This paper concerns $d$-dimensional cellular automata with the von Neumann neighborhood that conserve the sum of the states of all their cells. These automata, called number-conserving or density-conserving cellular automata, are of particular interest to mathematicians, computer scientists and physicists, as they can serve as models of physical phenomena obeying some conservation law. We propose a new approach to study such cellular automata that works in any dimension $d$ and for any set of states $Q$. Essentially, the local rule of a cellular automaton is decomposed into two parts: a split function and a perturbation. This decomposition is unique and, moreover, the set of all possible split functions has a very simple structure, while the set of all perturbations forms a linear space and is therefore very easy to describe in terms of a basis. We show how this approach allows to find all number-conserving cellular automata in many cases of $d$ and $Q$. In particular, we find all three-dimensional number-conserving CAs with three states, which until now was beyond the capabilities of computers.

本文关注 $d$具有冯·诺依曼（von Neumann）邻域的三维细胞自动机，可保留所有细胞状态的总和。这些自动机称为保数或保密度细胞自动机，数学家，计算机科学家和物理学家特别感兴趣，因为它们可以充当遵守某些守恒定律的物理现象的模型。我们提出了一种研究这种细胞自动机的新方法，该方法可以在任何维度上工作$d$ 以及任何状态集 $问$。本质上，元胞自动机的局部规则被分解为两个部分：分裂函数和微扰。这种分解是唯一的，而且所有可能的分裂函数的集合都具有非常简单的结构，而所有扰动的集合都形成了线性空间，因此很容易根据基础进行描述。我们展示了这种方法如何在许多情况下找到所有保数的细胞自动机。$d$ 和 $问$。特别是，我们发现具有三个状态的所有三维数量守恒CA，直到现在还超出了计算机的能力。