Michael Monagan
Roman Pearce
ABSTRACT
We present two algorithms for simplifying rational expressions modulo an ideal of the polynomial ring k[x1, . . . , xn]. The first method generates the set of equivalent expressions as amodule over k[x1, . . . , xn] and computes a reduced Gröbner basis. From this we obtain a canonical form for the expression up to our choice of monomial order for the ideal. The second method constructs equivalent expressions by solving systems of linear equations over k, and conducts a global search for an expression with minimal total degree. Depending on the ideal, the algorithms may or may not cancel all common divisors. We also provide some timings comparing the efficiency of the algorithms in Maple.
- W. Adams, P. Loustaunau. An Introduction to Gröbner Bases. American Mathematical Society, 1996.]]Google Scholar
- T. Becker and V. Weispfenning. Gröbner Bases. Springer-Verlag, 1993.]]Google Scholar
- M. Caborara, C. Traverso. Efficient Algorithms for Ideal Operations. ISSAC 1998 Proceedings, 147--152.]] Google ScholarDigital Library
- D. Cox, J. Little, D. O'Shea. Ideals, Varieties, and Algorithms. Second Edition. Springer-Verlag, 1996.]] Google ScholarDigital Library
- D. Cox, J. Little, D. O'Shea. Using Algebraic Geometry. Second Edition. Springer-Verlag, 2005.]]Google Scholar
- R. Fröberg. An Introduction to Gröbner Bases. Wiley, 1997.]]Google Scholar
- J. Gutierrez, T. Recio. Advances on the Simplification of Sine-Cosine Equations. J. Symb. Comp. 26(1), 31--70, 1998.]] Google ScholarDigital Library
- M. van Hoeij. Rational Parametrizations of Algebraic Curves using a Canonical Divisor. J. Symb. Comp. 23, 209--227, 1997.]] Google ScholarDigital Library
- J. Mulholland, M. Monagan. Algorithms for Trigonometric Polynomials. ISSAC 2001 Proceedings, 245--252.]] Google ScholarDigital Library
- R. Pearce. Rational Expression Simplification with Polynomial Side Relations. M.Sc. Thesis, Simon Fraser University, 2005.]]Google Scholar
- J. Schicho. Rational Parametrization of Real Algebraic Surfaces. ISSAC 1998 Proceedings, 302--308.]] Google ScholarDigital Library
- Q. Tran. Efficient Groebner Walk Conversion for Implicitization of Geometric Objects. Computer Aided Geometric Design, 21(9), 837--857, 2004.]]Google ScholarDigital Library
Index Terms
Rational simplification modulo a polynomial ideal
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