Flip Graphs for Polynomial Multiplication

TL;DR

This work extends the flip-graph framework from matrix to polynomial multiplication tensors, showing that, over sufficiently large fields, the Toom-Cook representation of rank $n+m+1$ is reachable from the standard representation via a finite, cubic-bounded sequence of flips and reductions. The authors provide a constructive path length bound $F(n,m)=nm(2n+2m+1)$ and illustrate the method with a detailed $n=m=1$ example, then generalize by induction. They also investigate small-field behavior (notably $\mathbb{Z}_2$) using software-assisted search and SAT proofs, revealing that flip-graph-derived schemes can be optimal for several small cases but may require different strategies to prove optimality or lift schemes to other rings. The results illuminate the structure of flip graphs and their potential for discovering optimal, low-rank decompositions in polynomial multiplication and, by extension, in other bilinear maps, while highlighting open questions about scalability and applicability to matrix multiplication and non-commutative settings.

Abstract

Flip graphs were recently introduced in order to discover new matrix multiplication methods for matrix sizes. The technique applies to other tensors as well. In this paper, we explore how it performs for polynomial multiplication.

Theorems & Definitions (13)

• Definition 1
• Lemma 2
• proof
• Theorem 3
• Lemma 4
• proof
• Lemma 5
• proof
• proof : Proof of Thm. \ref{['mainthm']}
• Theorem 6