[Paper Review] On the number of nonnegative solutions of a system of linear Diophantine equations

Kamil Brádler|arXiv (Cornell University)|Oct 20, 2016

Algebraic Geometry and Number Theory5 references3 citations

TL;DR

This paper derives a closed-form expression for the number of nonnegative integer solutions to a system of linear Diophantine equations arising in quantum field theory, particularly in counting Feynman diagrams for $φ^n$ scalar field models. Using symmetry analysis and combinatorial enumeration, the authors obtain explicit polynomial formulas for even and odd values of the total interaction strength $π$, revealing a surprising connection to magic squares and Floyd’s triangle in the simplified case.

ABSTRACT

We derive a closed expression for the number of nonnegative solutions of a certain system of linear Diophantine equations. The motivation comes from high energy physics where the nonnegative solutions play a crucial role in the perturbative calculation for a class of Lagrangians describing the interaction of an atom with a boson field or a non-linear interaction of boson fields among themselves (the so-called interacting phi^n models). The linear system can be solved and the nonnegative solutions enumerated but a closed expression for the number of solutions is preferable to counting the solutions. Interestingly, the problem led to a construction of a simpler linear Diophantine system whose nonnegative number of solutions turns out to be the magic constant.

Motivation & Objective

• To derive a closed-form expression for the number of nonnegative integer solutions to a system of linear Diophantine equations motivated by perturbative quantum field theory.
• To address the challenge of counting solutions efficiently, avoiding brute-force enumeration or black-box algorithms.
• To explore the structural symmetries and invariants in the system to enable exact counting.
• To reveal a surprising connection between the solution count and the magic constant of normal magic squares in a simplified case.

Proposed method

• The authors analyze the system of equations $2\alpha_{ii} + \sum_{j \neq i} \alpha_{ij} = \ell_i$ for $1 \leq i \leq 4$, with $\ell_i = \ell$ even or odd.
• They exploit symmetry in the system by splitting solutions into cases based on the relative ordering of $\ell_{11}$ and $\ell_{22}$, and use combinatorial bounds to enumerate valid configurations.
• The counting is performed via nested summations over even integers, with bounds derived from the constraints $\ell_{11} < \ell_{22}$, $\ell_{11} + \ell_{22} \leq \ell$, and $\ell_{11} + \ell_{22} > \ell$, adjusted for parity.
• The method accounts for symmetric solution pairs by doubling counts in symmetric regions, reducing computation to a single case with multiplicative factors.
• The derivation uses case analysis based on $\ell \mod 4$, distinguishing $\ell \equiv 0 \pmod{4}$ and $\ell \equiv 2 \pmod{4}$, with adjusted summation bounds for even lattice points.
• The final closed forms are derived by summing polynomial expressions and simplifying to obtain rational functions in $\ell$.

Experimental results

Research questions

• RQ1 What is the exact number of nonnegative integer solutions to the system of linear Diophantine equations modeling a $\phi^n$ scalar field interaction?
• RQ2 How can the solution count be expressed in closed form rather than through enumeration or algorithmic computation?
• RQ3 What is the connection between the solution count of this system and combinatorial structures like magic squares or Floyd’s triangle?
• RQ4 Can the symmetry and invariant structure of the system be leveraged to derive a general counting formula for arbitrary $k$-field interactions?
• RQ5 Does the simplified system with $\alpha_{i4} = 0$ yield a solution count that matches known combinatorial sequences such as the magic constant?

Key findings

• For even $\ell$, the number of nonnegative solutions is given by $\mathsf{e}(\ell) = \frac{1}{576}(\ell+2)(\ell+4)\big{(}\ell(\ell+5)(\ell(\ell+4)+12)+72\big{)}$, a degree-6 polynomial.
• For odd $\ell$, the number of solutions is $\mathsf{d}(\ell) = \frac{1}{576}(\ell+1)(\ell+3)\big{(}\ell(\ell+5)(\ell(\ell+6)+17)+72\big{)}$, also a degree-6 polynomial.
• After reparametrization $\ell \mapsto 2\ell-2$, the even case yields the magic constant of a normal magic square of order $\ell$, confirming a deep combinatorial link.
• The simplified system with $\alpha_{i4} = 0$ for all $i$ produces a solution count matching the sequence A006003 (Floyd’s triangle), confirming the connection to triangular numbers.
• The solution count for the full system is invariant under permutation of the $\ell_i$ values when they are equal, reflecting the underlying symmetry of the interaction model.
• The derived formulas are verified through case analysis and summation of symmetric regions, with consistent results across different modular conditions on $\ell$.