’s paper titled “Some Sums over Irreducible Polynomials”

Let \mathcal{P} be the set of irreducible polynomials over F_2[T]. Then \sum\limits_{P\in \mathcal{P}}\frac{1}{1-P}=0. The paper lists certain examples of \frac{1}{1-P} below. These are all expanded as geometric series. As one can see only P=T, T+1 contribute to the coefficient of T^{-1} in the sum \sum\limits_{P\in \mathcal{P}}\frac{1}{1-P}=0. Why don’t the other irreducible polynomials do the same? This is because these are the only two linear polynomials in F_2[T]. All other polynomials are of higher degree. Moreover, all other irreducible polynomials have the constant term 1; otherwise they would be reducible, as T would be a common factor. Hence \frac{1}{P-1} would be of the form \frac{1}{T^{a_1}+T^{a_2}+\dots+T^{a_n}}, where a_1>1. Now divide both the numerator and denominator by T^{a_1}. So we get an expression of the form \frac{1}{T^{a_1}}(\frac{1}{1+T^{a_2-a_1}+T^{a_3-a_1}+\dots+T^{a_n-a_1}}). As a_i-a_1<0 for all i\neq 1, this is a power series expansion in negative powers of T. Also, as a_1\geq 2, all such negative exponents will be less than -1. This proves that only the polynomials T and T+1 contribute to the coefficient of T^{-1} in \sum\limits_{P\in \mathcal{P}}\frac{1}{1-P}=0.

We now try and understand Theorem 1.1 in this paper. Let \mathcal{P_1} be the set of monic irreducible polynomials in F_{2^n}[T]. Then \sum\limits_{P\in \mathcal{P_1}}\frac{1}{P^k-1}\in F_{2^n}(T) for any k\equiv 0(\mod 2^n-1).

A corollary of this is that \sum\limits_{P\in \mathcal{P}}\frac{1}{P^k-1} is in F_{2^n}(T)

Proof of corollary: We have rewritten \sum\limits_{P\in \mathcal{P}}\frac{1}{P^k-1} as \sum\limits_{P\in \mathcal{P}_1}\sum\limits_{a\in \Bbb{F}_q^\times}\frac{1}{(aP)^k-1}, where q=2^n. Why can we do that? This is because for any a\in\Bbb{F}_q^\times, a^{q-1}=1. Hence, we’re essentially counting the same thing as before. Aren’t we counting each term |\Bbb{F}_q^\times| times? Also, every irreducible polynomial is of the form aP for some P\in\mathcal{P_1}. Now consider the identity \sum\limits_{a\in \Bbb{F}_q^\times}\frac{1}{(aX)^k-1}=\frac{1}{(X)^{lcm(k,q-1)}-1} in \Bbb{F}_q(U). Why is this true? This is because \frac{1}{(aX)^k-1} can be written as \sum\limits_{j=1}^\infty\frac{1}{(ax)^{kj}} (just multiply and divide \frac{1}{(aX)^k-1} by \frac{1}{({aX})^k}).

Now, as \sum\limits_{a\in\Bbb{F}_q}a^m=1 if m\equiv 0 \mod q-1, and \sum\limits_{a\in\Bbb{F}_q}a^m=0 otherwise. This is because if m\equiv 0 \mod q-1, then \sum a^m is essentially adding 1 to itself q-1 times. As the characteristic of the field is 2, and as q-1 is essentially 2^m-1, this sum is equal to the inverse of 1, which is exactly 1. When q\not\equiv 0\mod q-1, then \sum a^m=0. This can be verified independently.

Introduction to Schemes

This is a short introduction to Scheme Theory, as modeled on the article by Brian Lawrence.

A variety here is a zero set that can be covered by a finite number of affine varieties. Hence, a morphism between varieties can be considered to be a bunch of affine morphisms, as long as they agree on the intersections.

We need a shift in perspective. What this means is that we need to start thinking about the coordinate ring rather than the points themselves.

Now let us think about the following example: the coordinate ring of y=0 in K^2 is K[x,y]/(y). However, the coordinate ring of y^2=0 is also K[x,y]/(y); it is not K[x,y]/(y^2). The reasons for this can be worked out easily. Hence, the variety in this case is not accurately recovered from the coordinate ring. We started off with the variety y^2=0, and got back y=0. We need a new concept, which would allow us to accurately get back the variety from the coordinate ring- something that would allow nilpotents.

An affine scheme, written as \text{Spec }A, is the data of a ring A. A morphism of affine schemes \text{Spec } A\to \text{Spec }B, is a morphism of rings B\to A. An affine scheme over a field k is a scheme \text{Spec }A where A is equipped with a k-Algebra structure.

Why are morphisms defined backwards here? In other words, why is \text{spec }A\to \text{spec }B defined as B\to A? This is because A,B are the coordinate rings. Let Var(A) be the variety corresponding to the coordinate ring A. Then a map Var(A)\to Var(B) defines a map B\to A, and vice-versa. Maybe \text{spec }A is a formal representation of Var(A). It is at least easy to remember which way the arrow goes this way.

How do we recover points from coordinate rings? Hilbert’s Nullstellensatz tells us that we can recover them using maximal ideals. Hence, our aim right now is to take an affine morphism, and construct a morphism between varieties. Hence, if the affine morphism is B\to A, we want to construct a map Var(A)\to Var(B).

Given a ring homomorphism \phi:R\to S, for any prime ideal p\in S, \phi^{-1}(p) is also prime. This is an elementary exercise in ring theory. It is however, not true in general that the inverse image of a maximal ideal is also maximal. For example, consider the map \psi:\Bbb{Z}[x]\to\Bbb{Q}[x] defined by inclusion. Then the only maximal ideal of \Bbb{Q}[x] is (x), the inverse of which is also just (x). It is easy to see that (x) is not a maximal ideal in \Bbb{Z}[x]. For instance, 2\notin (x), and 2+zx\neq 1 for any z\in\Bbb{Z}[x].

We define the points of the affine scheme to be prime ideals. Why? Let us work this through. We have a scheme morphsim \phi:B\to A, where both B and A are coordinate rings. Now let us take a prime ideal in A. From the discussion above, we know that \phi^{-1}(p) is a prime ideal in B. Hence, if prime ideals were points, we have taken a point in A, and mapped it to B. In a way, we have constructed a map from Var(A) to Var(B).

However, this is a little weird. Points correspond to maximal ideals, and not prime ideals. All maximal ideals are prime, but the converse is not true. Do we really have a map from Var(A)\to Var(B)? No. At least not in the traditional sense. What we have is a map from some “stuff” in A, which includes points, to “stuff” in B, which too includes points (possibly not all). Hence, something that’s not a point in A may map to a point in B, and a point in A may map to something that is not a point in B. We’re gonna call this “stuff” generic points. Hence, generic points in A go to generic points in B. This is a classic example of formulating new definitions to suit our world-view.

Now that we have the concept of “generic” points, we also need a name for “actual points”. This name is “classical points”. Hence, we’ll refer to maximal ideals in A as classical points.

So what exactly is a scheme? A scheme is a coordinate ring, whose prime ideals are its points. Simple. It generalizes the notion of a variety. How? A variety has a set of points and an associated coordinate ring. A scheme has a larger set of points, and an associated coordinate ring. Hence the generalization is in the set of points; at least in this instance. Also, as discussed before, although k[x,y]/(y) and k[x,y]/(y^2) correspond to different coordinate rings but the same variety, they correspond to different schemes. Why? If a scheme was the data of its “points” (read generic points), then the points of k[x,y]/(y^2) are different from those of k[x,y]/(y) (the cosets look different, for starters). Hence, we now allow for distinguishing between multiplicities.