Evaluating a polynomial on a set of points is a fundamental task in computer algebra. In this work, we revisit a particular variant called trimmed multipoint evaluation: given an $n$-variate polynomial with bounded individual degree $d$ and total degree $D$, the goal is to evaluate it on a natural class of input points. This problem arises as a key subroutine in recent algorithmic results [Dinur; SODA '21], [Dell, Haak, Kallmayer, Wennmann; SODA '25]. It is known that trimmed multipoint evaluation can be solved in near-linear time [van der Hoeven, Schost; AAECC '13] by a clever yet somewhat involved algorithm. We give a simple recursive algorithm that avoids heavy computer-algebraic machinery, and can be readily understood by researchers without specialized background.

翻译：多项式在点集上的求值是计算机代数中的基本任务。本文重新审视一种称为裁剪多点求值的特定变体：给定一个具有有界个体次数 $d$ 和总次数 $D$ 的 $n$ 元多项式，目标是在一类自然的输入点上对其进行求值。该问题作为近期算法结果[Dinur; SODA '21]、[Dell, Haak, Kallmayer, Wennmann; SODA '25]中的关键子程序出现。已知裁剪多点求值可以通过一种巧妙但略显复杂的算法在近线性时间内求解[van der Hoeven, Schost; AAECC '13]。我们提出了一种简单的递归算法，该算法避免了繁重的计算机代数工具，并且易于不具备专业背景的研究人员理解。