Normalization of Quaternionic Polynomials in Coordinate-Free Quaternionic Variables in Conjugate-Alternating Order

TL;DR

This work delivers the first readable certification that the conjectured reduced Gröbner basis BG defines the ideal ${\cal I}$ of coordinate-free quaternionic polynomials under the conjugate-alternating order. It introduces four novel reduction techniques for free associative algebras, expands BG via conjugation-based extensions to ${\bf BG}^{ext}$, and formulates the notion of clear S-polynomials to drastically reduce the S-polynomial workload. The Main Theorem asserts that BG is a reduced Gröbner basis of ${\cal I}$, with the certification achieved through systematic reductions and substitutional techniques. The results enable efficient normalization of quaternionic polynomials and illuminate how different orderings of quaternionic variables influence the algebraic structure underpinning coordinate-free quaternionic calculus.

Abstract

Quaternionic polynomials occur naturally in applications of quaternions in science and engineering, and normalization of quaternionic polynomials is a basic manipulation. Once a Groebner basis is certified for the defining ideal I of the quaternionic polynomial algebra, the normal form of a quaternionic polynomial can be computed by routine top reduction with respect to the Groebner basis. In the literature, a Groebner basis under the conjugate-alternating order of quaternionic variables was conjectured for I in 2013, but no readable and convincing proof was found. In this paper, we present the first readable certification of the conjectured Groebner basis. The certification is based on several novel techniques for reduction in free associative algebras, which enables to not only make reduction to S-polynomials more efficiently, but also reduce the number of S-polynomials needed for the certification.

Figures (3)

• Figure 1: Double non-decreasing structure of quaternionic monomials in normal form, with peak sequence ${\bf q}_{p1}{\bf q}_{p2}\cdots {\bf q}_{pr}{\bf e}_1{\bf e}_2\cdots {\bf e}_{s+1}$, floor sequence ${\bf q}_{f1}{\bf q}_{f2}\cdots {\bf q}_{f(r+s)}$, and ceiling sequence ${\bf A}_1{\bf q}_{p1}{\bf A}_2{\bf q}_{p2}\cdots {\bf A}_{r+1}{\bf e}_1{\bf e}_2\cdots {\bf e}_{s+1}$.
• Figure 2: Leading terms of BG. Here ${\bf e}\succ {\bf e}'$ are letters in $\cal E$, ${\bf 1}\prec {\bf 2}\prec \ldots \prec {\bf 5}$ are letters in $\cal Q$, and ${\bf A}_4$ is a non-decreasing monomial satisfying ${\bf 5}\succ_M {\bf A}_4\succeq {\bf 3}$. The notation ${\bf e},{\bf 2}$ at a vertex stands for two possible evaluations at the vertex: ${\bf e}$ or ${\bf 2}$. The notation ${\bf e} / {\bf 3}$ stands for two different combinations: those letters on the left the slash symbol at each vertex form the first group, and those on the right of the slash form the second group.
• Figure 3: Leaders of clear S-quadruplets with a generator of type ${\bf N}$. Here ${\bf e}\in \cal Q$, ${\bf 1}\prec {\bf 2}\prec \cdots \prec {\bf 8}$ are letters in ${\cal Q}$, and for $l=5,6,7$, ${\bf A}_l$ is a non-decreasing monomial satisfying ${\bf (l-1)} \preceq_M {\bf A}_l\prec_M {\bf l}$.

Theorems & Definitions (66)

• Definition 2.1
• Proposition 2.1
• proof
• Corollary 2.2
• proof
• Theorem 2.3: Normal form
• Theorem 2.4: Main Theorem
• Proposition 2.5
• proof
• Proposition 3.1