Wednesday, 30 September 2009

Finite geometries, part 4: Lines in PG(n,Fq)

[Apologies about the formatting - blogger seems to have won the struggle this time.]

Last time we saw that the points in the projective geometry PG(n,Fq) are defined to be the lines (through the origin) in the vector space Fq^(n+1). What about lines in PG(n,Fq)? Well, it's simple - they are defined as the planes (through 0) in Fq^(n+1).

Let's look at an example, to try to clarify our intuitions. Here are the points of PG(2,F3) again.

> :load Math.Combinatorics.FiniteGeometry
> ptsPG 2 f3

(Recall that each point in PG(2,F3) represents a line through 0 in F33. We have chosen the representative which has 1 as its first non-zero coordinate. This has the consequence that the coordinates in the diagram are best read back-to-front - they're zyx instead of xyz.)

So the lines in PG(2,F3) correspond to the planes through 0 in F33. For example, the plane x = 0 (the left hand face of the cube) gives rise to a line in PG(2,F3) containing the points 010, 100, 110, 120. The plane x = z (a diagonal cut through the cube from front left to back right) gives rise to the line containing the points 010, 101, 111, 121. Just to be clear, we are only looking at the planes in F33 that contain 0. So for example, 012, 101, 111, 121 - the plane x+z=2 - is not a line in PG(2,F3). Also, remember that F3 wraps around - so for example 010, 102, 112, 122 is a line in PG(2,F3).

Okay, onwards. As we did with AG(n,Fq), we can represent a line in PG(n,Fq) by any pair of points on it. We would then like to be able to do the following:
  1. Given a line of PG(n,Fq), list the points of PG(n,Fq) which are on it
  2. Given a line of PG(n,Fq), find a canonical representation for it
  3. Find all lines of PG(n,Fq)
(These are basically just the same things that we did when we looked at AG(n,Fq) a couple of weeks back.)

Okay, so suppose that we are given two distinct points of PG(n,Fq), representing a line, and we are asked to list all the points of PG(n,Fq) that are on the line. Well, the two points are also points in Fqn+1, and as they are distinct in PG(n,Fq), they are linearly independent in Fqn+1 - hence they generate a plane through 0. We are being asked to find all lines through 0 that are contained in the plane. Well, that's easy:
  1. Generate all linear combinations of the two points - all points in the plane in Fqn+1.
  2. Each non-zero point in the plane represents a line through 0, and hence a point of PG(n,Fq)
  3. However, we can filter out all except those that are in the canonical form for pts of PG(n,Fq) - having a 1 as their first non-zero coordinate - in order to avoid double counting the lines.
So we need an auxiliary function to detect whether a point is in canonical form - what I call "projective normal form" or PNF:

ispnf (0:xs) = ispnf xs
ispnf (1:xs) = True
ispnf _ = False

Then, given two distinct points, to list all points on the line they generate:

linePG [p1,p2] = toListSet $ filter ispnf [(a *> p1) <+> (b *> p2) | a <- fq, b <- fq]
where fq = eltsFq undefined

(Recall that c *> v is scalar multiplication of a vector, and u <+> v is addition of vectors. Note that in HaskellForMaths <= 0.1.9, you must use the more general "closurePG" function instead, which has the same behaviour.)

For example, here are the two lines of PG(2,F3) that we discussed earlier:

> linePG [[0,1,0],[1,0,0]] :: [[F3]]
> linePG [[0,1,0],[1,0,1]] :: [[F3]]

A line in PG(n,Fq) can be represented by any pair of distinct points on the line. Is there a canonical pair to pick? Yes, there is. Given any pair of points of PG(n,Fq), we can consider them as the rows of a matrix. Any matrix which can be obtained from this matrix by elementary row operations represents the same plane in Fqn+1. For our canonical form, we can take the reduced row echelon form for the matrix (see wikipedia). However, I'm not going to dwell on this - use the "reducedRowEchelonForm" function if you'd like to experiment.

What I'd like to consider instead is how we can list all the lines in PG(n,Fq). The answer is simply to list all the reduced row echelon forms. We can do this in two steps:
  1. List all the reduced row echelon "shapes"
  2. Fill in all possible combinations of values from Fq, to get all reduced row echelon forms
An example will make this clearer. Suppose we want to find the lines in PG(3,Fq). Thus we are looking for the 2-dimensional subspaces of Fq^4. The reduced row echelon shapes are the following:

[1 0 * *] [1 * 0 *] [1 * * 0]
[0 1 * *] [0 0 1 *] [0 0 0 1]

[0 1 0 *] [0 1 * 0] [0 0 1 0]
[0 0 1 *] [0 0 0 1] [0 0 0 1]

To get the reduced row echelon forms, we take each shape in turn, and fill in the stars with the values from Fq in all possible ways.

The Haskell code finds row echelon shapes for k-dimensional subspaces of Fq^n. (For lines, set k = 2):

data ZeroOneStar = Zero | One | Star deriving (Eq)

instance Show ZeroOneStar where
show Zero = "0"
show One = "1"
show Star = "*"

rrefs n k = map (rref 1 1) (combinationsOf k [1..n]) where
rref r c (x:xs) =
if c == x
then zipWith (:) (oneColumn r) (rref (r+1) (c+1) xs)
else zipWith (:) (starColumn r) (rref r (c+1) (x:xs))
rref _ c [] = replicate k (replicate (n+1-c) Star)
oneColumn r = replicate (r-1) Zero ++ One : replicate (k-r) Zero
starColumn r = replicate (r-1) Star ++ replicate (k+1-r) Zero

Thus we can calculate the shapes we saw above as follows:

> mapM_ print $ rrefs 4 2

Next, to find all lines in PG(n,Fq), we need to substitute values from Fq for the stars. The following code does the trick, although I suspect that there might be a better way to write it:

flatsPG n fq k = concatMap substStars $ rrefs (n+1) (k+1) where
substStars (r:rs) = [r':rs' | r' <- substStars' r, rs' <- substStars rs]
substStars [] = [[]]
substStars' (Star:xs) = [x':xs' | x' <- fq, xs' <- substStars' xs]
substStars' (Zero:xs) = map (0:) $ substStars' xs
substStars' (One:xs) = map (1:) $ substStars' xs
substStars' [] = [[]]

linesPG n fq = flatsPG n fq 1

For example:

> mapM_ print $ linesPG 2 f3

It occurs to me that I should explain why PG(n,Fq) is worth looking at. Well, I mentioned before that the symmetry groups of PG(n,Fq) are some of the "atoms of symmetry", from which all symmetry groups are composed. In addition, projective geometry is in fact more fundamental than affine geometry (at least from the point of view of algebraic geometry, although I'm not sure how you would justify that statement in general). Well, maybe you'll just have to take my word for it that PG(n,Fq) is a beautiful thing, for the moment.

Next time, symmetries of PG(n,Fq).

Friday, 25 September 2009

Finite geometries, part 3: Points in PG(n,Fq)

[New release HaskellForMaths-0.1.8 - contains bugfix to Fractional instance for ExtensionField, plus documentation improvements.]

Over the last two weeks, we have looked at the finite affine geometries AG(n,Fq). Hopefully, this has all been fairly straightforward - affine geometry is just the familiar Euclidean geometry of points and lines, but without angles or distances, and all we have done is replace the reals R by the finite fields Fq.

This week we're going to look at finite projective geometries. Now, I know from my own experience that it can take a little while to "get" projective geometry. There are probably several reasons for this, but one of them is that mathematicians define projective geometry one way, but then most of the time think about it in a different way. So, I'll do my best to explain it, but don't worry if you don't get it at first, one day it will all click into place.

Okay, so let's start with the points. The points of PG(n,Fq) are defined to be the lines through the origin (ie the one-dimensional subspaces) in Fq^(n+1). That should sound a bit strange at first reading - how can the points (in PG(n,Fq)) be lines (in Fq^(n+1))? Bear with me - we'll see why it makes sense to think of them as points.

For example, the points of PG(2,F3) are the lines through 0 in F3^3. So the line {(0,0,0), (0,1,0), (0,2,0)} is a point of PG(2,F3), as is the line {(0,0,0), (1,0,2), (2,0,1)}. (Recall that arithmetic in F3 is modulo 3.) As a line in F3^3 is a one-dimensional subspace, it is generated by any non-zero point on the line, and consists of all scalar multiples of such a point. So we could represent the two lines we just mentioned as <(0,1,0)> and <(1,0,2)>, with angle brackets meaning "the line generated by". Any non-zero point on the line will do to represent the line, but it would be good to have a way of choosing a canonical representative. For the moment, let's say that we will choose the point whose last non-zero coordinate is 1. Here, then, are the points of PG(2,F3):
So the blue points are the points of the form (x,y,1). Every point of the form (x,y,1) represents a line through 0 in F3^3. However, this is not all the lines through 0, for it misses out those lines which are in the plane z = 0. The green points are the points of the form (x,1,0). Each point of the form (x,1,0) represents a line through 0 in the plane z = 0. However, this is still not all, for we are missing the line y = z = 0. This is represented by the red point (1,0,0). And that's it - these are all the "points" of PG(2,F3). (Confirm for yourself that every line through zero in F3^3 is represented by one of these "points".)

Since any scalar multiple of a point in F3^3 represents the same line through 0, and hence the same point of PG(2,F3), mathematicians often write the representatives as (x:y:z), with the colons indicating that it is only the ratios we are interested in. Thus (0:1:0) and (0:2:0) represent the same line in F3^3, and hence the same point of PG(2,F3). (Note that (0:0:0) is not a point of PG(2,F3).)

Okay, so I mentioned that mathematicians define PG(n,Fq) one way, but then think about it another way. So how do mathematicians think about it?

Well, the blue points in PG(2,F3) look just like a copy of AG(2,F3), don't they. Then the green points are called "the line at infinity". Finally, the red point is called "the point at infinity". So mathematicians think of PG(2,F3) as being like AG(2,F3), but with some additional points "at infinity". In particular, although PG(2,F3) was defined in terms of lines through 0 in F3^3, mathematicians actually think of it as consisting of points (not lines).

The idea that the additional points are "at infinity" comes from perspective drawing. Imagine an artist sitting in front of a canvas, looking along the z-axis. Their eye is the origin. The canvas is the plane {(x,y,1)}. To draw what they see, the artist needs to project a line from their eye, through the canvas, until it hits something. Given a large enough canvas, the artist can draw anything that is in front. However, things that are above or to the sides would have to be drawn "at infinity".

One thing I should emphasize is that, contrary to appearance, the line and point at infinity are no different from the other points. Indeed, I could have had my artist looking along the x-axis instead of the z-axis. In that case, the embedded affine plane would have been the points {(1,y,z)}, the line at infinity would have been {(0,1,z)}, and the point at infinity would have been (0,0,1). If we revert to thinking about lines in F3^3 for a moment, it should be clear that the points at infinity are not distinguished in any way from other lines in F3^3. Their appearance of being distinguished has arisen solely from our choice of coordinate system.

Okay, time for some code. In the above, we said that the canonical representative for a line through 0 would be the point with 1 as its last non-zero coordinate - corresponding to looking along the z-axis. That made things easier to explain. However, it turns out that it's usually more convenient to choose the point with 1 as its first non-zero coordinate (ie, looking along the x-axis). Here is the code to list the points of PG(n,Fq):

ptsPG 0 _ = [[1]]
ptsPG n fq = map (0:) (ptsPG (n-1) fq) ++ map (1:) (ptsAG n fq)

For example:

> ptsPG 2 f3
> map reverse it

If we reverse the coordinates, you can see that we have the point at infinity, the line at infinity, and the embedded copy of AG(2,F3), as before.

That's probably enough to take in at one sitting. Next time, we'll look at lines in PG(n,Fq).

Friday, 18 September 2009

Finite geometries, part 2: Symmetries of AG(n,Fq)

Last time, we met the finite affine geometries AG(n,Fq), which are analogous to n-dimensional Euclidean geometry, but defined over the finite fields Fq instead of the reals R. This time I want to talk about the symmetries of AG(n,Fq).

Recall that when we were looking at graphs, we defined a symmetry of a graph as a permutation of the vertices which left the edges (collectively) in the same places. For the moment, we will think of a finite geometry as a configuration of points and lines, and define a symmetry as a permutation of the points which leaves the lines (collectively) in the same places.

Here's AG(2,F3) again:

There's an obvious four-fold rotational symmetry, which moves all the points except the middle point, moves all the lines, but leaves the lines collectively in the same places. Less obvious perhaps is that reflection in the middle line, or translating everything one point to the right (with wraparound), are also symmetries. Remember that the straightness or otherwise of the lines doesn't matter - the only thing we're interested in is which points they are incident with.

So how can we find all the symmetries? Well, when we were looking at graphs, we used depth-first search, trying to create a pairing of source and target vertices, and backtracking whenever the adjacency between the source vertices didn't match the adjacency between the target vertices. Unfortunately, this method doesn't easily generalize to finite geometries. Exercise: Why not?

Instead, we will change the problem into a problem about graphs. Given a collection of points and lines, we can create an "incidence graph". This is the bipartite graph constructed as follows. On the left, we have a vertex for every point. On the right, we have a vertex for every line. We have an edge joining a point on the left to a line on the right just in the case that the point is incident with the line in our geometry.

Here's the code to create the incidence graph of AG(n,Fq):
incidenceGraphAG n fq = G vs es where
    points = ptsAG n fq
    lines = linesAG n fq
    vs = L.sort $ map Left points ++ map Right lines
    es = L.sort [ [Left x, Right b] | b <- lines, x <- closureAG b]

(Recall that a line is represented by two distinct points on it. "closureAG" calculates all the points on the line.)

Now, the trick is, that we can find the symmetries of the finite geometry by finding the symmetries of its incidence graph. For clearly, any symmetry of the finite geometry permutes the points among themselves, permutes the lines among themselves, and preserves incidence - hence it gives rise to a symmetry of the incidence graph. So in principle, we can find the symmetries of the finite geometry by doing the following:

> mapM_ print $ graphAuts $ incidenceGraphAG 2 f3
[[Left [0,0],Left [0,1]],[Left [1,0],Left [2,2],Left [1,1],Left [2,1],Left [1,2],Left [2,0]],[Right [[0,0],[1,0]],Right [[0,1],[1,0]],Right [[0,0],[1,1]],Right [[0,1],[1,1]],Right [[0,0],[1,2]],Right [[0,1],[1,2]]],[Right [[0,2],[1,0]],Right [[0,2],[1,2]],Right [[0,2],[1,1]]],[Right [[1,0],[1,1]],Right [[2,0],[2,1]]]]
[[Left [0,0],Left [0,2],Left [0,1]],[Left [2,0],Left [2,1],Left [2,2]],[Right [[0,0],[1,0]],Right [[0,2],[1,0]],Right [[0,1],[1,0]]],[Right [[0,0],[1,1]],Right [[0,2],[1,1]],Right [[0,1],[1,1]]],[Right [[0,0],[1,2]],Right [[0,2],[1,2]],Right [[0,1],[1,2]]]]

However, there are a couple of minor problems with this:
  1. We only really want to know what happens to the points. What happens to the lines follows from that. So we only want to know about the Left parts, not the Right parts. (Furthermore, if we were to show only the Left part, we then wouldn't need all those Lefts and Rights which are cluttering up the output.)
  2. It is just possible that the incidence graph might have some symmetries which interchange points and lines. This is a very interesting situation, which we'll discuss in due course. But it is a symmetry of the graph which does not correspond to a symmetry of the geometry.
For these reasons, HaskellForMaths provides an "incidenceAuts" function, which works just like graphAuts, except that it knows that the input is an incidence graph, so it avoids wasting time looking for point-line crossover symmetries, and outputs only the permutations of the points.

> mapM_ print $ incidenceAuts $ incidenceGraphAG 2 f3

This is a strong generating set for the symmetries of AG(2,F3). We could now go on to investigate further. We could ask what different types of symmetries there are, using the conjClassReps function, or how many symmetries there are, using the orderSGS function.

For example, among the elements of the SGS, some have obvious interpretations:

[[[1,0],[2,0]],[[1,1],[2,1]],[[1,2],[2,2]] - remember that this means that it swaps [1,0] and [2,0], swaps [1,1] and [2,1], and swaps [1,2] and [2,2]. So it can be thought of as a reflection in the line x = 0, or as a 2* stretch of the x-coordinate. It is given by the matrix [[2,0],[0,1]]. (In F3, 2 = -1, which is why this can be thought of as either a reflection or a stretch.)

[[[1,0],[1,1],[1,2]],[[2,0],[2,2],[2,1]]] is the shear given by the matrix [[1,1],[0,1]].
[Later: You might question whether a shear is a symmetry. We have defined a symmetry informally as a change that leaves things looking the same. But, for example, a shear in R^2 doesn't leave the unit square looking the same. Is it really a symmetry? Well, when we're studying geometries within combinatorics, we're only interested in the incidence between points and lines, and not, for example, distances. From this point of view, a shear is indeed a symmetry, because it preserves incidence. We will later look at what happens when we require symmetries to preserve more than just incidence.]

[[[0,0],[1,0],[0,1]],[[0,2],[2,0],[2,2]],[[1,1],[2,1],[1,2]]] cannot be represented by a 2*2 matrix, since it moves the origin. However, it can be represented by a 3*3 matrix, as follows:
[x']   [2 2 1] [x]
[y'] = [1 0 0] [y]
[ 1]   [0 0 1] [1]
This is the same as saying:
[x'] = [2 2] [x] + [1]
[y']   [1 0] [y]   [0]
So we see that it is a composite of some sort of shear, followed by a translation.

Indeed, we could conjecture that all symmetries of AG(n,Fq) are of this form - a linear transformation followed by a translation. Let's see whether we can confirm this conjecture by counting symmetries.

We have seen that some symmetries can be represented by a 2*2 matrix. In fact, the matrix must be non-singular, otherwise we won't have a permutation. The group of 2*2 non-singular matrices over Fq is the general linear group GL(2,Fq). How many elements does it have? Well, for the first row, we can choose any non-zero vector in Fq^2, of which there are q^2-1. For the second row, we can choose any vector in Fq^2 which is not linearly dependent on the one we already chose, of which there are q^2-q. Generalising to GL(n,Fq), the following code calculates the number of elements of GL(n,Fq):
orderGL n q = product [q^n - q^i | i <- [0..n-1] ]

Then, in addition to these linear transformations, all of which leave the origin fixed, we can do a translation. The number of translations of Fq^n is of course q^n.

A transformation consisting of a translation and a linear transformation is called an affine transformation. The group of all such is called the affine group, Aff(n,Fq). It is straightforward to calculate the number of elements of this group:
orderAff n q = q^n * orderGL n q
(Note that by multiplying the number of translations by the number of linear transformations, I have made an assumption that they are "semi-independent" of one another. This assumption is valid, but at this stage I don't want to spell out exactly what it means.)

Okay, so our conjecture is that all symmetries of AG(n,Fq) are affine transformations. Let's test our conjecture:
> orderSGS $ incidenceAuts $ incidenceGraphAG 2 f3
> orderAff 2 3
> orderSGS $ incidenceAuts $ incidenceGraphAG 2 f4
> orderAff 2 4
Uh-oh - what's going on here? We seem to have twice as many symmetries as we expected.

It took me quite a while to figure this one out actually. What's going on?

Well, remember that when we looked at extension fields, I mentioned that there can be field automorphisms. (For example, in the complex numbers, we have complex conjugation.) When we looked at the finite fields Fq, for q = p^n a prime power, I mentioned the Frobenius automorphism x -> x^p. Okay, well what's going on is that the field automorphisms of Fq give rise to further symmetries of AG(n,Fq). If q = p^n, then the number of such automorphisms is n, so the total number of symmetries of AG(n,Fq) will be n * orderAff n q.

And now that really does account for all symmetries of AG(n,Fq). Apart from the following exceptions:
> orderSGS $ incidenceAuts $ incidenceGraphAG 3 f2
> orderAff 3 2

What's going on here? Well the problem is that AG(n,F2) is degenerate. A line in AG(n,F2) has just two points - and every pair of points forms a line. So in fact, AG(n,F2) is just the complete graph K (2^n), and hence has (2^n)! symmetries.

That's it for now. Next time, finite projective geometries.

Sunday, 13 September 2009

Finite geometries, part 1: AG(n,Fq)

Time to recap a little. Through June and July, we looked at graphs and their symmetries. Then through August, we looked at finite fields. Now I want to look at finite geometries. Why?

Well, it's a continuation of my earlier investigation of symmetry. I chose graphs as the starting point because they're easy to understand. However, graphs are not the only combinatorial structure having symmetry - far from it. Finite geometries are important for several reasons:
1. We'll see that all symmetry groups turn out to be composed out of "atoms of symmetry", called "simple groups". There are several infinite families of finite simple groups - and a few "sporadic" ones that don't fit into any family. One of the infinite families are symmetries of finite projective geometries, which we will look at in due course. (I got the term "atoms of symmetry" from Ronan, Symmetry and the Monster, which is a nice easy read on this stuff.)
2. The symmetry groups of infinite geometries are very important in physics (especially particle physics). So if you like, you can pretend that what we're looking at here will help you understand physics.

This time, I want to look at affine geometries, but what we're aiming towards is projective geometries.

Within mathematics, there are probably many different definitions of geometry. Within combinatorics, we're just interested in the combinatorial structure, so roughly speaking, a geometry is just a collection of points, lines, planes and so on, together with a relation, called "incidence", saying which points are on which lines, and so on. Most of the time it's natural to think of a line as just being a set of points, in which case we can think of the incidence relation as just being the membership relation. However, this isn't always the best way to represent things in the code (or in the maths), as we shall see.

The most familiar geometry is the Euclidean plane, R^2. In this geometry, every pair of lines meets in at most one point. Then of course, there are Euclidean spaces of three or more dimensions. There are also non-Euclidean geometries. For example, spherical geometry is geometry on the surface of a sphere, with the lines being great circles. In this case, every pair of lines meets in exactly two points.

If you know about vector spaces, then you might think that Euclidean geometry is basically about the vector space R^n, with points, lines, planes being the zero-, one-, two-dimensional subspaces, and so on. However, this isn't quite right, because a vector space has a distinguished point, the origin, and every subspace of a vector space contains the origin. In Euclidean geometry there is no such distinguished point - we can have points, lines, planes which don't contain the origin. So we need to take the vector space R^n, and then "forget" the origin. If we do this, we get what is called an affine space. The subspaces of an affine space are the subspaces of the underlying vector space and their translates. The collection of points, lines, planes, and so on in an affine space, together with the incidence relation, is called an affine geometry.

We can construct finite affine geometries simply by replacing R by a finite field Fq in the above. In this way, we obtain the affine geometries AG(n,Fq). So in AG(n,Fq), we will have points, lines, planes and so on, but there will be only a finite number of each. Let's start with the points. Well, that's easy. The points of AG(n,Fq) are just the points Fq^n. The HaskellForMaths code goes like this:

ptsAG 0 fq = [[]]
ptsAG n fq = [x:xs | x <- fq, xs <- ptsAG (n-1) fq]

Actually, I think the stylish way to do this in Haskell would be:

ptsAG n fq = sequence $ replicate n fq

In any case, here's an example:

> :load Math.Combinatorics.FiniteGeometry
> ptsAG 2 f3

In other words, the points of AG(2,F3) form a 3*3 grid.

What about the lines? Well, the points of AG(2,F3) are the vector space F3^2. The lines are just the one-dimensional subspaces of this vector space, and their translates. Here's a picture:

Don't worry about the fact that the lines aren't straight. All we're concerned about is which points are incident with which lines. You might just want to convince yourself that every one-dimensional subspace of F3^2 is shown, as well as every translate.

Hopefully this picture makes the analogy between finite geometries and graphs pretty obvious. Points are analogous to vertices; lines are analogous to edges. The main difference is that unlike edges, lines can have more than two points on them. Indeed, if we consider just the points and lines (and not planes, hyperplanes, etc), then a finite geometry gives rise to a "hypergraph" - a graph, but where the "hyperedges" are allowed to contain more than two points. However, finite geometries have some additional structure, since they can also contain planes, hyperplanes etc.

Anyway, we want to calculate the symmetries of finite geometries, and in order to do that, we're going to need to work out what the lines are. First, let's think about how to represent the lines in code. There are several ways we could do it, including at least the following:
- We could represent a line as a list/set of the points on it. We will occasionally want to list all the points on a line, but as a way to represent a line, it's not very efficient.
- A line in the plane has an equation ax+by=c. We could represent a line by the triple (a,b,c). However, that would only work in two dimensions.
- We could represent a line by any pair of distinct points on the line. This will work in any number of dimensions, and will generalise to planes (represented by three points), and so on.

This last approach is the one we will take. So we would like two functions:
- given a pair of points in AG(n,Fq), return all points on the line that they generate
- given n, Fq, return all lines in AG(n,Fq) (represented as a pair of points)

The first is straightforward:

lineAG [p1,p2] = L.sort [ p1 <+> (c *> (p2 <-> p1)) | c <- fq ] where
    fq = eltsFq undefined

There are a couple of unfamiliar things here. <+>, <->, *> are the HaskellForMaths functions for adding two vectors, subtracting, and multiplying by a scalar. The second line is just a little bit of phantom type trickery, using type inference to magic up the elements of Fq. Note that the points are returned sorted.

Now to find all the lines. The naive way to do this is as follows:
- list all pairs of points in AG(n,Fq)
- but any pair of points on a line generates the same line, so each line will be listed many times
- so keep only those pairs of points which are the first two points on the line

Here's the code:

linesAG1 n fq = [ [x,y] | [x,y] <- combinationsOf 2 (ptsAG n fq),
                          [x,y] == take 2 (lineAG [x,y]) ]
(This is not a very efficient way to find the lines, but we don't yet have the background to do it more efficiently.)

For example:

> mapM_ print $ linesAG1 2 f3

Or we can list all the points on all the lines:

> mapM_ (print . lineAG) $ linesAG1 2 f3

You can verify for yourself that these are the lines shown in the picture above.

That'll do for now. Next time we'll look at how to work out the symmetries of AG(n,Fq).

Wednesday, 2 September 2009

Finite fields, part 2

Okay, so last time, we saw how to extend Q, the field of rational numbers, by adjoining a new element which is the zero of a polynomial in Q[X]. For example, we can adjoin sqrt 2, a zero of X^2-2, to obtain the field Q(sqrt2).

The time before that, we saw that for each prime p, there is a finite field Fp, consisting of the set {0,1,...,p-1}, with the arithmetic operations carried out modulo p.

This week we're going to bring these two ideas together, and look at algebraic extensions of Fp.

Algebraic extensions of Fp work just the same way as algebraic extensions of Q. First, find an irreducible polynomial over the base field K - that means, a polynomial f(x) in K[x], which can't be expressed as f(x) = g(x) * h(x), for g(x), h(x) in K[x] of lower degree. For example, whenever d is an element of K without a square root in K, then x^2-d is irreducible. Then, form the quotient ring K[x]/ - that means, work in K[x], but use the division with remainder algorithm to set f=0, which has the effect of making x act like a zero of f. For f(x)=x^2-d, this means x ends up acting like sqrt d.

For example, in F3, and also in F5, there is no square root of 2. So we can form the fields F3(sqrt2) or F5(sqrt2). We need to be a bit careful about terminology. 2 in F3, 2 in F5, and 2 in Q, are all different things - consequently sqrt2 in F3, in F5, and in Q are all different things. We mustn't make the mistake of thinking they're the same thing.

We saw last time that Q(sqrt2) consists of { a + b sqrt2 | a, b in Q }. Similarly, F5(sqrt2) consists of { a + b sqrt2 | a, b in F5 }. In particular, it follows that F5(sqrt2) has 25 elements. In general, a simple algebraic extension of Fp will have p^n elements, where n is the degree of the irreducible polynomial.

Over Q, we were able to form the fields Q(sqrt2), Q(sqrt3), Q(i), and so on. Somewhat surprisingly, over Fp, it turns out that any algebraic extension of degree n is isomorphic to any other. For example, F5(sqrt2) is isomorphic to F5(sqrt3). For this reason, we typically just talk about F25 (meaning, the field with 25 elements) - or in general, Fq, where q = p^n is a prime power.

However, in order to do arithmetic in F25 or in Fq, we clearly need to have some particular irreducible polynomial in mind. Otherwise, we won't know whether to reduce x^2 to 2, or to 3, for example. So for each p, n, we need to agree on an irreducible polynomial of degree n over Fp. There doesn't appear to be any natural way to choose, so an artificial way to choose has been devised, called the Conway polynomials.

So, using a little phantom type trickery as before, here are the first few finite fields of prime power order, as defined in the HaskellForMaths library:

data ConwayF4
instance PolynomialAsType F2 ConwayF4 where pvalue _ = convert $ x^2+x+1
type F4 = ExtensionField F2 ConwayF4
f4 = map Ext (polys 2 f2) :: [F4]
a4 = embed x :: F4

data ConwayF8
instance PolynomialAsType F2 ConwayF8 where pvalue _ = convert $ x^3+x+1
type F8 = ExtensionField F2 ConwayF8
f8 = map Ext (polys 3 f2) :: [F8]
a8 = embed x :: F8

data ConwayF9
instance PolynomialAsType F3 ConwayF9 where pvalue _ = convert $ x^2+2*x+2
type F9 = ExtensionField F3 ConwayF9
f9 = map Ext (polys 2 f3) :: [F9]
a9 = embed x :: F9

data ConwayF16
instance PolynomialAsType F2 ConwayF16 where pvalue _ = convert $ x^4+x+1
type F16 = ExtensionField F2 ConwayF16
f16 = map Ext (polys 4 f2) :: [F16]
a16 = embed x :: F16

data ConwayF25
instance PolynomialAsType F5 ConwayF25 where pvalue _ = convert $ x^2+4*x+2
type F25 = ExtensionField F5 ConwayF25
f25 = map Ext (polys 2 f5) :: [F25]
a25 = embed x :: F25

As we did with the fields Fp, we provide the functions f4, f8, f9, etc, which just return a list of the elements of the corresponding fields. a4, a8, a9 etc return the element that has been adjoined to the underlying field to make the extension field.

For example:

> f8
[0,a^2,a,a+a^2,1,1+a^2,1+a,1+a+a^2] :: [F8]
> a8 ^ 3
1+a :: F8

We can partially verify the claim that F25 = F5(sqrt2) = F5(sqrt3), by exhibiting square roots of 2 and 3 in F25:

> (2+a25)^2
2 :: F25
> (1+3*a25)^2
3 :: F25

Last time, I mentioned that extension fields can have automorphisms, without really spelling out what that means. Well, if L is an extension of K, then f is an automorphism of the extension if for a,b in L, c in K, f(a+b) = f(a)+f(b), f(a*b) = f(a)*f(b), etc, and f(c*a)= c*f(a). In other words, f(L) looks the same as L, viewed from K.

The finite fields Fq (q = p^n) have an automorphism called the Frobenius automorphism, defined by f(x)=x^p. For example:

> f9
> map frobeniusAut f9

Finally, a word about efficiency. The HaskellForMaths implementation of fields Fq is very natural from a mathematical point of view, but it isn't very efficient.

Firstly, we are working with polynomials with coefficients in Fp. To multiply two degree n polynomials together, we have to do O(n^2) additions and multiplications of coefficients. If the coefficients are in Fp, each one of these involves a (`mod` p) operation. It would be better to do the polynomial arithmetic over the integers, and only do the `mod` p at the end. We would then only have to do O(n) (`mod` p) operations.

Secondly, we could consider implementing the fields Fq using Zech logarithms instead.

Anyway, it doesn't matter too much for the moment, as I only want to use the finite fields to construct combinatorial structures. But sometime I might try to write a more efficient implementation.

Next time: finite geometries.
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