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Greatest common divisor of several polynomials

Published online by Cambridge University Press:  24 October 2008

S. Barnett
S. Barnett
Affiliation:
School of Mathematics, University of Bradford, Yorkshire
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Abstract

Given a polynomial a(λ) with degree n, and polynomials b1(λ), …, bm(λ) of degree not greater than n – 1, then the degree k of the greatest common divisor of the polynomials is equal to the rank defect of the matrix R = [b1(A), b2(A), …, bm(A)], where A is a suitable companion matrix of a(λ). Furthermore, it is shown that if the first k rows of R are expressed as linear combinations of the remaining n – k rows (which are linearly independent) then the greatest common divisor is given by the coefficients of row k + 1 in these expressions. A simple expression is derived for R and a permutation of the columns of this matrix establishes a direct connexion with controllability of a constant linear control system. Finally, when m = 1 a relationship between the corresponding R and Sylvester's matrix is exhibited.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 70 , Issue 2 , September 1971 , pp. 263 - 268
Copyright
Copyright © Cambridge Philosophical Society 1971

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References

REFERENCES

(1)Barnett, S.Degrees of greatest common divisors of invariant factors of two regular polynomial matrices. Proc. Cambridge Philos. Soc. 66 (1969), 241–245.CrossRefGoogle Scholar
(2)Barnett, S.Greatest common divisor of two polynomials, Linear Algebra and Appl. 3 (1970), 7–9.CrossRefGoogle Scholar
(3)Bôcher, M.Introduction to higher algebra (New York, 1964).Google Scholar
(4)Fryer, W. D.Applications of Routh's algorithm to network-theory problems, Trans. I.R.E. CT-6 (1959), 144–149.Google Scholar
(5)Fuller, A. T.Stability criteria for linear systems and realizability criteria for RC networks, Proc. Cambridge Philos. Soc. 53 (1957), 878–896.CrossRefGoogle Scholar
(6)Laidacker, M. A.Another theorem relating Sylvester's matrix and the greatest common divisor. Math. Mag. 42 (1969), 126–128.CrossRefGoogle Scholar
(7)MacDuffee, C. C.Some applications of matrices in the theory of equations, Amer. Math. Monthly 57 (1950), 154–161.Google Scholar
(8)Rosenbrock, H. H.State-8pace and multivariabie theory (London, 1970).Google Scholar
(9)Zadee, L. A. and Desoer, C. A.Linear system theory (New York, 1963).Google Scholar