Using discrete Darboux polynomials to detect and determine preserved measures and integrals of rational maps

In this Letter we propose a systematic approach for detecting and calculating preserved measures and integrals of a rational map. The approach is based on the use of cofactors and Discrete Darboux Polynomials and relies on the use of symbolic algebra tools. Given sufficient computing power, all rational preserved integrals can be found. We show, in two examples, how to use this method to detect and determine preserved measures and integrals of the considered rational maps.

💡 Research Summary

The paper introduces a systematic, algorithmic method for detecting and computing preserved measures and first‑ and second‑order integrals of rational maps. The core idea is to transfer the classical Darboux polynomial theory—originally devised for continuous polynomial ODEs—to the discrete setting. In the continuous case a Darboux polynomial (P(x)) satisfies (\dot P = C(x)P) with a polynomial cofactor (C(x)); two Darboux polynomials sharing the same cofactor yield a rational first integral via their ratio. In the discrete case the definition becomes (P(x_{n+1}) = C(x_n)P(x_n)) where now (C) is a rational function. A crucial observation is that, unlike the additive cofactor rule for ODEs, the cofactor of a product of discrete Darboux polynomials is the product of the individual cofactors. This multiplicative property makes the factorisation of cofactors unique and turns the search for Darboux polynomials into a purely linear problem.

The authors propose a concrete ansatz for admissible cofactors. Let (D(x)) be the common denominator of the rational map (\phi) and let (J(x)) be its Jacobian determinant. Writing (J(x)=D(x)^{-m}\prod_i K_i(x)^{b_i}) with polynomial factors (K_i), the ansatz restricts cofactors to the form \