The minimum root separation of a polynomial
HTML articles powered by AMS MathViewer
 by George E. Collins and Ellis Horowitz PDF
 Math. Comp. 28 (1974), 589597 Request permission
Abstract:
The minimum root separation of a complex polynomial A is defined as the minimum of the distances between distinct roots of A. For polynomials with Gaussian integer coefficients and no multiple roots, three lower bounds are derived for the root separation. In each case, the bound is a function of the degree n of A and the sum d of the absolute values of the coefficients of A. The notion of a seminorm for a commutative ring is defined, and it is shown how any seminorm can be extended to polynomial rings and matrix rings, obtaining a very general analogue of Hadamard’s determinant theorem.References

G. E. Collins, "Computing time analyses for some arithmetic and algebraic algorithms," Proc. 1968 Summer Institute on Symbolic Mathematical Computation, IBM Corp., Cambridge, Mass., 1961, pp. 197231.
 George E. Collins, The calculation of multivariate polynomial resultants, J. Assoc. Comput. Mach. 18 (1971), 515–532. MR 298921, DOI 10.1145/321662.321666
 Lee E. Heindel, Integer arithmetic algorithms for polynomial real zero determination, J. Assoc. Comput. Mach. 18 (1971), 533–548. MR 300434, DOI 10.1145/321662.321667
 Donald E. Knuth, The art of computer programming. Vol. 2: Seminumerical algorithms, AddisonWesley Publishing Co., Reading, Mass.LondonDon Mills, Ont., 1969. MR 0286318 R. G. K. Loos, "A constructive approach to algebraic numbers," Math. of Comp. (submitted.)
 Henryk Minc and Marvin Marcus, Introduction to linear algebra, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1965. MR 0188221
 Michael T. McClellan, The exact solution of systems of linear equations with polynomial coefficients, J. Assoc. Comput. Mach. 20 (1973), 563–588. MR 347068, DOI 10.1145/321784.321787 D. R. Musser, Algorithms for Polynomial Factorization, Univ. of Wisconsin Comp. Sci. Dept. Technical Report No. 134 (Ph.D Thesis), Sept. 1971, 174 pp. J. R. Pinkert, Algebraic Algorithms for Computing the Complex Zeros of Gaussian Polynomials, Univ. of Wisconsin Comp. Sci. Dept. Ph.D. Thesis, May 1973, Technical Report No. 188, July 1973. B. L. van der Waerden, Moderne Algebra. Vol. I, Springer, Berlin, 1930; English transl., Ungar, New York, 1949. MR 10, 587.
Additional Information
 © Copyright 1974 American Mathematical Society
 Journal: Math. Comp. 28 (1974), 589597
 MSC: Primary 12D10; Secondary 30A08
 DOI: https://doi.org/10.1090/S0025571819740345940X
 MathSciNet review: 0345940