A quasi-optimal lower bound for skew polynomial multiplication

TL;DR

This work analyzes the complexity of multiplying skew polynomials over a field by establishing a lower bound for the bilinear map μ$_d$ that matches the conjectured upper bound up to a logarithmic factor: C$_{\mathcal{k}}$(μ$_d$) ≥ rank$_{\mathcal{k}}$(μ$_d$) = Ω(d min\{d,r\}^{ω-2} r). For d ≥ r, the bound tightens to rank$_{\mathcal{k}}$(μ$_d$) ≥ d r^{ω-1}$ over algebraically closed fields. The authors provide quasi-optimal algorithms for low-degree cases in several special étale/Galois algebras (totally split, Kummer, Artin) and in towers, achieving cost ~ Ŝ(d^{ω-1} r). They also derive an upper bound on average bilinear complexity, showing A-rank$_{\mathcal{k}}$(N) = Ŝ(d^{ω-1} r) for N = Ω(r), and discuss extensions to towers of Galois algebras, highlighting the broader implications for skew polynomial multiplication and the matrix-multiplication exponent ω.

Abstract

We establish a lower bound for the complexity of multiplying two skew polynomials. The lower bound coincides with the upper bound conjectured by Caruso and Borgne in 2017, up to a log factor. We present algorithms for three special cases, indicating that the aforementioned lower bound is quasi-optimal. In fact, our lower bound is quasi-optimal in the sense of bilinear complexity. In addition, we discuss the average bilinear complexity of simultaneous multiplication of skew polynomials and the complexity of skew polynomial multiplication in the case of towers of extensions.

Theorems & Definitions (27)

• Definition 1
• Lemma 2
• Theorem 3
• Theorem 4: primitive element theorem
• Lemma 5
• Lemma 6
• Lemma 7
• Lemma 8
• Lemma 9
• Lemma 10