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Ben Green and I have just uploaded our joint paper, “The distribution of polynomials over finite fields, with applications to the Gowers norms“, to the arXiv, and submitted to Contributions to Discrete Mathematics. This paper, which we first announced at the recent FOCS meeting, and then gave an update on two weeks ago on this blog, is now in final form. It is being made available simultaneously with a closely related paper of Lovett, Meshulam, and Samorodnitsky.
In the previous post on this topic, I focused on the negative results in the paper, and in particular the fact that the inverse conjecture for the Gowers norm fails for certain degrees in low characteristic. Today, I’d like to focus instead on the positive results, which assert that for polynomials in many variables over finite fields whose degree is less than the characteristic of the field, one has a satisfactory theory for the distribution of these polynomials. Very roughly speaking, the main technical results are:
- A regularity lemma: Any polynomial can be expressed as a combination of a bounded number of other polynomials which are regular, in the sense that no non-trivial linear combination of these polynomials can be expressed efficiently in terms of lower degree polynomials.
- A counting lemma: A regular collection of polynomials behaves as if the polynomials were selected randomly. In particular, the polynomials are jointly equidistributed.
Recently, I had tentatively announced a forthcoming result with Ben Green establishing the “Gowers inverse conjecture” (or more accurately, the “inverse conjecture for the Gowers uniformity norm”) for vector spaces over a finite field , in the special case when p=2 and when the function for which the inverse conjecture is to be applied is assumed to be a polynomial phase of bounded degree (thus , where is a polynomial of some degree ). See my FOCS article for some further discussion of this conjecture, which has applications to both polynomiality testing and to various structural decompositions involving the Gowers norm.
This conjecture can be informally stated as follows. By iterating the obvious fact that the derivative of a polynomial of degree at most d is a polynomial of degree at most d-1, we see that a function is a polynomial of degree at most d if and only if
for all . From this one can deduce that a function bounded in magnitude by 1 is a polynomial phase of degree at most d if and only if the Gowers norm
is equal to its maximal value of 1. The inverse conjecture for the Gowers norm, in its usual formulation, says that, more generally, if a function bounded in magnitude by 1 has large Gowers norm (e.g. ) then f has some non-trivial correlation with some polynomial phase g (e.g. for some ). Informally, this conjecture asserts that if a function has biased derivatives, then one should be able to “integrate” this bias and conclude that the function is biased relative to a polynomial of degree d. The conjecture has already been proven for . There are analogues of this conjecture for cyclic groups which are of relevance to Szemerédi’s theorem and to counting linear patterns in primes, but I will not discuss those here.
At the time of the announcement, our paper had not quite been fully written up. This turned out to be a little unfortunate, because soon afterwards we discovered that our arguments at one point had to go through a version of Newton’s interpolation formula, which involves a factor of d! in the denominator and so is only valid when the characteristic p of the field exceeds the degree. So our arguments in fact are only valid in the range , and in particular are rather trivial in the important case ; my previous announcement should thus be amended accordingly.
Earlier this month, in the previous incarnation of this page, I posed a question which I thought was unsolved, and obtained the answer (in fact, it was solved 25 years ago) within a week. Now that this new version of the page has better feedback capability, I am now tempted to try again, since I have a large number of such questions which I would like to publicise. (Actually, I even have a secret web page full of these somewhere near my home page, though it will take a non-trivial amount of effort to find it!)
Perhaps my favourite open question is the problem on the maximal size of a cap set – a subset of ( being the finite field of three elements) which contains no lines, or equivalently no non-trivial arithmetic progressions of length three. As an upper bound, one can easily modify the proof of Roth’s theorem to show that cap sets must have size (see e.g. this paper of Meshulam). This of course is better than the trivial bound of once n is large. In the converse direction, the trivial example shows that cap sets can be as large as ; the current world record is , held by Edel. The gap between these two bounds is rather enormous; I would be very interested in either an improvement of the upper bound to , or an improvement of the lower bound to . (I believe both improvements are true, though a good friend of mine disagrees about the improvement to the lower bound.)