Vol. 74, No. 1, 1978

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Algorithms for localizing roots of a polynomial and the Pisot Vijayaraghavan numbers

R. J. Duffin

Vol. 74 (1978), No. 1, 47–56

Pisot and Vijayaraghavan studied numbers whose m-th power is nearly an integer ín the sense that the discrepancy vanishes as m becomes infinite. One plus square root two is an example. Algebraic numbers of this type are characterized as algebraic integers whose conjugate roots each have absolute value less than one. This note develops a test for this property. An algorithm is given which determines whether or not one root of a polynomiaI has absolute value greater than one and all the other roots have absolute value less than one. If n is the degree of the polynomial, this algorithm involves only n rational steps.

Mathematical Subject Classification
Primary: 12A15
Received: 15 November 1976
Revised: 9 August 1977
Published: 1 January 1978
R. J. Duffin