Summary
The current proposals for applying the so called “fast” O(N loga N) algorithms to multivariate polynomials is that the univariate methods be applied recursively, much in the way more conventional algorithms are used. Since the size of the problems is rather large for which a “fastrd algorithm is more efficient than a classical one, the recursive approach compounds this size completely out of any practical range”.
The degree homomorphism is proposed here as an alternative to this recursive approach. It is argued that methods based on the degree homomorphism and a “fast” algorithm may be viable alternatives to more conventional algorithms for certain multivariate problems in the setting of algebraic manipulation. Several such problems are discussed including: polynomial multiplication, powering, division (both exact and with remainder), greatest common divisors and factoring.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Pollard, J. M.: The fast Fourier transform in a finite field. Mathematics of Computation 25, 365–374 (1971)
Fateman, R. J.: Polynomial multiplication, powers and asymptotic analysis: Some comments. SIAM J. of Computing 3, 196–213.(1974)
Strassen, V.: Die Berechnungskomplexität von elementar symetrischen Funktionen und von Interpolationskoeffizienten. Numer. Math. 20, 238–251 (1973)
Moenck, R.: Studies in fast algebraic algorithms. University of Toronto, Ph. D. Thesis, Sept 1973
Borodin, A., Moenck, R.: Fast modular transforms. J. Computer and System Sciences 18, 366–387 (1974)
Moenck, R.: On the efficiency of algorithms for polynomial factoring. To appear in Mathematics of Computation
Bonneau, R. J.: Fast polynomial operations using the fast Fourier transform. Pre-print MIT 1973
Lipson, J. D.: Chinese remainder and interpolation algorithms. In: Petrick, S. R. (ed.): Proc. 2nd Symp. on Symbolic and Algebraic Manipulation. New York: ACM 1971, p. 188–194
Horowitz, E.: Modular arithmetic and finite field theory: A tutorial. In: Petrick, S. R. (ed.): Proc. 2nd Symp. on Symbolic and Algebraic Manipulation. New York: ACM 1971, p. 188–194
Lang, S.: Algebra. Reading (Mass.): Addison-Wesley 1971
Brown, W. S., Traub, J. F.: On Euclid's algorithm and the theory of subresultants. J. ACM 18, 505–514 (1971)
Brown, W. S.: On Euclid's algorithm and the computation of polynomial greatest common divisors. J. ACM 18, 478–504 (1971)
Yun, D. Y. Y.: The Hensel lemma in algebraic manipulation. MIT, Cambridge (Mass.), Ph. D. Thesis, Nov 1974
Hall, A. D.: The Altran system for rational function manipulation — A survey. Comm. ACM 14, 517–521 (1971)
Williams, L. H.: Algebra of polynomials in several variables for a digital computer. J. ACM 9, 29–40 (1962)
Author information
Authors and Affiliations
Additional information
This research was supported by NRC Grant A9284.
Rights and permissions
About this article
Cite this article
Moenck, R.T. Another polynomial homomorphism. Acta Informatica 6, 153–169 (1976). https://doi.org/10.1007/BF00268498
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00268498