Scope and Limits

8.1 What the three lemmas explain

The table of §1.2 can now be read with the proofs in hand. Each theorem of Chapters 3 to 6 is a counting step, which produces a polynomial (or, in Chapter 3, an explicit product polynomial that needs no counting), followed by a zeros step or a rank step, which shows the polynomial impossible, with one problem-specific bridge between them. The bridges are:

• Sums of residues: the product $\prod_{c\in C}(x+y-c)$ vanishes on $A\times B$ exactly when $C\supseteq A+B$, and a single binomial coefficient of the product is computable. The zeros step is the Nullstellensatz.
• Kakeya: a polynomial vanishing on a Kakeya set vanishes on a line in every direction, so its top homogeneous part vanishes at every direction, i.e. everywhere. The zeros step is Schwartz–Zippel on the grid $\mathbb F_q^n$, with multiplicity to remove the $n!$.
• Joints: a polynomial vanishing on many joints per line vanishes on the lines, so its gradient vanishes at the joints, so its derivatives vanish on the same set with lower degree. The zeros step is minimality of degree.
• Cap sets: on a cap set the matrix $P(\mathbf x+\mathbf y)$ is diagonal. The step is rank, not zeros.

Two features of this pattern deserve comment. First, the zeros step has two forms, a grid form (Schwartz–Zippel) and a minimality form (Chapter 5), and they have different domains of validity: the grid form needs a finite grid and gives bounds in terms of the field size; the minimality form works over any field and gives bounds in terms of the configuration alone, but needs the set of vanishing polynomials to be closed under a degree-lowering operation. The Kakeya problem over $\mathbb R$ has neither a grid nor such an operation, and this is why Dvir's argument does not touch it (§8.2). Second, the rank step is genuinely different from the zeros step: it does not show that the polynomial is zero, only that it has a large "effective dimension", and it therefore gives bounds of the form $c^n$ with $c<q$ rather than bounds of the form $q^n/C$. This is why the cap-set problem, where the truth is exponentially smaller than the ambient space, needed a different lemma from the Kakeya problem, where the truth is a constant fraction of it.

8.2 Where they stop

No structure. The method bounds sizes; it does not describe extremal configurations. Vosper's theorem, that equality in Cauchy–Davenport holds only for arithmetic progressions, has no polynomial proof of the kind given here, and the corresponding structural questions for the Erdős–Heilbronn bound, for Kakeya sets, and for cap sets are not answered by these arguments. The reason is visible in the proofs: the contradiction is derived from the existence of a nonzero coefficient or a nonzero polynomial, and nothing in the argument records which configurations make that coefficient vanish.

Euclidean Kakeya. Dvir's argument requires a polynomial vanishing on the set, and a set of Lebesgue measure zero in $\mathbb R^n$, or even a finite set of points, imposes no useful constraint on a polynomial: there is always one of degree about $|K|^{1/n}$ vanishing on any finite $K$, and a Besicovitch set is uncountable. The finite-field problem was solvable because $\mathbb F_q^n$ is a grid on which Schwartz–Zippel bites. The Euclidean conjecture remains open in dimensions four and higher; the three-dimensional case has recently been settled by Wang and Zahl (2025), by methods that go well beyond the three lemmas of this dissertation, though the polynomial partitioning of Guth and Katz (2015) and Guth (2016) plays a role in the modern theory.

Incidence bounds over finite fields. The Szemerédi–Trotter theorem (1983) bounds incidences between $N$ points and $N$ lines in $\mathbb R^2$ by $O(N^{4/3})$, and Guth and Katz's distinct-distances theorem rests on an incidence bound in $\mathbb R^3$ proved by polynomial partitioning. Over $\mathbb F_q$ the Szemerédi–Trotter bound is false: all $q^2$ points and all $q^2+q$ lines of $\mathbb F_q^2$ have $q^3+q^2$ incidences, far more than $N^{4/3}$. Nontrivial incidence bounds over finite fields exist (Bourgain, Katz, and Tao 2004) but rest on sum–product estimates, not on the polynomial method. Thus the method is not uniformly stronger over finite fields; it is stronger for Kakeya-type problems, where the grid structure is an asset, and weaker for incidence problems, where the topology of $\mathbb R^2$ is what the Euclidean proofs use.

Constants. The constants produced by the method are rarely sharp. Dvir's $1/n!$ was improved to $2^{-n}$ by multiplicities, and the truth is between $2^{-n}$ and $2^{-(n-1)}$; the joints constant $\sqrt{48}$ is a factor of about $36$ above the grid example; the cap-set base $2.7551$ is a factor of $1.24$ above the best constructions, per dimension. The one place where the method is nearly sharp is the plane Kakeya problem, where Dvir's bound is $(q-1)/2$ below the truth, an additive error in a quantity of order $q^2/2$. The lesson of the computations in Appendix A is that the method is sharp exactly when the bridge loses nothing, which in the plane means that a Kakeya set is very nearly the zero set of a polynomial of degree $q$.

The slice-rank barrier. For cap sets, the base $c=2.7551$ is the best that Lemma 2.10 can give, since the slice rank of the relevant tensor is $c^{n-o(n)}$ (Blasiak et al. 2017). Improving the base requires a new idea, not a sharper computation.

8.3 Open problems

The following are open at the time of writing, to the author's knowledge.

1. The cap-set exponent. $2.2202\le\limsup r(n)^{1/n}\le2.7552$. The limit $\lim r(n)^{1/n}$ exists by Lemma 6.11 and Fekete's lemma, and its value is unknown.

2. The Kakeya constant over $\mathbb F_q$. For fixed $n\ge3$, $\liminf_{q\to\infty}q^{-n}\min|K|$ lies in $[2^{-n},2^{-(n-1)}]$; its value is unknown, and the exact minimum $\min|K|$ is unknown for every $q$ when $n\ge3$.

3. The joints constant. The optimal constant in Theorem 5.5 is not determined by the argument here; the grid example gives $n^{-n/(n-1)}$ as a lower bound on it. Recent work has narrowed the gap and extended the problem from lines to varieties (Tidor, Yu, and Zhao 2022).

4. Restricted sums in general groups. Károlyi (2004) extended the Erdős–Heilbronn bound to abelian groups whose smallest prime factor is $p$; the structure of the extremal sets, and analogues for non-abelian groups, are open in general. Snevily's conjecture on Latin transversals in abelian groups of odd order was settled by Arsovski (2011) with the Nullstellensatz; Kemnitz's conjecture was settled by Reiher (2007) with Chevalley–Warning. Higher-dimensional analogues of Erdős–Ginzburg–Ziv beyond the plane are open.

5. Euclidean Kakeya in dimension at least four. The polynomial method as developed here does not apply, for the reason given in §8.2.

6. The exact values of $r(n)$ for $n\ge7$. The best lower bounds come from the constructions of Edel (2004) and their successors, the exhaustive methods of Appendix A are hopeless at $n=7$, and the best upper bound is Theorem 6.5, which at $n=7$ gives $822$ against a lower bound of a few hundred.