Twisted Chowla Method Proves Strong Van der Waerden for Degrees 40 and Above
A new twisted function-field version of Chowla's conjecture yields the required density-zero bound for irreducible non-$S_n$ polynomials in every degree $n \geq 40$.
Underlying Paper
The strong form of Van der Waerden's conjecture via twisted Chowla
Determining the properties of a random polynomial has fuelled significant investigation over the past century. One driving force of this research is a 1936 paper of Van der Waerden. Fix $n \ge 3$ and let $E_n(B)$ be the number of monic, irreducible, non-$S_n$ polynomials $f = X^n + a_1 X^{n-1} + \cdots + a_n$ with $|a_j| \leq B$ for all $j$. A recent breakthrough of Bhargava bounds $E_n(B) \ll B^{n-1}$. This spectacularly resolves a conjecture of Van der Waerden, but leaves open its stronger form, namely that $E_n(B) = o(B^{n-1})$. Inspired by recent progress, we now prove this stronger form for all $n \ge 40$. Bhargava's result, together with work of Chow and Dietmann, essentially reduces the strong Van der Waerden conjecture to the claim that the number of polynomials $f$ with Galois group $A_n$ is $o(B^{n-1})$. We connect this claim to Chowla's conjecture, an area driving major mathematical progress. Indeed, we formulate a twisted function field version of Chowla's conjecture, and show how Van der Waerden's conjecture reduces to this conjecture for $n \ge 8$. Finally, we make significant progress towards this twisted Chowla conjecture, leading to our resolution of the strong form of Van der Waerden's conjecture for all $n \geq 40$.
Van der Waerden's conjecture concerns how often a monic integer polynomial of fixed degree fails to have the full symmetric group as its Galois group. Let count irreducible, non- polynomials with coefficients bounded by . Bhargava recently proved the boundary-order estimate . The stronger form asks for .
Core Contribution
Anderson, Chow, Dietmann, Hokken, Koymans, Lemke Oliver, Sawin, and Shusterman prove this stronger statement for every . Their result turns the prior upper bound into a density-zero conclusion: among the polynomial families counted at this scale, the exceptional irreducible polynomials become negligible as grows.
The paper builds on reductions that focus attention on polynomials with alternating-group Galois group . It connects the resulting counting problem to a twisted function-field version of Chowla's conjecture. The authors show that the strong Van der Waerden problem reduces to this twisted Chowla statement for .
Technical Approach
The work introduces and advances a twisted function-field Chowla framework tailored to the polynomial-counting problem. Rather than proving the full conjecture in complete generality, the authors establish sufficient progress in the needed setting to derive the strong Van der Waerden conclusion from degree 40 onward.
This approach links questions about Galois groups of random polynomials with cancellation phenomena central to Chowla-type conjectures. The result combines the reduction to alternating-group cases with estimates obtained from the twisted function-field formulation.
Results and Scope
The principal theorem establishes
for every . This improves on a bound of order by proving a genuine asymptotic saving.
The result does not claim the strong form for every degree. Although the reduction to the twisted Chowla problem applies from , the stated resolution of the original conjecture begins at degree 40. The full twisted Chowla conjecture also remains broader than the range required for this theorem.
Scope and Caveats
This is a theoretical mathematics result rather than an empirical study: it provides no benchmark data set or computational evaluation. Its significance rests on the proof connecting the twisted Chowla framework to the polynomial-counting problem. Degrees below 40 are outside the stated main theorem.
Evidence Box
strongKey Claims
- •The strong form of Van der Waerden's conjecture holds for every degree n ≥ 40
- •The alternating-group case is connected to a twisted function-field Chowla conjecture
- •The reduction to the twisted Chowla statement applies for n ≥ 8
Key Results
- •Eₙ(B)=o(Bⁿ⁻¹) for every n ≥ 40
- •Bhargava's earlier result gives Eₙ(B) ≪ Bⁿ⁻¹
- •The paper makes progress on the twisted Chowla conjecture sufficient for the main theorem
Limitations & Caveats
- •The stated strong-form theorem begins at n ≥ 40
- •Degrees below 40 are not covered by the stated main theorem
- •The full twisted Chowla conjecture is not established
