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Let be a field. A definable set over is a set of the form

where is a natural number, and is a predicate involving the ring operations of , the equality symbol , an arbitrary number of constants and free variables in , the quantifiers , boolean operators such as , and parentheses and colons, where the quantifiers are always understood to be over the field . Thus, for instance, the set of quadratic residues

is definable over , and any algebraic variety over is also a definable set over . Henceforth we will abbreviate “definable over ” simply as “definable”.

If is a finite field, then every subset of is definable, since finite sets are automatically definable. However, we can obtain a more interesting notion in this case by restricting the *complexity* of a definable set. We say that is a *definable set of complexity at most * if , and can be written in the form (1) for some predicate of length at most (where all operators, quantifiers, relations, variables, constants, and punctuation symbols are considered to have unit length). Thus, for instance, a hypersurface in dimensions of degree would be a definable set of complexity . We will then be interested in the regime where the complexity remains bounded, but the field size (or field characteristic) becomes large.

In a recent paper, I established (in the large characteristic case) the following regularity lemma for dense definable graphs, which significantly strengthens the Szemerédi regularity lemma in this context, by eliminating “bad” pairs, giving a polynomially strong regularity, and also giving definability of the cells:

Lemma 1 (Algebraic regularity lemma)Let be a finite field, let be definable non-empty sets of complexity at most , and let also be definable with complexity at most . Assume that the characteristic of is sufficiently large depending on . Then we may partition and with , with the following properties:

My original proof of this lemma was quite complicated, based on an explicit calculation of the “square”

of using the Lang-Weil bound and some facts about the étale fundamental group. It was the reliance on the latter which was the main reason why the result was restricted to the large characteristic setting. (I then applied this lemma to classify expanding polynomials over finite fields of large characteristic, but I will not discuss these applications here; see this previous blog post for more discussion.)

Recently, Anand Pillay and Sergei Starchenko (and independently, Udi Hrushovski) have observed that the theory of the étale fundamental group is not necessary in the argument, and the lemma can in fact be deduced from quite general model theoretic techniques, in particular using (a local version of) the concept of stability. One of the consequences of this new proof of the lemma is that the hypothesis of large characteristic can be omitted; the lemma is now known to be valid for arbitrary finite fields (although its content is trivial if the field is not sufficiently large depending on the complexity at most ).

Inspired by this, I decided to see if I could find yet another proof of the algebraic regularity lemma, again avoiding the theory of the étale fundamental group. It turns out that the spectral proof of the Szemerédi regularity lemma (discussed in this previous blog post) adapts very nicely to this setting. The key fact needed about definable sets over finite fields is that their cardinality takes on an essentially discrete set of values. More precisely, we have the following fundamental result of Chatzidakis, van den Dries, and Macintyre:

Proposition 2Let be a finite field, and let .

- (Discretised cardinality) If is a non-empty definable set of complexity at most , then one has
where is a natural number, and is a positive rational number with numerator and denominator . In particular, we have .

- (Definable cardinality) Assume is sufficiently large depending on . If , and are definable sets of complexity at most , so that can be viewed as a definable subset of that is definably parameterised by , then for each natural number and each positive rational with numerator and denominator , the set
is definable with complexity , where the implied constants in the asymptotic notation used to define (4) are the same as those that appearing in (3). (Informally: the “dimension” and “measure” of depends definably on .)

We will take this proposition as a black box; a proof can be obtained by combining the description of definable sets over pseudofinite fields (discussed in this previous post) with the Lang-Weil bound (discussed in this previous post). (The former fact is phrased using nonstandard analysis, but one can use standard compactness-and-contradiction arguments to convert such statements to statements in standard analysis, as discussed in this post.)

The above proposition places severe restrictions on the cardinality of definable sets; for instance, it shows that one cannot have a definable set of complexity at most and cardinality , if is sufficiently large depending on . If are definable sets of complexity at most , it shows that for some rational with numerator and denominator ; furthermore, if , we may improve this bound to . In particular, we obtain the following “self-improving” properties:

- If are definable of complexity at most and for some , then (if is sufficiently small depending on and is sufficiently large depending on ) this forces .
- If are definable of complexity at most and for some and positive rational , then (if is sufficiently small depending on and is sufficiently large depending on ) this forces .

It turns out that these self-improving properties can be applied to the coefficients of various matrices (basically powers of the adjacency matrix associated to ) that arise in the spectral proof of the regularity lemma to significantly improve the bounds in that lemma; we describe how this is done below the fold. We also make some connections to the stability-based proofs of Pillay-Starchenko and Hrushovski.

I’ve just uploaded to the arXiv my article “Algebraic combinatorial geometry: the polynomial method in arithmetic combinatorics, incidence combinatorics, and number theory“, submitted to the new journal “EMS surveys in the mathematical sciences“. This is the first draft of a survey article on the polynomial method – a technique in combinatorics and number theory for controlling a relevant set of points by comparing it with the zero set of a suitably chosen polynomial, and then using tools from algebraic geometry (e.g. Bezout’s theorem) on that zero set. As such, the method combines algebraic geometry with combinatorial geometry, and could be viewed as the philosophy of a combined field which I dub “algebraic combinatorial geometry”. There is also an important extension of this method when one is working overthe reals, in which methods from algebraic topology (e.g. the ham sandwich theorem and its generalisation to polynomials), and not just algebraic geometry, come into play also.

The polynomial method has been used independently many times in mathematics; for instance, it plays a key role in the proof of Baker’s theorem in transcendence theory, or Stepanov’s method in giving an elementary proof of the Riemann hypothesis for finite fields over curves; in combinatorics, the nullstellenatz of Alon is also another relatively early use of the polynomial method. More recently, it underlies Dvir’s proof of the Kakeya conjecture over finite fields and Guth and Katz’s near-complete solution to the Erdos distance problem in the plane, and can be used to give a short proof of the Szemeredi-Trotter theorem. One of the aims of this survey is to try to present all of these disparate applications of the polynomial method in a somewhat unified context; my hope is that there will eventually be a systematic foundation for algebraic combinatorial geometry which naturally contains all of these different instances the polynomial method (and also suggests new instances to explore); but the field is unfortunately not at that stage of maturity yet.

This is something of a first draft, so comments and suggestions are even more welcome than usual. (For instance, I have already had my attention drawn to some additional uses of the polynomial method in the literature that I was not previously aware of.)

Once again it is time to roll over the previous discussion thread, which has become rather full with comments. The paper is nearly finished (see also the working copy at this subdirectory, as well as the rest of the directory), but several people are carefully proofreading various sections of the paper. Once all the people doing so have signed off on it, I think we will be ready to submit (there appears to be no objection to the plan to submit to Algebra and Number Theory).

Another thing to discuss is an invitation to Polymath8 to write a feature article (up to 8000 words or 15 pages) for the Newsletter of the European Mathematical Society on our experiences with this project. It is perhaps premature to actually start writing this article before the main research paper is finalised, but we can at least plan how to write such an article. One suggestion, proposed by Emmanuel, is to have individual participants each contribute a brief account of their interaction with the project, which we would compile together with some additional text summarising the project as a whole (and maybe some speculation for any lessons we can apply here for future polymath projects). Certainly I plan to have a separate blog post collecting feedback on this project once the main writing is done.

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