Algebraic Geometry
Polynomial
Implement Sturm's Theorem of computing the number of real roots within an interval bound of a univariate polynomial.
MCCGB
Implementation of Minimal Canonical Comprehensive Gröbner Basis (MCCGB) algorithms:
- One first computes a Comprehensive Gröbner System with minimal number of segments, then construct a MCCGB from it.
- One directly computes a MCCGB from a given parametric multivariate polynomial system.
Implementation
- Written in SINGULAR (developed at technischer Universität Kaiserslautern), a computer algebra system for symbolic computation, which is the fastest one computing Gröbner bases.
I also plan to implement it using Python in SymPy, which is a light-weighted open-source symbolic computation algebra system used by industrial people.
References
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Deepak Kapur, and Yiming Yang, "An Algorithm to Check Whether a Basis of a Parametric Polynomial System is a Comprehensive Gröbner Basiss and the Associated Completion Algorithm." In Proceedings of the 2015 ACM on International Symposium on Symbolic and Algebraic Computation, pp. 243-250, ACM, 2015.
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Deepak Kapur, and Yiming Yang. "An algorithm for computing a minimal comprehensive Gröbner basis of a parametric polynomial system." In EACA, vol. 2014, p.21, 2014.