Finiteness criteria for the solutions of a sequence of decomposable form inequalities

1. Problem Statement

The paper addresses a central problem in Diophantine approximation: determining the finiteness of solutions to sequences of decomposable form inequalities over number fields.

• Context: Let $k$ be a number field, $S$ a finite set of places containing all archimedean places, and $O_S$ the ring of $S$-integers. The classical problem involves finding solutions to inequalities of the form:
  $$0 < \prod_{v \in S} \|F(x_0, \dots, x_m)\|_v \leq c H_S^\lambda(x_0, \dots, x_m)$$
  where $F$ is a decomposable form (factorizable into linear factors over a finite extension), $H_S$ is the $S$-height, and $c, \lambda$ are constants.
• Limitations of Previous Work:
  • Győry and Ru (2004): Proved finiteness for sequences of decomposable forms $F_n$ that factorize into $q$ linear factors in general position, provided $\lambda < q - 2m$.
  • Ji, Yan, and Yu (2020): Generalized this to forms factorizing into homogeneous polynomials $Q_{j,n}$, but relied on specific geometric configurations (general position).
• The Gap: There was a need to generalize these results to sequences of forms where the factors are semi-decomposable (products of non-constant homogeneous polynomials) and where the factors are not necessarily in general position, but satisfy a weaker geometric condition known as having a bounded "distributive constant."

2. Methodology

The author employs a sophisticated blend of Diophantine geometry and Nevanlinna theory techniques, specifically adapting the Subspace Theorem to moving targets.

Key Concepts Introduced/Utilized:

1. Semi-$q$-decomposable Forms: A form $F$ is semi-$q$-decomposable if it factors into $q$ non-constant homogeneous polynomials $Q_1, \dots, Q_q$ over $\bar{k}$.
2. Distributive Constant ($\Delta$): A crucial geometric invariant introduced by the author (in previous works) to measure the "position" of a family of hypersurfaces relative to a subvariety. It generalizes the concept of "general position" and "$\ell$-subgeneral position."
   • For a family of moving hypersurfaces $\mathcal{Q} = \{Q_1, \dots, Q_q\}$, $\Delta_{\mathcal{Q}, V}$ is defined based on the dimension of the intersection of subsets of these hypersurfaces with a variety $V$.
3. Moving Hypersurfaces: The paper treats the factors $Q_{j,n}$ as a sequence of moving hypersurfaces indexed by $n \in \mathbb{N}$.
4. Corvaja-Zannier Filtration: The proof utilizes a filtration of the vector space of homogeneous polynomials vanishing on the image of the sequence of points. This technique, originally developed by Ji, Yan, and Yu, is adapted here to handle the moving targets and the distributive constant.

Proof Strategy (Theorem 3.4):

The core of the paper is the proof of a Schmidt's Subspace Theorem for arbitrary families of moving hypersurfaces (Theorem 3.4). The strategy involves three steps:

1. Hypersurface Replacement: Estimating the sum of Weil functions for the given $q$ moving hypersurfaces by bounding them with the sum of Weil functions of $m+1$ new moving hypersurfaces in general position within the variety $V$.
2. Reduction to Subgeneral Position: Applying the replacement method again to reduce the problem to $\ell+1$ hypersurfaces in general position with respect to the ideal generated by polynomials vanishing on the sequence of points.
3. Filtration and Estimation: Constructing a Corvaja-Zannier type filtration for the vector space of polynomials. By analyzing the dimensions of the quotient spaces in this filtration, the author derives an upper bound for the sum of Weil functions involving the distributive constant $\Delta$.

3. Key Contributions

A. Main Theorem (Theorem 1.1)

The paper establishes a finiteness criterion for a sequence of semi-$q$-decomposable forms $F_n = Q_{1,n} \cdots Q_{q,n}$.

• Condition: Let $d = \max(\deg Q_{j,n})$ and $\Delta$ be the distributive constant of the family $\{Q_{1,n}, \dots, Q_{q,n}\}$. If the degree $\ell$ of $F_n$ satisfies:
  $$\ell > d \Delta \left(\frac{m}{2} + 1\right)^2$$
  and the exponent $\lambda$ in the inequality satisfies:
  $$\lambda < \ell - d \Delta \left(\frac{m}{2} + 1\right)^2$$
  Then, there are only finitely many $O_S^*$-non-proportional solutions to the inequality.

B. Generalization of Previous Results

• Improvement over Győry-Ru: The result removes the requirement that factors be linear and in general position. It allows for higher-degree factors and relaxes the geometric condition to a bounded distributive constant.
• Improvement over Ji-Yan-Yu: It generalizes their result on homogeneous polynomial factors by introducing the distributive constant, which handles cases where the factors are not in general position.

C. Corollary 1.4 (Fixed Forms)

The paper derives a corollary for a fixed semi-$q$-decomposable form $F = Q_1 \cdots Q_q$. It proves that if $\deg F > d \Delta (m/2 + 1)^2$, the inequality has only finitely many non-proportional solutions. This improves upon Theorem 4.2 in a previous work [2].

D. Application to Equations (Theorem 4.5)

The author extends the inequality results to equations of the form $F_n(x) = G_n(x)$, where $G_n$ is a polynomial of lower degree. This provides finiteness criteria for solutions to sequences of semi-decomposable form equations.

4. Results and Significance

• Mathematical Significance: The paper represents a significant advancement in the quantitative theory of Diophantine approximation. By introducing the distributive constant into the context of moving targets and sequences of forms, it bridges the gap between classical Subspace Theorems (fixed targets, general position) and modern Nevanlinna-theoretic approaches (moving targets).
• Optimality of Bounds: The bound $\ell > d \Delta (m/2 + 1)^2$ is derived using the specific geometry of the filtration. While the term $(m/2 + 1)^2$ suggests a potential for improvement (as classical results often have linear dependence on $m$), the result is currently the strongest known for the general case of moving hypersurfaces with arbitrary distributive constants.
• Unification: The framework unifies various known results (Győry-Ru, Ji-Yan-Yu) under a single theorem, demonstrating that the "general position" assumption is merely a special case where the distributive constant is small.
• Future Directions: The methodology suggests that the distributive constant is a robust tool for analyzing Diophantine problems involving moving targets, potentially applicable to other problems in arithmetic geometry involving families of varieties.

Conclusion

Si Duc Quang's paper provides a powerful generalization of the Schmidt Subspace Theorem. By defining and utilizing the distributive constant, the author successfully extends finiteness criteria to sequences of decomposable forms with non-linear factors and relaxed geometric configurations. This work deepens the understanding of the interplay between the algebraic structure of the forms, the geometric arrangement of their zero sets, and the distribution of integer solutions.