We introduce a new framework term coding for extremal problems in discrete mathematics and information flow, where one chooses interpretations of function symbols so as to maximise the number of satisfying assignments of a finite system of term equations. We then focus on dispersion, the special case in which the system defines a term map $Θ^\mathcal I:\A^k\to\A^r$ and the objective is the size of its image. Writing $n:=|\A|$, we show that the maximum dispersion is $Θ(n^D)$ for an integer exponent $D$ equal to the guessing number of an associated directed graph, and we give a polynomial-time algorithm to compute $D$. In contrast, deciding whether \emph{perfect dispersion} ever occurs (i.e.\ whether $\Disp_n(\mathbf t)=n^r$ for some finite $n\ge 2$) is undecidable once $r\ge 3$, even though the corresponding asymptotic rate-threshold questions are polynomial-time decidable.

翻译：我们引入了一种新的框架——项编码，用于解决离散数学与信息流中的极值问题。该框架的核心在于选择函数符号的解释，以最大化有限项方程系统的可满足赋值数量。随后，我们聚焦于离散度这一特例：此时系统定义了一个项映射 $Θ^\mathcal I:\A^k\to\A^r$，目标为其像集的大小。记 $n:=|\A|$，我们证明最大离散度为 $Θ(n^D)$，其中整数指数 $D$ 等于一个关联有向图的猜测数，并给出了计算 $D$ 的多项式时间算法。相比之下，判定是否会出现完美离散度（即是否存在某个有限 $n\ge 2$ 使得 $\Disp_n(\mathbf t)=n^r$）在 $r\ge 3$ 时是不可判定的，尽管对应的渐近速率阈值问题是多项式时间可判定的。