Basis-Free Quaternionic/Clifford Polynomial Manipulations: Mathematical Completeness

Presenter: Yue Liu, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, China

Joint work with Hongbo Li, Lei Huang, Shoubin Yao, Ge Li

In geometric computing, directions, lines, spheres, etc. may be represented by single vector variables, and symbolic manipulations of these variables may not resort to their explicit coordinate forms. This raises the following question: if a geometric conclusion can be derived from the coordinate forms of the variables, is it always true that the conclusion is derivable from some coordinate-free manipulations of the variables in coordinate-free form? If so, how to derive them in the latter approach? This is the mathematical completeness problem that must be answered by any coordinate-free symbolic algebraic system to be used in geometric computing. This talk introduces our recent work started in 2013 on the the mathematical completeness of basis-free (hence coordinate-free) symbolic manipulations of quaternionic polynomials in quaternionic variables, and of Clifford polynomials in vector variables taking values in a general 3D non-degenerate inner-product space. By now some striking results have been obtained. The conclusions are rather simple to describe, but the proofs are highly complicated.