A Simple Algorithm for Trimmed Multipoint Evaluation

TL;DR

This paper introduces a simple recursive algorithm for trimmed multipoint evaluation of multivariate polynomials, improving understandability and avoiding complex algebraic techniques, with applications in recent algorithmic developments.

Contribution

A straightforward recursive algorithm for trimmed multipoint evaluation that simplifies previous approaches and is accessible to researchers without specialized background.

Findings

Achieves near-linear time complexity

Simplifies the algorithmic process

Avoids heavy algebraic machinery

Abstract

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.