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Exact calculation of degrees for lattice equations: a singularity approach

Takafumi Mase

TL;DR

This work develops a rigorous, Halburd-inspired framework to compute exact degree growth for lattice (quad) equations, addressing the longstanding challenge of determining algebraic entropy in discrete lattice systems. By introducing lattice analogues of singularity patterns, past/future light cones, and a divisor-based degree counting mechanism, the authors prove that movable singular patterns are governed by corresponding basic patterns under the $\partial$-factor and basic pattern conditions. The main theorems connect the first singularity, movable patterns, and degree sequences, enabling exact degree computations from singularity structure alone when solvability conditions hold. The approach is demonstrated across multiple lattice equations, including polynomial and Laurent-property cases, and the authors discuss extensions to more general lattices and non-constant singularities, with implications for integrability tests in discrete systems.

Abstract

The theory of degree growth and algebraic entropy plays a crucial role in the field of discrete integrable systems. However, a general method for calculating degree growth for lattice equations (partial difference equations) is not yet known. Here we propose a method to rigorously compute the exact degree of each iterate for lattice equations. Halburd's method, which is a novel approach to computing the exact degree of each iterate for mappings (recurrence relations, typically from ordinary difference equations) from the singularity structure, forms the basis of our idea. The strategy is to extend this method to lattice equations. First, we illustrate, without rigorous details, how to calculate degrees for lattice equations using the lattice version of Halburd's method, and outline the issues that must be resolved to make the method rigorous. We then provide a framework to ensure that all calculations are accurate and rigorous. We further address how to detect the singularity structure in lattice equations. Our method is not only accurate and rigorous but can also be easily applied to a wide range of lattice equations.

Exact calculation of degrees for lattice equations: a singularity approach

TL;DR

This work develops a rigorous, Halburd-inspired framework to compute exact degree growth for lattice (quad) equations, addressing the longstanding challenge of determining algebraic entropy in discrete lattice systems. By introducing lattice analogues of singularity patterns, past/future light cones, and a divisor-based degree counting mechanism, the authors prove that movable singular patterns are governed by corresponding basic patterns under the -factor and basic pattern conditions. The main theorems connect the first singularity, movable patterns, and degree sequences, enabling exact degree computations from singularity structure alone when solvability conditions hold. The approach is demonstrated across multiple lattice equations, including polynomial and Laurent-property cases, and the authors discuss extensions to more general lattices and non-constant singularities, with implications for integrability tests in discrete systems.

Abstract

The theory of degree growth and algebraic entropy plays a crucial role in the field of discrete integrable systems. However, a general method for calculating degree growth for lattice equations (partial difference equations) is not yet known. Here we propose a method to rigorously compute the exact degree of each iterate for lattice equations. Halburd's method, which is a novel approach to computing the exact degree of each iterate for mappings (recurrence relations, typically from ordinary difference equations) from the singularity structure, forms the basis of our idea. The strategy is to extend this method to lattice equations. First, we illustrate, without rigorous details, how to calculate degrees for lattice equations using the lattice version of Halburd's method, and outline the issues that must be resolved to make the method rigorous. We then provide a framework to ensure that all calculations are accurate and rigorous. We further address how to detect the singularity structure in lattice equations. Our method is not only accurate and rigorous but can also be easily applied to a wide range of lattice equations.
Paper Structure (10 sections, 31 theorems, 247 equations, 9 figures)

This paper contains 10 sections, 31 theorems, 247 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Lattice version of Halburd's method
  3. How to calculate degrees for lattice equations
  4. Proofs
  5. Examples
  6. Conclusions and Discussion
  7. Essential choices of domain
  8. Substitution and derivatives
  9. Algebraic lemmas
  10. Divisors on P1(K)

Key Result

Theorem 3.14

Suppose that a quad equation $\Phi$ satisfies the $\partial$-factor condition (Definition definition:d_factor_condition). Let $H \subset \mathbb{Z}^2$ be a domain and let $z$ be one of the initial variables. Let $z^{*} \in \mathbb{P}^1 \left( \overline{\mathbb{K}} \right)$ and consider the constant has a minimum element with respect to the product order $\le$ on $\mathbb{Z}^2$. In particular, the

Figures (9)

  • Figure 1: Example of an undesirable singularity pattern, where a single value $z^{*}$ generates two or more independent singularity blocks. Here, we call two singularity blocks independent if they cannot be compared under the product order on $\mathbb{Z}^2$. Such a phenomenon can never occur in the case of mappings, since all points are totally ordered on $\mathbb{Z}$.
  • Figure 2: Similar to Figure \ref{['figure:inexistent_pattern_1']}, but two singularity blocks are not independent, as after a singularity block is confined, another block appears. Such a phenomenon is common in the case of mappings since two or more patterns are sometimes combined and generate this type of pattern. A cyclic pattern of a mapping falls into this category, too.
  • Figure 3: Similar to Figure \ref{['figure:inexistent_pattern_1']}, but one singularity block has two or more starting points. Such a phenomenon can never occur in the case of mappings.
  • Figure 4: Domain on which we consider basic patterns.
  • Figure 5: Situation in Proposition \ref{['proposition:key_lemma']}. If the points marked with "$\boxplus$" are not constant singularities, then those marked with "$+$" depend on $w$ and are not constant singularities, either. This procedure can easily be repeated and thus none of the points east to but not higher than the $\boxplus$-wall is a constant singularity. To prove this proposition, it is essential that the points marked with "$+$" lie outside the past light cone emanating from $(t, n)$.
  • ...and 4 more figures

Theorems & Definitions (110)

  • Example 1.1
  • Example 2.1
  • Definition 3.1: Past and Future Light Cones
  • Definition 3.2
  • Remark 3.3
  • Definition 3.4: $\mathbb{K}$
  • Definition 3.5: Constant Singularity, Constant Singularity Pattern
  • Remark 3.6
  • Definition 3.7: Basic Pattern
  • Remark 3.8
  • ...and 100 more