Reasoning about Surfaces Using Differential Zero and Ideal Decomposition

Abstract

This paper presents methods for zero and ideal decomposition of partial differential polynomial systems and the application of these methods and their implementations to deal with problems from the local theory of surfaces. We show how to prove known geometric theorems and to derive unknown relations automatically. In particular, an algebraic relation between the first and the second fundamental coefficients in a very compact form has been derived, which is more general and has smaller degree than a relation discovered previously by Z. Li. Moreover, we provide symmetric expressions for Li’s relation and clarify his statement. Some examples of theorem proving and computational difficulties encountered in our experiments are also discussed.KeywordsPrime IdealFundamental FormTheorem ProveLocal TheoryDecomposition AlgorithmThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.