I haven’t posted for a while; I have been in China where the firewall allows me to read WordPress but not to post. Normal service now resumed (hopefully).
Given four real polynomials, all of which vanish at the origin, suppose that p1(x) < p2(x) < p3(x) < p4(x) for small negative values of x. What are the possible orderings of the four polynomials for small positive values of x? Perhaps one should think of four marathon runners going into a tunnel; when they emerge, how will their positions have changed?
You might think that this problem should have been solved centuries ago. But apparently it wasn’t; the solution was given by Maxim Kontsevich in 2009.
To save writing, let me give the initial order as 1234. As a warmup, let me note that, in the case of three polynomials, all permutations of the initial order are possible. But for four polynomials, this is not the case; Kontsevich showed that the orders 2413 and 3142 are impossible, while all other orders are possible.
This is not very difficult: Kontsevich proved it on a Paris Métro ticket. You might like to try it yourself.
At this stage, my friends who work on permutation patterns are probably pricking up their ears, noticing that this means that the permutations realised by n polynomials avoid these two 4-element patterns, and possibly even conjecturing that, for general n, any permutation which avoids these two patterns can be realised.
This is indeed the case. Indeed this is a very interesting class of permutations: it consists of all those permutations obtained from the permutation 1 by direct sum and skew sum, or those whose permutation graph is a cograph (I prefer to call these graphs N-free, since they do not contain an induced path of length 3, and are closely related to the N-free posets, which arise in experimental design as the structures which can be obtained by nesting and crossing). Their occurrence in the polynomial interchange problem is due to Étienne Ghys.
I have just described to you the first 20 pages or so of a remarkable book by Ghys, entitled A singular mathematical promenade. Two truly remarkable things about this book: the author is not afraid of either algebraic geometry or combinatorics, and is happy to mix them together; and he has put the book on the arXiv (number 1612.06373), from where it can be freely downloaded.
There is, of course, much more in the book’s 300 pages than just the above. The main concern is the structure of an algebraic curve in the neighbourhood of a singular point. In such a neighbourhood, the curve consists of a number of smooth branches, so already we have something resembling four curves y = Pi(x) passing through the origin). Ghys’ aim is to give a complete description of how the branches of a real algebraic curve can intersect a small circle around a singular point, where the first forbidden pattern occurs for five branches. By analogy with N-free graphs, you may now suspect that certain ternary structures like pentagon-free two-graphs will arise.
Indeed this is so, but I don’t want to give too much away. You should read the book yourself!
But just to continue a little more. One of the great puzzles for historians of Greek mathematics is the mysterious statement by Hipparchus, that the number of propositions which can be formed from 10 simple propositions is 103,049 “on the affirmative side” and 310,952 “on the negative side”. It has been claimed that no sense can be extracted from these figures; but 103,049 is the tenth “small Schroeder number”, and these numbers doubled count certain kinds of trees which represent compound propositions formed from ten simple propositions using only the connectives “and” and “or”. (There is an obvious duality given by swapping the two connectives, and the number given counts the dual pairs.)
As Ghys points out,
Most mathematicians, including myself, have a naive idea about Greek mathematics. We believe that it only consists of Geometry, in the spirit of Euclid. The example of the computation by Hipparchus of the tenth Schroeder number may be a hint that the Ancient Greeks had developed a fairly elaborate understanding of combinatorics.
This example is intended to give you some of the flavour of the book. As the title describes, it is a promenade, around a singular point of a plane curve or around some related and attractive areas of mathematics with surprising interconnections.
So go for a little stroll with Étienne Ghys; he is an excellent guide!
To conclude, I am very grateful to my friend Yaokun Wu for drawing my attention to this remarkable book, during my recent stay in Shanghai.