An Implementation of the Characteristic Set Method in Maple
The method of characteristic sets was introduced by J. F. Ritt (1932, 1950) in the context of his work on differential algebra in the early 1930s and was revitalized and further developed by Wen-tsün Wu (1984a, b, 1986) through his recent work on mathematics-mechanization. In addition to being a powerful tool for Wu’s general theory and method of mechanical theorem proving, the characteristic set method has proved efficient for solving a wide class of problems in geometry and algebra (see the series of work in “Mathematics-Mechanization Research Preprints” 1–11, 1987–1994, for example). It has been partially implemented by different research groups in China, USA and Austria (Chou 1988, Ko 1986, Kusche et al. 1989) for geometry theorem proving and solving other relevant problems. The author has learned that an implementation of this method in the Reduce system is ongoing at the University of Bath, England. However, to the best of our knowledge neither a complete implementation exists nor a partial implementation has been generally available in current symbolic and algebraic computation systems. The incompleteness of the existing implementations was mainly due to difficulties in polynomial factorization over successive algebraic extension fields (for which there was a lack of general and efficient procedures) and the determination of prime bases of ideals from their characteristic sets (for which no simple and practical method was available). We have overcome the difficulties through the discovery of a new method for polynomial factorization and the application of Gröbner bases for determining the prime bases.
KeywordsDecomposition Tree Differential Algebra Polynomial Factorization Characteristic Series Algebraic Number Field
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