Définition tendancieuse

(I guess the title of this post would translate as something like “Biased definition” in English; according to the OED, “tendencious” does exist, but is ascribed as coming from the German “tendenziös”)

My son is currently reading an abridged version of Les Misérables for his French class. This is a text intended for schools and comes (among other things) with explanations of “hard words”. While glancing through it recently, I noticed the following striking instance:

Le hasard, c’est-à-dire la providence(1)

where the footnote translates, in lapidary style:

1. Providence = chance

(in English: Providence = luck). I may not know a lot about Victor Hugo, but it’s as clear as day to me that nothing could be further away from his use of the word “providence” than the idea that this is mere luck.

This reminded me of another definition I have seen in the French Larousse Universel encyclopedic dictionary from 1922 concerning the German language (see here in the middle of the page):

Langue: … une langue laborieuse… de là un certain manque de rapidité et de précision dans l’expression de la pensée.

(Or: … a clumsy language… from this comes a certain lack of speed and precision in the expression of one’s thoughts.)

This is actually a very nice book overall, with wonderfully useful illustrations to understand what, say, a “face-à-main” is, or to remind yourself of the important classification of “chapeaux bicornes”

Bicornes
Bicornes

(the scan I am linking to does not do justice to the book; one can download the PDFs of the two volumes, but each is a huge file of at least 250 MB, and the quality is also not so great — but the books become searchable).

Condorcet, Dedekind, Minkowski

One of my great pleasures in life is to walk leisurely down from my office about 30 minutes before the train (to Paris, or Göttingen, or Basel, or what you will) starts, browse a few minutes in one of the second-hand bookstores on the way, and get on the train with some wonderfully surprising book, known or not.

A few months ago, I found “Condorcet journaliste, 1790-1794”,

Condorcet
Condorcet

which one cannot call a well-known book. It is the printed version of the 1929 thesis (at the École des Hautes Études Sociales) of Hélène Delsaux, and its main goal is to survey and discuss in detail all the journal articles that Condorcet, that particularly likable character of the French revolution (about the only one to be happily married, one of the very few in favor of a Republic from the outset, and — amid much ridicule — a supporter of vote for women), wrote during those years.

Condorcet was also known at the time as a mathematician; hence this remarkable quote from the book in question:

Il est généralement admis que rien ne dessèche le coeur comme l’étude approfondie des mathématiques…

or in a rough translation

It is a truth universally acknowledged that nothing shrivels the heart more than the deep study of mathematics… [Ed. Note: what about real estate?]

This book cost me seven Francs. More recently, my trip to the bookstore was crowned by the acquisition of a reprint of R. Dedekind’s Stetigkeit und irrationale Zahlen” and “Was sind und was sollen die Zahlen” (five Francs)

Dedekind
Dedekind

and of a first edition (Teubner Verlag, Leipzig, 1907) of Minkowski’s “Diophantische Approximationen”

Minkowski
Minkowski

for the princely sum of thirty-eight Francs.

The content of Minkowski’s book is not at all what the title might suggest. There are roughly two parts, one concerned with the geometry of numbers, and the second with algebraic number theory. In both cases, the emphasis is on dimensions 2 and (indeed, especially) 3, so cubic fields are at the forefront of the discussion in the second part. This leads to a much greater number of pictures (there are 82) than a typical textbook of algebraic number theory would have today. Here are two examples,

Minkowski
Minkowski
Minkowski
Minkowski

and here is Minkowski’s description of the Minkowski functional (or gauge) of a convex set:

Minkowski
Minkowski

AMS Open Math Notes

When I was attending the conference in honor of Alex Lubotzky’s 60th birthday, Karen Vogtmann, who was also there, told me of the Open Math Notes repository, a new project of the AMS that she was involved with. This is meant to be a collection of (mostly) lecture notes, such as many mathematicians write for a course, but which are not published (nor necessarily meant to be published). So they can be incomplete, they might contain mistakes, and may more generally be subject to all the slings and arrows that mathematical writing is heir to. (See the web site for more information, submission guidelines, etc…)

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…

Unicyclic groups

When working on large finite field versions of conjectures like Bunyakowski’s or Schinzel’s or Bateman-Horn’s, one small point that always somewhat tickled me was that the answer, which follows from the computation of some Galois group, seems to be weaker than this actual computation. For instance, to get the “right” answer in Bunyakowski’s Conjecture, one computes a Galois group G that is in a natural way a subgroup of the symmetric group on n letters, and it is enough to show that the proportion of n-cycles in G is exactly 1/n. It is of course sufficient to show that G is the full symmetric group (which contains the right number of n-cycles), but a priori, this is a weaker condition.

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 1/n of n-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 G of \mathrm{Sym}_n, is equivalent with G containing a unique conjugacy class of n-cycles (simply because the centralizer of an n-cycle, either in G or in the bigger symmetric group, is the cyclic subgroup of order n 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 31 (even 32 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 20, there exist (up to isomorphism) 1117 transitive permutation groups, and 332 of them contain at least one 20-cycle, and 35 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 \mathrm{Sym}_n, or there are two unicyclic groups. Amusingly the second case occurs if and only if n 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 n-cycle. It is restrictive enough that one can then relatively easily exclude all groups except the symmetric groups and, when n is prime, the group of transformations x\mapsto ax+b of the finite field \mathbf{F}_n.

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 2-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 7-transitive or so, and from the Classification of Finite Simple Groups, deduces that the group contains the alternating group). Hence, if the degree n is not prime, granted that one first applies Entin’s method to check primitivity, having Galois group \mathrm{Sym}_n is equivalent to the natural version of Bunyakowski’s conjecture in the large finite field limit…
If n is prime, well one can say that if G is not solvable, then it must be the symmetric group (assuming n is at least 5…), although that is not as good — is there a better way to do it?