A Generalization of Sárközy's theorem in function fields

Technical Summary: A Generalization of Sárközy's Theorem in Function Fields

Problem Statement
The paper addresses the problem of finding dense subsets in the ring of polynomials over a finite field, $\mathbb{F}_q[t]$, that avoid specific polynomial configurations. This is the function field analog of Sárközy's theorem in integers, which states that any subset of $\mathbb{Z}$ with positive upper asymptotic density contains two distinct elements whose difference is a perfect square (or more generally, the value of an intersective polynomial).

In the context of $\mathbb{F}_q[t]$, previous work by Green established strong quantitative bounds for sets avoiding differences of the form $F(b(t))$, where $F$ is a polynomial with zero constant term. However, Green's result relied on a technical condition requiring the number of roots of $F$ in $\mathbb{F}_q[t]$ to be coprime to $q$. While Li and Sauermann recently removed this condition for the single-variable case, the problem of extending these results to equations involving multiple variables and more complex linear combinations remained open.

Methodology
The authors employ the Croot-Lev-Pach polynomial method, a technique that utilizes the "slice rank" of functions to bound the size of sets avoiding certain configurations. The core of their approach involves:

1. Reduction to a Single Root: Following the observation by Li and Sauermann, the authors note that for any polynomial $F(x) \in \mathbb{F}_q[t][x]$ with zero constant term, one can replace $F(x)$ with $F(t^M x)$ for a sufficiently large integer $M$. This transformation ensures that the resulting polynomial has only the trivial root $x=0$ in $\mathbb{F}_q[t]$, thereby satisfying the root condition required by the polynomial method without loss of generality.
2. Polynomial Mapping and Slice Rank: The problem is framed using two polynomial maps:
   • $\Psi: (\mathbb{F}_q^n)^k \to \mathbb{F}_q^{n'}$, representing the linear combination $c_1 a_1^r + \dots + c_k a_k^r$.
   • $\Phi: \mathbb{F}_q^m \to \mathbb{F}_q^{n'}$, representing the polynomial $F(b)$.
     The authors construct a composite function $f(a_1, \dots, a_k) = P(\Psi(a_1, \dots, a_k))$, where $P$ is a polynomial supported on the image of $\Phi$ (guaranteed to exist by a lemma of Green).
3. Bounding via Slice Rank:
   • Lower Bound: By the properties of the maps and the assumption that the set $A$ contains no non-trivial solutions, the function $f$ is non-zero if and only if $a_1 = \dots = a_k$. By a lemma of Tao, such a diagonal function has a slice rank equal to $|A|$.
   • Upper Bound: The slice rank is also bounded by the number of monomials in the expansion of $f$ that satisfy specific degree constraints. This is estimated using a pigeonhole principle argument and generating functions, leading to an inequality involving the size of the set $|A|$.

Key Contributions and Results
The primary contribution is Theorem 5, which generalizes Sárközy's theorem to equations in $k$ variables over $\mathbb{F}_q[t]$. Specifically, the authors prove:

• Theorem 5: Let $r, k, d, \ell$ be integers satisfying $k > 2r^2(1 - 1/\ell^2)$. Let $c_1, \dots, c_k \in \mathbb{F}_q[t]$ have degree at most $d$ with $\sum c_i = 0$. Let $F(x) \in \mathbb{F}_q[t][x]$ be a polynomial of degree $\ell$ with zero constant term. Then there exist constants $c \in (0, 1)$ and $c' > 0$ such that any subset $A \subset G_n$ (polynomials of degree $< n$) with cardinality $|A| > c' q^{cn}$ contains a non-trivial solution to:
  $$c_1 a_1^r + \dots + c_k a_k^r = F(b)$$
  for some $b \in \mathbb{F}_q[t]$. A solution is trivial if $a_1 = \dots = a_k$ and $b=0$.

This result subsumes previous theorems:

• It recovers Green's result (Theorem 3) and the Li-Sauermann improvement (Theorem 4) when $r=1$ and $k=2$.
• It extends the work of the first author (Theorem 6) regarding the homogeneous equation $c_1 a_1^r + \dots + c_k a_k^r = 0$ to the inhomogeneous case involving $F(b)$.

The paper also provides Proposition 7, a general statement combining the slice rank method with polynomial maps, from which Theorem 5 is deduced.

Significance
The paper claims significance in two main areas:

1. Generalization to Multiple Variables: It extends the scope of Sárközy-type theorems in function fields from differences of two elements to linear combinations of $k$ elements raised to the power $r$, equated to a polynomial value.
2. Removal of Technical Conditions: By integrating the simple observation regarding the transformation $F(t^M x)$, the authors demonstrate that the restrictive condition on the number of roots of $F$ (coprime to $q$) is unnecessary for the qualitative validity of the theorem in this generalized setting.

The authors note that while Theorem 6 (by Bienvenu) provides a stronger conclusion for the homogeneous case, it requires a larger number of variables ($k \ge 2r^2 + 1$) compared to Theorem 5 ($k > 2r^2(1 - 1/\ell^2)$). Thus, the two theorems are complementary rather than one subsuming the other. The work relies entirely on the Croot-Lev-Pach polynomial method and does not introduce new experimental proposals or unverified applications.