Preview
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.
Baur, W. oral communication (1980).
Baur, W. & Strassen, V. The complexity of partial derivatives (1982) Theoret. Comput. Sci. 22 (1982) 317–330.
Belaga, E.G. Evaluation of polynomials of one variable with preliminary processing of the coefficients (1961) Problemy Kibernetiki 5 (1961) 7–15.
Bini, D., Capovani, M., Lotti, G. & Romani, F. O(n2.7799) complexity for matrix multiplication (1979) Inform. Proc. Letters 8 (1979) 234–235.
Büchi, W. & Clausen, M. On a Class of Primary Algebras of Minimal Rank (1985) Linear Algebra and its Appl. 69 (1985) 249–268.
Coppersmith, D. & Winograd, S. On the asymptotic complexity of matrix multiplication (1982) SIAM J. Comput. 11 (1982) 472–492.
Coppersmith, D. & Winograd, S. Matrix multiplication via Behrend's Theorem (1986) Preprint, IBM Thomas J. Watson Research Center Yorktown Heights (1986).
Feig, E. On systems of bilinear forms whose minimal divisor-free algorithms are all bilinear (1981) J. of Algorithms 2 (1981) 261–281.
Fellmann, A. Optimal algorithms for the multiplication in simply generated local algebras (1985) Interner Bericht 4/1985, FB Informatik, Universität Frankfurt/M.
Fellmann, A. Algorithmenvarietäten der kommutativen Algebren minimalen Ranges (1988) Ph.-D. thesis, Universität Frankfurt/M.
Fiduccia, C. & Zalcstein, I. Algebras having linear multiplicative complexity (1977) J. of the ACM 24 (1977) 311–331.
von zur Gathen, J. & Strassen, V. Some polynomials that are hard to compute (1980) Theoret. Comput. Sci. 11/3 (1980) 331–336.
Giraldo, L.E. Ph.-D. thesis, Universidad Buenos Aires.
Grigor'ev, D.Yu. Notes of Scientific Seminars of LOMI 118 (1982) 25–82.
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.
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.
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.
de Groote, H.F. Lectures on the complexity of bilinear problems (1987) LN Comput. Sci. 245, Springer, Berlin Heidelberg (1987).
de Groote, H.F. & Heintz, J. Commutative algebras of minimal rank (1983) Linear Algebra and its Appl. 55 (1983) 37–68.
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.
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.
Heintz, J. A Polynomials with Symmetric Galois Group which are Easy to Compute (1986) Theoret. Comput. Sci. 47 (1986) 99–105.
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.
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.
Heintz, J. & Sieveking, M. Lower bounds for polynomials with algebraic coefficients (1980) Theoret. Comput. Sci. 11 (1980) 321–330.
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.
Keller, W. Asymptotisch schnelle Algorithmen der linearen Algebra (1982) Master's thesis, Universität Zürich.
Lafon, J.C. & Winograd, S. A lower bound for the multiplicative complexity of the product of two matrices (1978) (unpublished) manuscript.
Lang, S. Diophantine Geometry (1982) Interscience Publ. Wiley, New York.
Lipton, R.J. & Stockmeyer, L.J. Evaluation of polynomials with superpreconditioning (1976) Proc. 8th Annual ACM Symp. on Theory of Computing (1976) 174–180.
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.
Möhler, St. Komplexitätstheorie auflösbarer Lie Algebren (1987) Master's thesis, Universität Frankfurt/M.
Morgenstern, J. Preprint (1984), University of Nice.
Motzkin, T.S. Evaluation of polynomials and evaluation of rational functions (1955) Bull. AMS 61 (1955) 163.
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.
Pan, V.Ya. Methods of computing values of polynomials (1966) Russian Math. Surveys 21/1 (1966) 105–136.
Paterson, M.S. & Stockmeyer, L.J. On the number of nonscalar multiplications necessary to evaluate polynomials (1973) SIAM J. Comput. 2 (1973) 60–66.
Risler, J.J. Additive complexity and zeros of real polynomials (1985) SIAM J. Comput. 14 (1985) 178–183.
Schmidt, H.D. Generische Unteralgebren und untere Schranken für die Komplexität der einfachen klassischen Lie Algebren (1987) Master's thesis, Universität Frankfurt/M.
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.
Schnorr, C.P. An extension of Strassen's degree bound (1981) SIAM J. Comput. 10 (1981) 371–382.
Schönhage, A. An elementary proof for Strassen's degree bound (1976) Theoret. Comput. Sci. 3 (1976) 267–272.
Schönhage, A. Partial and total mairix multiplication (1981) SIAM J. Comput. 10 (1981) 434–455.
Stoss, H.J. Untere Schranken für die Komplexität rationaler Funktionen (1984) Preprint, Universität Konstanz.
Strassen, V. Gaussian elimination is not optimal (1969) Numer. Mathematik 13 (1969) 354–356.
Strassen, V. Evaluation of rational functions (1972) in: Complexity of Computer Computations, R. Miller and J. Thatcher (Eds.), Plenum Press (1972) 1–10.
Strassen, V. Vermeidung von Divisionen (1973 (a)) Crelles Journal für die reine und angew. Mathematik 264 (1973) 184–202.
Strassen, V. Die Berechnungskomplexität von elementarsymmetrischen Funktionen und von Interpolationskoeffizienten (1973 (b)) Numer. Mathematik 20/3 (1973) 238–251.
Strassen, V. Polynomials with rational coefficients which are hard to compute (1974) SIAM J. Comput. 3/2 (1974) 128–149.
Strassen, V. Algebaische Berechnungskomplexität (1984) in: Perspectives in Mathematics, Anniversary of Oberwolfach 1984, Birkhäuser Verlag, Basel, 509–550.
Strassen, V. Relative bilinear complexity and matrix multiplication (1986) Preprint, University Zurich (1986).
van der Waerden, B.L. Die Seltenheit der Gleichungen mit Affekt (1934) Math. Annalen 109 (1934) 13–16.
van der Waerden, B.L. Algebra I (1960) 5. Auflage der Modernen Algebra, Springer Verlag, Berlin.
Winograd, S. On computing the discrete Fourier transform (1976) Proc. Nat. Acad. Sci. USA 73 (1976) 1005–1006.
Winograd, S. Some bilinear forms whose multiplicative complexity depends on the field of constants (1977) Math. Systems Theory 10 (1977) 169–180.
Author information
Authors and Affiliations
Editor information
Rights 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