Abstract
Gröbner Bases [Buc70] and Cylindrical Algebraic Decomposition [Col75,CMMXY09] are generally thought of as two, rather different, methods of looking at systems of equations and, in the case of Cylindrical Algebraic Decomposition, inequalities. However, even for a mixed system of equalities and inequalities, it is possible to apply Gröbner bases to the (conjoined) equalities before invoking CAD. We see that this is, quite often but not always, a beneficial preconditioning of the CAD problem.
It is also possible to precondition the (conjoined) inequalities with respect to the equalities, and this can also be useful in many cases.
The examples used in this paper are available in [Wil12]. This work was partially supported by the U.K.’s EPSRC under grant number EP/J003247/1.
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References
Aubry, P., Lazard, D., Moreno Maza, M.: On the Theories of Triangular Sets. J. Symbolic Comp. 28, 105–124 (1999)
Böge, W., Gebauer, R., Kredel, H.: Gröbner Bases Using SAC2. In: Caviness, B.F. (ed.) ISSAC 1985 and EUROCAL 1985. LNCS, vol. 204, pp. 272–274. Springer, Heidelberg (1985)
Buchberger, B., Hong, H.: Speeding-up Quantifier Elimination by Gröbner Bases. Technical Report 91-06 (1991)
Brown, C.W.: QEPCAD B: a program for computing with semi-algebraic sets using CADs. ACM SIGSAM Bulletin 37(4), 97–108 (2003)
Brown, C.W.: Tutorial handout (2004), http://www.cs.usna.edu/~wcbrown/research/ISSAC04/handout.pdf
Brown, C.W.: SLFQ — simplifying large formulas with QEPCAD B (2005), http://www.cs.usna.edu/~qepcad/SLFQ/Home.html
Buchberger, B.: Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystem (English translation in [Buc98]). Aequationes Mathematicae 4, 374–383 (1970)
Buchberger, B.: An Algorithmic Criterion for the Solvability of a System of Algebraic Equations. In: Gröbner Bases and Applications, pp. 535–545 (1998)
Collins, G.E., Hong, H.: Partial Cylindrical Algebraic Decomposition for Quantifier Elimination. J. Symbolic Comp. 12, 299–328 (1991)
Chen, C., Moreno Maza, M., Xia, B., Yang, L.: Computing Cylindrical Algebraic Decomposition via Triangular Decomposition. In: May, J. (ed.) Proceedings ISSAC 2009, pp. 95–102 (2009)
Collins, G.E.: Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition. In: Proceedings 2nd GI Conference Automata Theory & Formal Languages, pp. 134–183 (1975)
Dolzmann, A., Seidl, A., Sturm, T.: Efficient Projection Orders for CAD. In: Gutierrez, J. (ed.) Proceedings ISSAC 2004, pp. 111–118 (2004)
Gianni, P.: Properties of Gröbner Bases Under Specializations. In: Davenport, J.H. (ed.) ISSAC 1987 and EUROCAL 1987. LNCS, vol. 378, pp. 293–297. Springer, Heidelberg (1989)
Kalkbrener, M.: Solving Systems of Algebraic Equations by Using Gröbner Bases. In: Davenport, J.H. (ed.) ISSAC 1987 and EUROCAL 1987. LNCS, vol. 378, pp. 282–292. Springer, Heidelberg (1989)
Moreno Maza, M.: On Triangular Decompositions of Algebraic Varieties (2005), http://www.csd.uwo.ca/~moreno/Publications/M3-MEGA-2005.pdf
Phisanbut, N.: Practical Simplification of Elementary Functions using Cylindrical Algebraic Decomposition. PhD thesis, University of Bath (2011)
Platzer, A., Quesel, J.-D., Rümmer, P.: Real World Verification. In: Schmidt, R.A. (ed.) CADE-22. LNCS, vol. 5663, pp. 485–501. Springer, Heidelberg (2009)
Tarski, A.: A Decision Method for Elementary Algebra and Geometry, 2nd edn. Univ. Cal. Press (1951)
Wilson, D.J.: Real Geometry and Connectness via Triangular Description: CAD Example Bank (2012), http://opus.bath.ac.uk/29503
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Wilson, D.J., Bradford, R.J., Davenport, J.H. (2012). Speeding Up Cylindrical Algebraic Decomposition by Gröbner Bases. In: Jeuring, J., et al. Intelligent Computer Mathematics. CICM 2012. Lecture Notes in Computer Science(), vol 7362. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31374-5_19
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DOI: https://doi.org/10.1007/978-3-642-31374-5_19
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