A Simple Algorithm for Trimmed Multipoint Evaluation
Nick Fischer, Melvin Kallmayer, Leo Wennmann
TL;DR
The paper addresses efficient trimmed multipoint evaluation for $n$-variate polynomials with individual degree $d$ and total degree $D$ on trimmed grids. It introduces a simple recursive algorithm that achieves near-linear time $O^*(\binom{n}{\le D}_{d})$ by interleaving coefficient polynomials through an LU factorization of a Vandermonde matrix, preserving degree bounds via $Q_j = \sum_{i=j}^{d} U_{j,i} P_i$ and recovering $P(Z_\ell)$ from $L$-combinations. The method is designed to be accessible and avoids heavy computer-algebra machinery, while providing a straightforward interpolation counterpart. This trimmed evaluation/subroutine is valuable for exponential-time algorithms solving systems of polynomial equations and related problems, offering a practical and implementable approach with clear theoretical guarantees.
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.