I think that this is a great idea, and am very happy that, as the web site is now public, two of my own lecture notes can be found among the inaugural set! The highlight of the current selection is however undoubtedly “A singular mathematical promenade”, by Étienne Ghys, his beautiful book on graphs of polynomials, Newton’s method, Puiseux expansions, divergent series, and much much else that I have yet to see (I’m only one-third through looking at it…)

Hopefully, the Open Math Notes collection will grow to contain many further texts. The example of the book of Ghys is already an illustration of how useful this may be — although it is also available on his home page, one doesn’t necessarily visit it frequently enough to notice it…

Two final whimsical remarks to conclude: (1) among the six authors currently represented [Update (four hours later): this has already changed!], three [Update: four] (at least) are French; (2) one of my set of notes promises a randonnée, and Ghys’s book is a promenade — clearly, one can think of mathematics as a journey…

]]>Although I mentioned this briefly in my talk in the recent conference in honor of A. Lubotzky’s 60th birthday (where I was greatly honored to be invited!), it’s only last week or so that I somehow finally did the obvious thing, namely some experiments with **Magma** to see if the property of having proportion of -cycles is widespread.

When doing this, the first thing to realize (which I only did when P.P. Palfy pointed it out during that conference…) is that this condition, for a subgroup of , is equivalent with containing a unique conjugacy class of -cycles (simply because the centralizer of an -cycle, either in or in the bigger symmetric group, is the cyclic subgroup of order that it generates). So we can coin a solid decent name for these groups: we call them the unicyclic permutation groups.

**Magma** has a database of all transitive permutation groups of degree up to (even if one installs an extra specific database). Experimentation shows that there often exist unicyclic groups that are not the symmetric group. For instance, for degree , there exist (up to isomorphism) transitive permutation groups, and of them contain at least one -cycle, and of them are unicyclic.

However, the same experiments show that if we restrict our attention to primitive subgroups, then the situation is very different: either there is only the symmetric group , or there are two unicyclic groups. Amusingly the second case occurs if and only if is prime (an amusing primality test…)

One can indeed prove that these facts (which I did first experimentally notice) are correct, but it is not cheap. More precisely, from the Classification of Finite Simple Groups, Feit and later Jones (see for instance this paper which has a slightly more general result) deduced a full classification of primitive permutation groups that contain an -cycle. It is restrictive enough that one can then relatively easily exclude all groups except the symmetric groups and, when is prime, the group of transformations of the finite field .

That’s a nice result, but is it relevant for our original motivation? I think so, because (as Will Sawin pointed out), one can prove in considerable generality that the Galois group in (say) the Bunyakowski problem is primitive, before or without computing it exactly, because it suffices to prove that it is -transitive, and this is accessible to the diophantine interpretation using Chebotarev’s Density Theorem, as in Entin’s approach. (Indeed, Entin goes on to check that the Galois group is often -transitive or so, and from the Classification of Finite Simple Groups, deduces that the group contains the alternating group). Hence, if the degree is not prime, granted that one first applies Entin’s method to check primitivity, having Galois group is equivalent to the natural version of Bunyakowski’s conjecture in the large finite field limit…

If is prime, well one can say that if is not solvable, then it must be the symmetric group (assuming is at least …), although that is not as good — is there a better way to do it?

We consider random permutations in the symmetric group , and write for the number of cycles in the disjoint cycle representation of . It is a well-known fact (which may be due to Feller) that, viewed as a random variable on (with uniform probability measure), there is a decomposition in law

where is a Bernoulli random variable taking value with probability and with probability , and moreover the ‘s are independent.

This decomposition is (in the sources I know) usually proved using the “Chinese Restaurant Process” to describe random permutations ( guests numbered from to enter a restaurant with circular tables; they successively sit either at the next available space of one of the tables already occupied, or pick a new one, with the same probability; after all guests are seated, reading the cycles on the occupied tables gives a uniformly distributed element of .) If one is only interested in , then the decomposition above is equivalent to the formula

for the probability generating function of . (This formula comes up in mod-poisson convergence of and in analogies with the Erdös-Kac Theorem, see this blog post).

Here is an algebraic proof of this generating function identity. First, is a polynomial, so it is enough to prove the formula when is a non-negative integer. We can do this as follows: let be a vector space of dimension over . Then define and view it as a representation space of where the symmetric group permutes the factors in the tensor product. Using the “obvious” basis of , it is elementary that the character of this representation is the function on (the matrix representing the action of is a permutation matrix). Hence, by character theory, is the dimension of the -invariant subspace of . Since the representation is semisimple, this is the dimension of the coinvariant space, which is the -th symmetric power of This in turn is well-known (number of monomials of degree in variables), and we get

as claimed!

where is the parameter: if we consider the number (say ) of solutions over a finite field with elements, then we have

provided is not in the set . This was a fact that Fouvry, Michel and myself found out when working on our paper on algebraic twists of modular forms.

At the time, I knew one proof, based on computations with Magma: the curves above “are” elliptic curves, and Magma found an isogeny between the curves with parameters and , which implies that they have the same number of points by elementary properties of elliptic curves over finite fields.

By now, however, I am aware of three other proofs:

- The most elementary, which motivates this post, was just recently put on arXiv by T. Budzinski and G. Lucchini Arteche; it is based on the methods of Chevalley’s proof of Warning’s theorem: it computes modulo , proving the desired identity modulo , and then concludes using upper bounds for the number of solutions, and for its parity, to show that this is sufficient to have the equality as integers. Interestingly, this proof was found with the help of high-school students participating in the Tournoi Français des Jeunes Mathématiciennes et Mathématiciens. This is a French mathematical contest for high-school students, created by D. Zmiaikou in 2009, which is devised to look much more like a “real” research experience than the Olympiads: groups of students work together with mentors on quite “open-ended” questions, where sometimes the answer is not clear (see for instance the 2016 problems here).
- A proof using modular functions was found by D. Zywina, who sent it to me shortly after the first post (I have to look for it in my archives…)
- Maybe the most elegant argument comes by applying a more general result of Deligne and Flicker on local systems of rank on the projective line minus four points with unipotent tame ramification at the four missing points (the first cohomology of the curves provide such a local system, the missing points being ). Deligne and Flicker prove (Section 7 of their paper, esp. Prop. 7.6), using a very cute game with products of matrices and invariant theory, that if is such a local system and is any automorphism of the projective line that permutes the four points, then is isomorphic to .

Not too bad a track record for such a simple-looking question… Whether there is a bijective proof remains open, however!

]]>The last issue has a special focus on Open Science in its various forms. For some reason, although there is no discussion of Polymath *per se*, the editors decided to have a picture of a mathematician working on Polymath as an illustration, and they asked me if they could make such a picture with me, and in fact two of them (the photographer is Valérie Chételat) appear in the magazine. Readers may find it amusing to identify which particular comment of the Polymath 8 blog I am feigning to be studying in those pictures…

Besides (and of greater import than) this, I recommend looking at the illustration pages 6 and 7,

which is a remarkably precise computer representation of the 44000 trees in a forest near Baden, each identified and color-coded according to its species… (This is done by the team of M. Schaepman at the University of Zürich).

]]>which does not mean that geckos were not displaying themselves most beautifully also…

]]>is a Jordan block of size 2 with respect to the eigenvalue ?

I have the vague impression that most elementary textbooks in Germany (I taught linear algebra last year…) use , but for instance Bourbaki (Algèbre, chapitre VII, page 34, définition 3, in the French edition) uses , and so does Lang’s “Algebra”. Is it then a cultural dichotomy (again, like spines of books)?

I have to admit that I tend towards myself, because I find it much easier to remember a good model for a Jordan block: over the field , take the vector space , and consider the linear map defined by . Then the matrix of with respect to the basis is the Jordan block in its lower-triangular incarnation. The point here (for me) is that passing from to is nicely “inductive”: the formula for the linear map is “independent” of , and the bases for different are also nicely meshed. (In other words, if one finds the Jordan normal form using the classification of modules over principal ideal domains, one is likely to prefer the lower-triangular version that “comes out” more naturally…)

]]>you may find interesting to know that the first five volumes of the definitive catalogue of his paintings are freely available online on the Rembrandt Database website.

]]>for some (explicitly predicted) constant , called sometimes the “singular series”.

Except if has degree one, this problem is very much open. But it makes sense to translate it to a more geometric setting of polynomials over finite fields, and this leads (as is often the case) to problems that are more tractable. The translation is straightforward: instead of , one considers the ring of polynomials over a finite field with elements, instead of , one considers a polynomial , and then the question is to determine asymptotically how many polynomials of given degree are such that is an irreducible polynomial.

The reason the problem becomes more accessible is that there is an algebraic criterion for a polynomial with coefficients in a finite field to be irreducible: if we look at the natural action of the Frobenius automorphism on the set of roots of the polynomial, then is irreducible if and only if this action “is” a cycle of length . This is especially useful for the variant of the Schinzel problem where the *size* of the finite field is varying, whereas the degree of the polynomials remains fixed, since in that case the variation of the action of the Frobenius on the roots of the polynomial is encoded in a group homomorphism from the Galois group of the function field of the parameter space to the symmetric group on letters. (This principle goes back at least to work of S.D. Cohen on Hilbert’s Irreducibility Theorem).

If we apply this principle in the Schinzel setting, this means that we consider specialized polynomials for some fixed polynomial , where runs over polynomials of a fixed degree , but ranges over powers of a fixed prime. “Generically”, the polynomial has some fixed degree , and is squarefree. If we interpret the parameter space geometrically, the content of the previous paragraph is that we have a group homomorphism

from the fundamental group of to the symmetric group. Then the Chebotarev Density Theorem solves, in principle, the problem of counting the number of irreducible specializations in the large limit: essentially (omitting the distinction between geometric and arithmetic fundamental groups), the asymptotic proportion of such that is irreducible converges as to the proportion, in the image of , of the elements that are -cycles in . If the homomorphism is surjective, then this means that the probability that is irreducible is about . This is the expected answer in many cases, because this is also the probability that a random polynomial of degree is irreducible.

All this has been used by a number of people (including Hall, Pollack, Bary-Soroker, and most successfully Entin). However, there is a nice geometric interpretation that I haven’t seen elsewhere. To see it, we go back to and the action of Frobenius on its roots that will determine if is irreducible. A root of is an element such that

where we view as a two-variable polynomial. In other words, is the first coordinate of a point that belongs to the intersection of the graph of in the plane, and the plane affine curve with equation . Since the Frobenius will permute these intersection points in the same way that it permutes the roots of , we can interpret the Schinzel Problem, in that context, as asking about the “variation” of this Galois action as varies and the curve is fixed.

This point of view immediately suggests some generalizations: there is no reason to work over a finite field (any field will do), the base curve (which is implicitly the affine line where polynomials live) can be changed to another (open) curve ; the point at infinity, where polynomials have their single pole, might also be changed to any effective divisor with support the complement of in its smooth projective model (e.g., allowing poles at and on the projective line); and may be any (non-vertical) curve in . For instance (to see that this generalization is not pointless), take any curve , and define . Then the intersection of the graph of a function on and is the set of zeros of . The problem becomes something like figuring out the “generic” Galois group of the splitting field of this set of zeros. (E.g., the Galois group of a complicated elliptic function defined over …)

In fact this special case was (with different motivation and terminology) considered by Katz in his book “Twisted *L*-functions and monodromy” (see Chapter 9). Katz shows that if the (fixed) effective divisor used to define the poles of the functions considered has degree , where is the genus of the smooth projective model of , then the image of Galois is the full symmetric group (his proof is rather nice, using character sums on the Jacobian…)

The general case, on the other hand, does not seem to have been considered before. In the recent note that I’ve written on the subject, I use quite elementary arguments with Lefschetz pencils / Morse-like functions (again inspired by results of Katz and Katz-Rains) to show that in very general conditions, the image of the fundamental group is again the full symmetric group. This gives the asymptotic for this geometric Schinzel problem in this generality over finite fields. (In the classical case, this was essentially done by Entin, though the conditions of applicability are not exactly the same).

I recently gave a talk about this in Berlin, and the slides might be a good introduction to the ideas of the proof for interested readers…

As I mention at the end of those slides, the next step is of course to think about the fixed finite field case, where the degree of the polynomials tends to infinity. This seems, even geometrically, to be quite an interesting problem…

[Update: after I wrote this post, I remembered that in fact the (qualitative) problem of representing primes with one polynomial that I consider here is actually *Bunyakowski’s Problem*, and that the Schinzel Hypothesis is the qualitative statement for a finite set of polynomials… The quantitative versions of both are usually called the Bateman-Horn conjecture. So my terminology is multiply inaccurate…]