[Paper Review] Darboux theory of integrability in the sparse case

Guillaume Chèze|arXiv (Cornell University)|Oct 29, 2012

Advanced Differential Equations and Dynamical Systems20 references10 citations

TL;DR

This paper generalizes Darboux and Jouanolou's theorems on first integrals of polynomial vector fields by replacing the classical degree-based bounds with bounds based on the number of integer points in the Newton polytope of the vector field's associated Laurent polynomial. It proves that B + 1 Darboux polynomials imply a Darboux first integral, and B + n imply a rational first integral, where B is the number of integer points in the Newton polytope; the bound is shown to be optimal via explicit counterexamples.

ABSTRACT

Darboux's theorem and Jouanolou's theorem deal with the existence of first integrals and rational first integrals of a polynomial vector field. These results are given in terms of the degree of the polynomial vector field. Here we show that we can get the same kind of results if we consider the size of a Newton polytope associated to the vector field. Furthermore, we show that in this context the bound is optimal.

Motivation & Objective

• To extend classical Darboux and Jouanolou theorems on first integrals to the sparse case, where polynomial sparsity is captured by Newton polytopes.
• To replace the degree-based bounds in these theorems with bounds based on the number of integer points in the Newton polytope of the vector field.
• To demonstrate that the proposed bounds are optimal by constructing explicit counterexamples where the bound is tight.
• To provide a framework for analyzing integrability in polynomial systems where coefficients are zero, using geometric and algebraic tools from sparse polynomial theory.

Proposed method

• Define the Newton polytope ND as the convex hull of the exponents of the Laurent polynomial ∑ᵢ xᵢ Aᵢ / Xᵢ, where xᵢ are generic complex numbers.
• Introduce a weighted degree degν(f) for a polynomial f with respect to a vector ν ∈ ℤⁿ, which measures the maximum value of ν·m over the Newton polytope N(f).
• Prove that for any Darboux polynomial f with cofactor g, the Newton polytope N(g) is contained in ND ∩ ℤⁿ, establishing sparsity of cofactors.
• Use the fact that all cofactors lie in a complex vector space of dimension B, where B is the number of integer points in ND ∩ ℤⁿ, to derive linear dependence relations among cofactors.
• Leverage the strategy from [LZ10] to show that n linearly dependent cofactor relations with rational coefficients imply the existence of a rational first integral.
• Construct a counterexample derivation with no rational first integral but having exactly B + n − 1 Darboux polynomials, proving the bound is optimal.

Experimental results

Research questions

• RQ1 Can the classical degree-based bounds in Darboux and Jouanolou theorems be replaced by bounds based on the size of the Newton polytope in the sparse case?
• RQ2 Is the proposed bound in terms of the number of integer points in the Newton polytope optimal for the existence of rational first integrals?
• RQ3 How does the sparsity of polynomial vector fields, as measured by Newton polytopes, affect the existence and structure of Darboux and rational first integrals?
• RQ4 Can the cofactors of Darboux polynomials be shown to be sparse, and how does this sparsity constrain the space of possible first integrals?

Key findings

• The number of integer points B in the Newton polytope ND of the Laurent polynomial ∑ᵢ xᵢ Aᵢ / Xᵢ determines the critical threshold for integrability.
• If a derivation has at least B + 1 Darboux polynomials, then it admits a Darboux first integral, which is a product of powers of these polynomials.
• If a derivation has at least B + n Darboux polynomials, then it admits a rational first integral, i.e., a first integral in ℂ(X₁, ..., Xₙ).
• The bound B + n for rational first integrals is optimal, as demonstrated by a construction of a derivation with no rational first integral but having exactly B + n − 1 Darboux polynomials.
• In the dense case, where all coefficients are non-zero, B equals the binomial coefficient (n + d − 1 choose n), recovering the classical bounds of Darboux and Jouanolou.
• For sparse systems, such as those with only three non-zero terms (e.g., X₁^e X₂^e, X₁^{e−1} X₂^e, X₁^e X₂^{e−1}, X₁^0 X₂^0), the bound B = 3e grows linearly with degree d = 2e, while the classical bound grows quadratically, showing a significant improvement.