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.

中文翻译：

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关于正负多项式零个数的确定及其隔离

摘要 提出了一种具有两种变体的新方法，该方法可以比笛卡尔符号规则提供的确定性更好地限制具有实系数和次数的多项式的正零和负零的数量。该方法还可以非常成功地隔离多项式的零点，并且所确定的根界明显窄于柯西界和拉格朗日界。该方法依赖于求解次数小于给定多项式的方程。人们可以通过分析确定一次多项式的正负零点的确切数量，达到并包括五个，并且还可以通过分析完全隔离多项式的零点，并使用该方法的一种变体，人们可以通过求解不超过 4 次的方程来分析地逼近 9 次以下的多项式。对于更高阶的多项式，应递归应用该方法的两种变体中的任何一种。给出了许多例子。展示了三次方程的根的完整分类，以及它们的隔离区间——对于方程的系数可能是模型参数的函数的各种科学模型尤其重要。该方法在具有可变系数的四次方程中的应用（源自对平床两层流体中分段恒定涡量的方位二维非粘性流的物理现实模型中波-流相互作用的研究和自由表面）已被证明。