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：近期，基于向量场极化技术，针对具有多项式向量场的常微分方程提出了一类非常规积分器。其中最简单的实例是现已著名的二次向量场Kahan离散化方法。此类积分器均展现出显著的守恒特性。特别地，已有证明表明，当底层常微分方程为哈密顿系统时，其极化离散化形式具有运动积分与不变体积形式。本文提出一种新的代数方法，用于推导极化离散化中的运动积分。