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Testing Positiveness of Polynomials

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Abstract

Many problems in mathematics, logic, computer science, and engineering can be reduced to the problem of testing positiveness of polynomials (over real numbers). Although the problem is decidable (shown by Tarski in 1930), the general decision methods are not always practically applicable because of their high computational time requirements. Thus several partial methods were proposed in the field of term rewriting systems. In this paper, we exactly determine how partial these methods are, and we propose simpler and/or more efficient methods with the same power.

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References

  1. Arnon, D. S.: Algorithms for the Geometry of Semi-Algebraic Sets, Ph.D. thesis, Comp. Sci. Dept., Univ. of Wisconsin-Madison, 1981, Tech. Report No. 436.

  2. Arnon, D. S., Collins, G. E., and McCallum, S.: Cylindrical algebraic decomposition II: An adjacency algorithm for the plane, SIAM J. Comp. 13 (1984), 878–889.

    Google Scholar 

  3. Arnon, D. S., Collins, G. E., and McCallum, S.: An adjacency algorithm for cylindrical algebraic decompositions of three-dimensional space, Journal of Symbolic Computation 5(1,2) (1988).

  4. Ben-Cherifa, A. and Lescanne, P.: Termination of rewriting systems by polynomial interpretations and its implementation, Science of Computer Programming 9(2) (1987), 137–159.

    Google Scholar 

  5. Brown, C. and Collins, G.: Simplification of truth invariant CAD's and solution formula construction, in Proceedings of IMACS-ACA'96, 1996.

  6. Canny, J. F.: Improved algorithms for sign and existential quantifier elimination, The Computer Journal 36 (1993), 409–418. In a special issue on computational quantifier elimination, edited by H. Hong.

  7. Collins, G. E.: Quantifier elimination for the elementary theory of real closed fields by cylindrical algebraic decomposition, in Lecture Notes In Computer Science 33, Springer-Verlag, Berlin, 1975, pp. 134–183.

    Google Scholar 

  8. Collins, G. E.: Quantifier elimination by cylindrical algebraic decomposition - 20 years of progress, in B. Caviness and J. Johnson (eds), Quantifier Elimination and Cylindrical Algebraic Decomposition, Texts and Monographs in Symbolic Computation, Springer-Verlag, 1994, to appear.

  9. Collins, G. E. and Hong, H.: Partial cylindrical algebraic decomposition for quantifier elimination, Journal of Symbolic Computation 12(3) (1991), 299–328.

    Google Scholar 

  10. Dershowitz, N.: Termination of rewriting, Journal of Symbolic Computation 3 (1987), 69–116.

    Google Scholar 

  11. Giesl, J.: Generating polynomial orderings for termination proofs, in Proceedings of the 6th International Conference on Rewriting Techniques and Applications, Lecture Notes in Computer Science 914, Springer-Verlag, 1995.

  12. González-Vega, L.: A combinatorial algorithm solving some quantifier elimination problems, in Quantifier Elimination and Cylindrical Algebraic Decomposition, Springer-Verlag, 1996.

  13. Grigor'ev, D. Yu.: The complexity of deciding Tarski algebra, Journal of Symbolic Computation 5(1,2) (1988), 65–108.

    Google Scholar 

  14. Heintz, J., Roy, M.-F., and Solernó, P.: On the complexity of semialgebraic sets, in G. X. Ritter (ed.), Proc. IFIP, North-Holland, 1989, pp. 293–298.

  15. Hong, H.: An improvement of the projection operator in cylindrical algebraic decomposition, in International Symposium of Symbolic and Algebraic Computation (ISSAC-90), ACM, 1990, pp. 261–264.

  16. Hong, H.: Improvements in CAD-based Quantifier Elimination, Ph.D. thesis, The Ohio State University, 1990.

  17. Hong, H.: Simple solution formula construction in cylindrical algebraic decomposition based quantifier elimination, in International Conference on Symbolic and Algebraic Computation ISSAC-92, 1992, pp. 177–188.

  18. Hong, H.: Parallelization of quantifier elimination on a workstation network, in G. Cohen, T. Mora, and O. Moreno (eds), AAECC-10, Lecture Notes in Computer Science 673, Springer-Verlag, 1993, pp. 170–179.

  19. Hong, H.: Quantifier elimination for formulas constrained by quadratic equations via slope resultants, The Computer Journal 36(5) (1993), 440–449.

    Google Scholar 

  20. Hong, H.: Approximate quantifier elimination, in Proceedings of SCAN'95, 1995. Invited talk.

  21. Hong, H.: Generic quantifier elimination, in Proceedings of IMACS-ACA'95, 1995.

  22. Hong, H.: Heuristic search and pruning in polynomial constraints satisfaction, Annals of Math. and AI 19(3,4) (1997), 319–334.

    Google Scholar 

  23. Hong, H. and Neubacher, A.: Approximate quantifier elimination, in Proceedings of IMACS-ACA'96, 1996.

  24. Lankford, D.: A finite termination algorithm, Technical report, Southwestern University, Georgetown, Texas, 1976.

    Google Scholar 

  25. Loos, R. and Weispfenning, V.: Applying linear quantifier elimination, Computer Journal 36(5) (1993), 450–462.

    Google Scholar 

  26. McCallum, S.: An Improved Projection Operator for Cylindrical Algebraic Decomposition, Ph.D. thesis, University of Wisconsin-Madison, 1984.

  27. McCallum, S.: An improved projection operator for cylindrical algebraic decomposition, Journal of Symbolic Computation 5(1,2) (1988).

  28. McCallum, S.: Solving polynomial strict inequalities using cylindrical algebraic decomposition, The Computer Journal 36(5) (1993), 432–438. In a special issue on computational quantifier elimination, edited by H. Hong.

  29. Renegar, J.: On the computational complexity and geometry of the first-order theory of the reals, Journal of Symbolic Computation 13(3) (1992), 255–300.

    Google Scholar 

  30. Saunders, B. D., Lee, H. R., and Abdali, S. K.: A parallel implementation of the cylindrical algebraic decomposition algorithm, in International Symposium of Symbolic and Algebraic Computation, 1990, pp. 298–307.

  31. Steinbach, J.: Proving polynomials positive, in Proceedings of the 12th FST&TCS, Lecture Notes in Computer Science 652, New Delhi, India, 1992.

  32. Steinbach, J.: Termination of Rewriting, Ph.D. thesis, 1994.

  33. Tarski, A.: The completeness of elementary algebra and geometry, 1930. Reprinted in 1967.

  34. Tarski, A.: A Decision Method for Elementary Algebra and Geometry, 2nd edn, Univ. of California Press, Berkeley, 1951.

    Google Scholar 

  35. Weispfenning, V.: The complexity of linear problems in fields, Journal of Symbolic Computation 5(1,2) (1988), 3–27.

    Google Scholar 

  36. Weispfenning, V.: Quantifier elimination for real algebra - the cubic case, in ISSAC-94, 1994, pp. 258–263.

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Authors and Affiliations

  1. Department of Mathematics, North Carolina State University, Raleigh, NC, 27695-8205

    Hoon Hong

  2. Research Institute for Symbolic Computation, Johannes Kepler University, A-4040, Linz, Austria

    Dalibor Jakuš

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  1. Hoon Hong
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  2. Dalibor Jakuš
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Hong, H., Jakuš, D. Testing Positiveness of Polynomials. Journal of Automated Reasoning 21, 23–38 (1998). https://doi.org/10.1023/A:1005983105493

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