Recently, a family of unconventional integrators for ODEs with polynomial vector fields was proposed, based on the polarization of vector fields. The simplest instance is the by now famous Kahan discretization for quadratic vector fields. All these integrators seem to possess remarkable conservation properties. In particular, it has been proved that, when the underlying ODE is Hamiltonian, its polarization discretization possesses an integral of motion and an invariant volume form. In this note, we propose a new algebraic approach to derivation of the integrals of motion for polarization discretizations.

翻译：摘要：近年来，针对具有多项式向量场的常微分方程组，提出了一类基于向量场极化的新型非标准积分器。其中最简实例便是如今著名的二次向量场Kahan离散化方法。所有这些积分器似乎都具备显著的守恒性质。特别地，已有证明表明：当底层ODE为哈密顿系统时，其极化离散化方法具有运动积分与不变体积形式。本文提出了一种推导极化离散化运动积分的新型代数方法。