Polylogarithmic motivic Chabauty-Kim for $\mathbb{P}^1 \setminus \{ 0,1,\infty \}$: the geometric step via resultants

Technical Summary: Polylogarithmic Motivic Chabauty–Kim for $\mathbb{P}^1 \setminus \{0, 1, \infty\}$: The Geometric Step via Resultants

1. Problem Statement
The paper addresses the explicit determination of the set of $S$-integral points $X(\mathbb{Z}_S)$ on the thrice-punctured projective line $X = \mathbb{P}^1 \setminus \{0, 1, \infty\}$, where $S$ is a finite set of primes. By Siegel's theorem, this set is finite and equivalent to the solutions of the $S$-unit equation $a + b = 1$ with $a, b \in \mathbb{Z}_S^\times$.

The authors focus on the polylogarithmic motivic Chabauty–Kim method, a refinement of Kim's non-abelian Chabauty program. This method constructs a nested sequence of sets converging to $X(\mathbb{Z}_S)$:
$$X(\mathbb{Z}_p) \supseteq X(\mathbb{Z}_p)^{PL}_{S,1} \supseteq X(\mathbb{Z}_p)^{PL}_{S,2} \supseteq \dots \supseteq X(\mathbb{Z}_S)$$
where $X(\mathbb{Z}_p)^{PL}_{S,d}$ is the vanishing locus of specific polynomials in $p$-adic single polylogarithms.

The computation of these sets is divided into two steps:

1. Geometric Step: Finding non-trivial elements in the kernel of a specific cocycle map $\theta^\#$, which defines "motivic Chabauty–Kim functions." These are polynomials in a weighted ring involving unipotent de Rham polylogarithms ($\log_u, \text{Li}_n$) with coefficients in the shuffle algebra $\mathcal{O}(U_S)$ of the pro-unipotent mixed Tate fundamental group.
2. Arithmetic Step: Evaluating these functions under the $p$-adic period map to obtain $p$-adic functions whose zero sets approximate $X(\mathbb{Z}_S)$.

While the geometric step was solved for $|S|=1$ in depth $d \le 4$, it becomes computationally intractable for $|S| \ge 2$ using previous methods. The paper specifically targets the case $|S|=2$, aiming to construct explicit motivic Chabauty–Kim functions in depths greater than 4.

2. Methodology
The authors propose a novel approach to the geometric step that avoids the direct computation of the kernel of a massive matrix over a polynomial ring, a task they demonstrate is infeasible for $|S|=2$ even with supercomputers.

• Coordinate Model: The Selmer scheme is modeled affinely using non-canonical coordinates. The map $\theta^\#$ is expressed as a homomorphism of graded algebras:
  $$\theta^\#: \mathcal{O}(U_S)[\log_u, \text{Li}_1, \dots, \text{Li}_d] \to \mathcal{O}(U_S)[\Phi]$$
  The goal is to find non-zero elements in the kernel of this map restricted to specific degrees.

• Upper Bound Algorithm: To establish non-existence results for low degrees, the authors implement an algorithm that evaluates the matrix representation of $\theta^\#_{d,v}$ at random integer points in the coefficient ring. By Proposition 5.5, if the kernel dimension is zero for a generic evaluation, the kernel over the polynomial ring is trivial. This allows them to prove the non-existence of such functions for degrees $v < 18$ when $|S|=2$.

• Resultant Method: For constructing functions where they exist, the authors introduce a method based on resultants.

  1. They analyze the equations defining $\theta^\#$ for $|S|=2$ (where $\Sigma_{-1} = \{\tau_p, \tau_q\}$).
  2. They perform an elimination process to express the variables $\Phi_{\sigma_{2n-1}}$ and $\Phi_{\tau_{e_1}}$ in terms of $\Phi_{\tau_{e_0}}$ and the images of the polylogarithms.
  3. This reduction yields two polynomial equations, $P_4(X)$ and $P_6(X)$, in a single variable $X = \Phi_{\tau_p}^{e_0}$, where $P_k(X) = \theta^\#(\nu_k(X))$.
  4. Since these polynomials share a common root ($\Phi_{\tau_p}^{e_0}$), their resultant must vanish: $\text{Res}(P_4, P_6) = 0$.
  5. Because $\theta^\#$ is a homomorphism, $\text{Res}(\nu_4, \nu_6)$ lies in the kernel of $\theta^\#$. The authors explicitly compute the polynomials $\nu_4(X)$ and $\nu_6(X)$ and their resultant.

3. Key Contributions and Results
The paper provides the first explicit construction of motivic Chabauty–Kim functions for $|S| \ge 2$ in depth $d > 4$.

• Non-Existence in Low Degrees: Using their upper bound algorithm, the authors prove that for $|S|=2$, there are no non-trivial motivic Chabauty–Kim functions of degree less than 18.
• Optimal Depth and Degree: They prove that the lowest depth at which a non-trivial function exists is depth 6, and the minimal degree is 18.
• Explicit Construction: They construct a specific non-trivial function, denoted $F^{|2|}_{6,18}$, of depth 6 and degree 18.
  • This function is derived from the resultant $\text{Res}(\nu_4(X), \nu_6(X))$.
  • The resultant is shown to factor as $(\log_u)^6 (\text{Li}_2 - \frac{1}{2}\log_u \text{Li}_1) F^{|2|}_{6,18}$.
  • The factor $(\text{Li}_2 - \frac{1}{2}\log_u \text{Li}_1)$ corresponds to the known solution for $|S|=1$, while $F^{|2|}_{6,18}$ is the new, non-trivial component specific to $|S|=2$.
• Uniqueness: They prove that any other motivic Chabauty–Kim function of depth $<19$ and degree 18 is a multiple of $F^{|2|}_{6,18}$ by an element of $\mathcal{O}(U_S)$.
• Surprising Existence: The authors highlight that the existence of $F^{|2|}_{6,18}$ is not guaranteed by simple dimension counting (the domain dimension 996 is smaller than the codomain dimension 4183 at this depth/degree), suggesting new phenomena in the theory not visible in the $|S|=1$ case.

4. Significance and Claims
The paper claims to be the first work to tackle explicit Chabauty–Kim in depth $>4$ for $|S| \ge 2$.

• Algorithmic Advancement: It transforms the geometric step from a brute-force linear algebra problem (infeasible for $|S| \ge 2$) into a manageable elimination theory problem using resultants.
• Theoretical Insight: The construction reveals that the coefficients of these functions, while guaranteed to lie in the Goncharov subalgebra, may require individual terms that are not in this subalgebra, offering hints about the structure of the arithmetic step.
• Generalization: The authors outline a strategy to extend the resultant method to $|S| > 2$ by iteratively taking resultants to eliminate variables, though they note that verifying non-triviality becomes significantly harder in the general case.

The work serves as a proof of concept that the geometric step can be made effective for larger sets $S$ and higher depths, providing the necessary tools to potentially verify the Chabauty–Kim conjecture for the $S$-unit equation in previously inaccessible regimes.