TL;DR
This paper introduces the concept of comprehensive involutive systems, extending involutive bases to parametric ideals, and presents an algorithm for their construction, implemented in Maple, with illustrative examples.
Contribution
It defines comprehensive involutive systems and develops an algorithm for their computation, adapting existing methods for Groebner bases to involutive bases.
Findings
Algorithm successfully constructs comprehensive involutive systems.
Implementation in Maple demonstrates practical applicability.
Illustrative example validates the proposed method.
Abstract
In this paper, we consider parametric ideals and introduce a notion of comprehensive involutive system. This notion plays the same role in theory of involutive bases as the notion of comprehensive Groebner system in theory of Groebner bases. Given a parametric ideal, the space of parameters is decomposed into a finite set of cells. Each cell yields the corresponding involutive basis of the ideal for the values of parameters in that cell. Using the Gerdt-Blinkov algorithm for computing involutive bases and also the Montes algorithm for computing comprehensive Groebner systems, we present an algorithm for construction of comprehensive involutive systems. The proposed algorithm has been implemented in Maple, and we provide an illustrative example showing the step-by-step construction of comprehensive involutive system by our algorithm.
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