A Simple Algorithm for Trimmed Multipoint Evaluation

2025-06-30

AI Summary

This work addresses the *trimmed multivariate multipoint evaluation problem*: evaluating an $n$-variate polynomial—of individual degree at most $d$ and total degree at most $D$—over a structured point set (e.g., a subset of a Cartesian product). Existing algorithms rely on heavy computer algebra machinery (e.g., Gröbner bases, sparse interpolation), hindering accessibility and practical implementation. We propose a simple, recursive divide-and-conquer algorithm grounded solely in elementary polynomial algebra and fast univariate polynomial operations. Our method avoids advanced algebraic tools entirely, substantially lowering both theoretical and implementation barriers. It achieves near-linear runtime $ ilde{O}(d^n D)$, matching the asymptotic efficiency of the best known algorithms. The core contribution is a minimalist design that attains state-of-the-art performance without sacrificing clarity—establishing a new paradigm for multivariate multipoint evaluation that is intuitive, reproducible, and readily extensible.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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.

Problem

• Develops a simple algorithm for trimmed multipoint evaluation
• Evaluates polynomials on specific input points efficiently
• Avoids complex machinery for broader accessibility

Innovation

• Simple recursive algorithm for evaluation
• Avoids heavy computer-algebraic machinery
• Accessible to non-specialized researchers