Abstract
Abstract A novel method with two variations is proposed with which the number of positive and negative zeros of a polynomial with real coefficients and degree n can be restricted with significantly better determinacy than that provided by the Descartes rule of signs. The method also allows the isolation of the zeros of the polynomial quite successfully, and the determined root bounds are significantly narrower than the Cauchy and the Lagrange bounds. The method relies on solving equations of degree smaller than that of the given polynomial. One can determine analytically the exact number of positive and negative zeros of a polynomial of degree up to and including five and also fully isolate the zeros of the polynomial analytically, and with one of the variations of the method, one can analytically approach polynomials of degree up to and including nine by solving equations of degree not more than four. For polynomials of higher degree, either of the two variations of the method should be applied recursively. Numerous examples are given. Presented is the full classification of the roots of the cubic equation, together with their isolation intervals – especially important for various scientific models for which the coefficients of the equation might be functions of the model parameters. An application of the method to a quartic equation with variable coefficients (resulting from the study of wave–current interactions in the physically realistic model of azimuthal two-dimensional non-viscous flows with piecewise constant vorticity in a two-layer fluid with a flat bed and a free surface) has been demonstrated. A step-by-step algorithm for each version of the method has also been presented.
Highlights
According to the Descartes’ rule of signs [1] from 1637, an algebraic equation with real coefficients cannot have more positive real roots than the variations of sign in the sequence of its coefficients
Gauss showed [2] in 1876 that the number of positive real roots is, more precisely, either equal to the number of variations of signs in the sequence of the coefficients or is equal to the number of variations of signs in the sequence of the coefficients reduced by an even number
Many extensions of the Descartes rule have been proposed – see [3], in which Marden has given a thorough summary of the results about polynomial roots
Summary
According to the Descartes’ rule of signs [1] from 1637, an algebraic equation with real coefficients cannot have more positive real roots than the variations of sign in the sequence of its coefficients. The second variation of the method splits pn(x) by selecting the very first and the very last of its terms and “propping” them against the remaining ones This split is studied in minute detail and the full classification of the roots of the cubic equation is presented, together with their isolation intervals and the criterion for the classification. On the number of positive and negative polynomial zeros and their isolation 1389 be the only source of beneficial information
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