Skip to main content

On the computational complexity of polynomials and bilinear mappings. A survey

  • Conference paper
  • First Online:
Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC 1987)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 356))

The German version of this paper has been presented as the author's Habilitationsschrift to the University of Frankfurt in 1986.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Alder, A. & Strassen, V. On the algorithmic complexity of associative algebras (1981) Theoret. Comput. Sci 15 (1981) 201–211.

    Article  Google Scholar 

  • Baur, W. oral communication (1980).

    Google Scholar 

  • Baur, W. & Strassen, V. The complexity of partial derivatives (1982) Theoret. Comput. Sci. 22 (1982) 317–330.

    Article  Google Scholar 

  • Belaga, E.G. Evaluation of polynomials of one variable with preliminary processing of the coefficients (1961) Problemy Kibernetiki 5 (1961) 7–15.

    Google Scholar 

  • Bini, D., Capovani, M., Lotti, G. & Romani, F. O(n2.7799) complexity for matrix multiplication (1979) Inform. Proc. Letters 8 (1979) 234–235.

    Article  Google Scholar 

  • Büchi, W. & Clausen, M. On a Class of Primary Algebras of Minimal Rank (1985) Linear Algebra and its Appl. 69 (1985) 249–268.

    Article  Google Scholar 

  • Coppersmith, D. & Winograd, S. On the asymptotic complexity of matrix multiplication (1982) SIAM J. Comput. 11 (1982) 472–492.

    Article  Google Scholar 

  • Coppersmith, D. & Winograd, S. Matrix multiplication via Behrend's Theorem (1986) Preprint, IBM Thomas J. Watson Research Center Yorktown Heights (1986).

    Google Scholar 

  • Feig, E. On systems of bilinear forms whose minimal divisor-free algorithms are all bilinear (1981) J. of Algorithms 2 (1981) 261–281.

    Article  Google Scholar 

  • Fellmann, A. Optimal algorithms for the multiplication in simply generated local algebras (1985) Interner Bericht 4/1985, FB Informatik, Universität Frankfurt/M.

    Google Scholar 

  • Fellmann, A. Algorithmenvarietäten der kommutativen Algebren minimalen Ranges (1988) Ph.-D. thesis, Universität Frankfurt/M.

    Google Scholar 

  • Fiduccia, C. & Zalcstein, I. Algebras having linear multiplicative complexity (1977) J. of the ACM 24 (1977) 311–331.

    Article  Google Scholar 

  • von zur Gathen, J. & Strassen, V. Some polynomials that are hard to compute (1980) Theoret. Comput. Sci. 11/3 (1980) 331–336.

    Article  Google Scholar 

  • Giraldo, L.E. Ph.-D. thesis, Universidad Buenos Aires.

    Google Scholar 

  • Grigor'ev, D.Yu. Notes of Scientific Seminars of LOMI 118 (1982) 25–82.

    Google Scholar 

  • de Groote, H.F. On varieties of optimal algorithms for the computation of bilinear mappings. I. The isotropy group of a bilinear mapping (1978 (a)) Theoret. Comput. Sci. 7 (1978) 1–24.

    Article  Google Scholar 

  • de Groote, H.F. On varieties of optimal algorithms for the computation of bilinear mappings. II. Optimal algorithms for 2 × 2 — matrix multiplication (1978 (b)) Theoret. Comput. Sci. 7 (1978) 127–148.

    Article  Google Scholar 

  • de Groote, H.F. Characterization of division algebras of minimal rank and the structure of their algorithms varieties (1983) SIAM J. of Comput. 12/1 (1983) 101–117.

    Article  Google Scholar 

  • de Groote, H.F. Lectures on the complexity of bilinear problems (1987) LN Comput. Sci. 245, Springer, Berlin Heidelberg (1987).

    Google Scholar 

  • de Groote, H.F. & Heintz, J. Commutative algebras of minimal rank (1983) Linear Algebra and its Appl. 55 (1983) 37–68.

    Article  Google Scholar 

  • de Groote, H.F. & Heintz, J. A lower bound for the bilinear complexity of some semisimple Liealgebras (1985) in: Proceedings of the AAECC — 3 Conference, Grenoble 1985. LN Comput. Sci. 229, Springer, Berlin Heidelberg (1987) 211–222.

    Google Scholar 

  • Heintz, J. Definability and fast quantifier elimination in algebraically closed fields (1983 (a)) Ph.-D. thesis, Universität Zürich 1982. Theoret. Comput. Sci. 24 (1983) 239–277. Russian translation in: Kiberneticeskij Sbornik, Novaja Serija, Vyp. 22, Ed. O.B. Lupanov, Mir Moscow (1985) 113–158.

    Article  Google Scholar 

  • Heintz, J. A Polynomials with Symmetric Galois Group which are Easy to Compute (1986) Theoret. Comput. Sci. 47 (1986) 99–105.

    Article  Google Scholar 

  • Heintz, J. & Morgenstern, J. On associative algebras of minimal rank (1985) in: Proceedings of the AAECC — 2 Conference, Toulouse 1984. LN Comput. Sci. 228, Springer, Berlin Heidelberg (1986) 1–24.

    Google Scholar 

  • Heintz, J. & Schnorr, C.P. Testing Polynomials which are Easy to Compute (1982) Logic and Algorithmic. An international Symposium held in honour of Ernst Specker. Monographie No. 30 de L'Enseignement Mathématique, Genève 1982, 237–254; also published in: 12th Annual Symposium ACM on Computing (1980) 262–280.

    Google Scholar 

  • Heintz, J. & Sieveking, M. Lower bounds for polynomials with algebraic coefficients (1980) Theoret. Comput. Sci. 11 (1980) 321–330.

    Article  Google Scholar 

  • Heintz, J. & Sieveking, M. Absolute Primality of Polynomials is Decidable in Random Polynomial Time in the Number of Variables (1981) 8th International Colloquium on Automata, Languages, and Programming ICALP 81. LN Comput. Sci. 115, Springer Verlag, Berlin, 16–28.

    Google Scholar 

  • Keller, W. Asymptotisch schnelle Algorithmen der linearen Algebra (1982) Master's thesis, Universität Zürich.

    Google Scholar 

  • Lafon, J.C. & Winograd, S. A lower bound for the multiplicative complexity of the product of two matrices (1978) (unpublished) manuscript.

    Google Scholar 

  • Lang, S. Diophantine Geometry (1982) Interscience Publ. Wiley, New York.

    Google Scholar 

  • Lipton, R.J. & Stockmeyer, L.J. Evaluation of polynomials with superpreconditioning (1976) Proc. 8th Annual ACM Symp. on Theory of Computing (1976) 174–180.

    Google Scholar 

  • Mirwald, R. Die algorithmische Struktur der sl(2,k) (1985) Master's thesis, Universität Frankfurt/M. an English abstract in: Proceedings of the AAECC — 3 Conference, Grenoble 1985. LN Comput. Sci. 229, Springer, Berlin Heidelberg (1986) 274–287.

    Google Scholar 

  • Möhler, St. Komplexitätstheorie auflösbarer Lie Algebren (1987) Master's thesis, Universität Frankfurt/M.

    Google Scholar 

  • Morgenstern, J. Preprint (1984), University of Nice.

    Google Scholar 

  • Motzkin, T.S. Evaluation of polynomials and evaluation of rational functions (1955) Bull. AMS 61 (1955) 163.

    Google Scholar 

  • Ostrowski, A.M. On two problems in abstract algebra connected with Horner's rule (1954) Studies in Math. and Mech. presented to Richard von Mises, Academic Press, New York (1954) 40–48.

    Google Scholar 

  • Pan, V.Ya. Methods of computing values of polynomials (1966) Russian Math. Surveys 21/1 (1966) 105–136.

    Google Scholar 

  • Paterson, M.S. & Stockmeyer, L.J. On the number of nonscalar multiplications necessary to evaluate polynomials (1973) SIAM J. Comput. 2 (1973) 60–66.

    Article  Google Scholar 

  • Risler, J.J. Additive complexity and zeros of real polynomials (1985) SIAM J. Comput. 14 (1985) 178–183.

    Article  Google Scholar 

  • Schmidt, H.D. Generische Unteralgebren und untere Schranken für die Komplexität der einfachen klassischen Lie Algebren (1987) Master's thesis, Universität Frankfurt/M.

    Google Scholar 

  • Schnorr, C.P. Improved lower bounds on the number of multiplications/divisions which are necessary to evaluate polynomials (1978) Theoret. Comput. Sci. 7 (1978) 251–261.

    Article  Google Scholar 

  • Schnorr, C.P. An extension of Strassen's degree bound (1981) SIAM J. Comput. 10 (1981) 371–382.

    Article  Google Scholar 

  • Schönhage, A. An elementary proof for Strassen's degree bound (1976) Theoret. Comput. Sci. 3 (1976) 267–272.

    Article  Google Scholar 

  • Schönhage, A. Partial and total mairix multiplication (1981) SIAM J. Comput. 10 (1981) 434–455.

    Article  Google Scholar 

  • Stoss, H.J. Untere Schranken für die Komplexität rationaler Funktionen (1984) Preprint, Universität Konstanz.

    Google Scholar 

  • Strassen, V. Gaussian elimination is not optimal (1969) Numer. Mathematik 13 (1969) 354–356.

    Google Scholar 

  • Strassen, V. Evaluation of rational functions (1972) in: Complexity of Computer Computations, R. Miller and J. Thatcher (Eds.), Plenum Press (1972) 1–10.

    Google Scholar 

  • Strassen, V. Vermeidung von Divisionen (1973 (a)) Crelles Journal für die reine und angew. Mathematik 264 (1973) 184–202.

    Google Scholar 

  • Strassen, V. Die Berechnungskomplexität von elementarsymmetrischen Funktionen und von Interpolationskoeffizienten (1973 (b)) Numer. Mathematik 20/3 (1973) 238–251.

    Article  Google Scholar 

  • Strassen, V. Polynomials with rational coefficients which are hard to compute (1974) SIAM J. Comput. 3/2 (1974) 128–149.

    Article  Google Scholar 

  • Strassen, V. Algebaische Berechnungskomplexität (1984) in: Perspectives in Mathematics, Anniversary of Oberwolfach 1984, Birkhäuser Verlag, Basel, 509–550.

    Google Scholar 

  • Strassen, V. Relative bilinear complexity and matrix multiplication (1986) Preprint, University Zurich (1986).

    Google Scholar 

  • van der Waerden, B.L. Die Seltenheit der Gleichungen mit Affekt (1934) Math. Annalen 109 (1934) 13–16.

    Article  Google Scholar 

  • van der Waerden, B.L. Algebra I (1960) 5. Auflage der Modernen Algebra, Springer Verlag, Berlin.

    Google Scholar 

  • Winograd, S. On computing the discrete Fourier transform (1976) Proc. Nat. Acad. Sci. USA 73 (1976) 1005–1006.

    Google Scholar 

  • Winograd, S. Some bilinear forms whose multiplicative complexity depends on the field of constants (1977) Math. Systems Theory 10 (1977) 169–180.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Llorenç Huguet Alain Poli

Rights and permissions

Reprints and permissions

Copyright information

© 1989 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Heintz, J. (1989). On the computational complexity of polynomials and bilinear mappings. A survey. In: Huguet, L., Poli, A. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 1987. Lecture Notes in Computer Science, vol 356. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-51082-6_85

Download citation

  • DOI: https://doi.org/10.1007/3-540-51082-6_85

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-51082-6

  • Online ISBN: 978-3-540-46150-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics