Abstract
We exhibit a new method for showing lower bounds for the time complexity of polynomial evaluation procedures. Time, denoted by L, is measured in terms of nonscalar arithmetic operations. The time complexity function considered in this paper is L 2. In contrast with known methods for proving lower complexity bounds, our method is purely combinatorial and does not require powerful tools from algebraic or diophantine geometry.
By means of our method we are able to verify the computational hardness of new natural families of univariate polynomials for which this was impossible up to now. By computational hardness we mean that the complexity function L 2 grows linearly in the degree of the polynomials of the family we are considering.
Our method can also be applied to classical questions of transcendence proofs in number theory and geometry. A list of (old and new) formal power series is given whose transcendency can be shown easily by our method.
Work partially supported by Spanish DGCYT grant PB 96-0671-C02-02.
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Aldaz M., Heintz J., Matera G., Montaña J.L., Pardo L.M.: Time-space tradeoffs in algebraic complexity theory. J. of Complexity (submitted to), 1996.
Baur W.: Simplified lower bounds for polynomials with algebraic coefficients. J. of Complexity 13(1) (1997) 38–41.
Belaga E.G.: Some problems involved in the computations of polynomials. Dokl. Akad. Nauk. SSSR 123 (1958) 775–777.
Bochnak J., Coste M., Roy M.-F.: Géométrie Algébrique Réelle. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Vol. 12. Springer, 1987.
Borodin A., Munro I.: The Computational Complexity of Algebraic and Numeric Problems. American Elsevier, 1975.
Bürgisser P., Clausen M., Shokrollahi A.: Algebraic Complexity Theory. A Series of comprehensive studies in mathematics 315. Springer, 1997.
Fitchas N., Galligo A., Morgenstern J.: Precise sequential and parallel complexity bounds for the quantifier elimination over algebraically closed fields. J. Pure Appl. Algebra 67 (1990) 1–14.
von zur Gathen J.: Algebraic complexity theory. Ann. Review of Comp. Sci. 3 (1988) 317–347.
von zur Gahen J., Strassen V.: Some polynomials that are hard to compute. Theoret. Comp. Sc. 11(3) (1980) 331–335.
Heintz J.: On the computational complexity of polynomials and bilinear mappings. A survey. L. Huget and A. Poli, editors, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. Lectures notes in Computer Science 356, 269–300. Springer, 1989.
Heintz J., Morgenstern J.: On the intrinsic complexity of elimination theory. J. of Complexity 9(4) (1993) 471–498.
Heintz J., Schnorr C.P.: Testing polynomials which are easy to compute. Logic and Algorithmic: An International Symposium held in honor of Ernst Specker. l'Enseignement de Mathématiques 30, 237–254. Genève, 1982.
Heintz J., Sieveking M.: Lower bounds for polynomials with algebraic coefficients. Theoret. Comp. Sc. 11(3) (1980) 321–330.
Krick T., Pardo L.M.: A computational method for diophantine approximation. L. González-Vega and T. Recio, editors, Algorithms in Algebraic Geometry and Applications. Proceedings of MEGA'94, Progress in Mathematics 143, 193–254. Birkhäuser, 1996.
Kung H.T., Traub J.F.: All algebraic functions can be computed fast. J. of the ACM 25(2) (1978) 245–260.
Motzkin T.S.: Evaluation of polynomials and evaluation of rational functions. Bull. Amer. Math. Soc. 61 (1955) 163.
Pardo L.M: How lower and upper complexity bounds meet in elimination theory. G. Cohen, M. Giusti and T. Mora, editors, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. Proceedings AAECC-11, Lecture Notes in Computer Science 948, 33–69. Springer, 1995.
Paterson M.S., Stockmeyer L.J.: On the number of nonscalar multiplications necessary to evaluate polynomials. SIAM J. of Computing 2(1) (1973) 60–66.
Puddu S., Sabia J.: An effective algorithm for quantifier elimination over algebraically closed fields using straight-line programs. J. of Pure Appl. Algebra (to appear).
Schnorr C.P.: Improved lower bounds on the number of multiplications/divisions which are necessary to evaluate polynomials. Theoret. Comp. Sc. 7(3) (1978) 251–261.
Shoup V., Smolensky R.: Lower bounds for polynomial evaluation and interpolation problems. Proceedings of the 32nd Annual Symposium FOCS, 378-383. IEEE Computer Society Press, 1991.
Stoss H.J.: On the representation of rational functions of bounded complexity. Theoret. Comp. Sc. 64(1) (1989) 1–13.
Strassen V.: Polynomials with rational coefficients which are hard to compute. SIAM J. of Computing 3 (1974) 128–149.
Strassen V.: Algebraic Complexity Theory. Handbook of Theoretical Computer Science, Vol. A, Chap. 11, 635–672. Elsevier Science Publishers, 1990.
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Aldaz, M., Heintz, J., Matera, G., Montaña, J.L., Pardo, L.M. (1998). Combinatorial hardness proofs for polynomial evaluation. In: Brim, L., Gruska, J., Zlatuška, J. (eds) Mathematical Foundations of Computer Science 1998. MFCS 1998. Lecture Notes in Computer Science, vol 1450. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0055765
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DOI: https://doi.org/10.1007/BFb0055765
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