[Paper Review] Asymptotic estimates for roots of the cuboid characteristic equation in the linear region

Руслан Шарипов|arXiv (Cornell University)|May 11, 2015

Algebraic Geometry and Number Theory39 references6 citations

TL;DR

This paper derives asymptotic estimates for roots of the tenth-degree cuboid characteristic equation in the linear region of the parameter space (p, q), proving that for large p and q with p/q bounded away from 0 and ∞, integer roots do not exist in a narrow strip near p = q. The key result shows that no perfect cuboids can exist in this subregion, significantly narrowing the search space for solutions to the perfect cuboid problem.

ABSTRACT

A perfect cuboid is a rectangular parallelepiped whose edges, whose face diagonals, and whose space diagonal are of integer lengths. The second cuboid conjecture specifies a subclass of perfect cuboids described by one Diophantine equation of tenth degree and claims their non-existence within this subclass. This Diophantine equation is called the cuboid characteristic equation. It has two parameters. The linear region is a domain on the coordinate plane of these two parameters given by certain linear inequalities. In the present paper asymptotic expansions and estimates for roots of the characteristic equation are obtained in the case where both parameters tend to infinity staying within the linear region. Their applications to the cuboid problem are discussed.

Motivation & Objective

• To analyze the asymptotic behavior of roots of the cuboid characteristic polynomial Q_{pq}(t) = 0 in the linear region defined by 1/59 < p/q < 59.
• To determine whether integer solutions exist for the Diophantine equation Q_{pq}(t) = 0 under the constraints of the second cuboid conjecture.
• To identify regions in the (p, q)-plane where no perfect cuboids can exist by proving the absence of integer roots in specific asymptotic intervals.
• To refine the search space for perfect cuboids by excluding a narrow, spiky subregion around the line p = q where no solutions are possible.

Proposed method

• Derives asymptotic expansions for the positive real root t of the tenth-degree polynomial Q_{pq}(t) = 0 as p, q → ∞ with p/q → θ ∈ (1/59, 59), θ rational.
• Transforms variables via t = q² + 5p q + 10p² + r, where r is a remainder term, to isolate the dominant behavior of t.
• Applies the theory of continued fractions and unimodular transformations to define asymptotic intervals for the root t in terms of p and q.
• Uses the Euclidean algorithm to construct integer matrices S and T that preserve the Diophantine structure of the parameter space.
• Applies inequalities from Theorem 1.1 (t > p², t > p q, t > q², (p² + t)(p q + t) > 2t²) to constrain the location of valid roots.
• Employs contradiction arguments based on bounds on q relative to |p| to prove the absence of integer roots in specific regions.

Experimental results

Research questions

• RQ1 Does the cuboid characteristic equation Q_{pq}(t) = 0 have integer roots for large coprime integers p and q in the linear region 1/59 < p/q < 59?
• RQ2 Can the asymptotic behavior of the positive real root t of Q_{pq}(t) = 0 be estimated with sufficient precision to rule out integer solutions?
• RQ3 Is there a subregion within the linear region where no perfect cuboids can exist due to the absence of integer roots in the characteristic equation?
• RQ4 What are the precise bounds on q relative to |p| that guarantee no integer roots exist in the asymptotic intervals derived from the polynomial?

Key findings

• For p < 0 and q ≥ 97|p|, the asymptotic interval for the root t contains no integer points, proving no perfect cuboids exist in this region.
• For p > 0 and q > 74|p|³, the asymptotic interval for t contains no integer points, ruling out solutions in this subregion.
• The combined constraints q − q/97 ≤ p ≤ q + min(q/97, ∛(q/74)) define a narrow, spiky subregion around p = q where no perfect cuboids can exist.
• The subregion defined by these inequalities is small and excludes a substantial portion of the linear region, but a significant area remains open for numerical search.
• The results confirm that the second cuboid conjecture (irreducibility of Q_{pq}(t)) implies no perfect cuboids exist in the asymptotic subregion near p = q.
• The contradiction derived from assuming integer roots in the case p < 0 and q ≥ 97|p| confirms Theorem 8.1: no perfect cuboids exist in that region.