Normalization of Polynomials in Algebraic Invariants of Three-Dimensional Orthogonal Geometry

TL;DR

This paper advances invariant theory in three-dimensional orthogonal geometry by developing methods for polynomial normalization and Gr"obner bases using Clifford algebra, enhancing symbolic computation.

Contribution

It introduces a novel approach to compute Gr"obner bases and normal forms for advanced invariants in 3D geometry using Clifford products, extending classical invariant methods.

Findings

Established Gr"obner bases for syzygies among advanced invariants

Derived normal forms for polynomials of advanced invariants

Extended Young tableau straightening to advanced invariants

Abstract

In classical invariant theory, the Gr\"obner base of the ideal of syzygies and the normal forms of polynomials of invariants are two core contents. To improve the performance of invariant theory in symbolic computing of classical geometry, advanced invariants are introduced via Clifford product. This paper addresses and solves the two key problems in advanced invariant theory: the Gr\"obner base of the ideal of syzygies among advanced invariants, and the normal forms of polynomials of advanced invariants. These results beautifully extend the straightening of Young tableaux to advanced invariants.