论文标题
四分之一方程的真正根部及其毕达哥拉斯的曲调的分类
Classification of the Real Roots of the Quartic Equation and their Pythagorean Tunes
论文作者
论文摘要
提出的是对通用四分之一方程的真实根的位置$ x^4 + ax^3 + ax^3 + bx^2 + cx + d = 0 $具有真实系数的分类，并且以$ a $，$ a $ a $，$ b $，$ c $和$ d $表示的根部分类，而无需使用任何数字近似值。与一般四分之一相关的是，有许多子公司二次方程（分解二次方程），其根允许这种系统化以及确定四分位数单个根的边界。在许多情况下，发现根部隔离间隔。分析的第二层使用两个子公司立方方程（辅助立方方程），并将其与某些分解二次方程一起求解，允许对一般四分位数的根部进行完整分类，并确定每个根的隔离间隔。这些隔离间隔涉及四分位数的固定点（除其他），并且通过求解一些分解二次方程，还可以确定四分位数的固定点的隔离间隔。每种可能的情况都经过了仔细研究和说明，其中包含其特定特征的描述，基于求解立方方程的分析和基于求解二次方程的分析。随着对四分之一方程的根的分析，可以通过研究“子四分之一”的交点$ x^4 + ax^3 + ax^3 + bx^2 $，并具有一组合适的平行线，可以在这些相交点上找到一个美丽的毕达哥拉斯的类比，并且可以在这些相交点和一方面的平行案例和及其他曲目的平行序列之间，并且代表了其他景点，并且在其他景点上代表了其他景点。曲调。
Presented is a two-tier analysis of the location of the real roots of the general quartic equation $x^4 + ax^3 + bx^2 + cx + d = 0$ with real coefficients and the classification of the roots in terms of $a$, $b$, $c$, and $d$, without using any numerical approximations. Associated with the general quartic, there is a number of subsidiary quadratic equations (resolvent quadratic equations) whose roots allow this systematization as well as the determination of the bounds of the individual roots of the quartic. In many cases the root isolation intervals are found. The second tier of the analysis uses two subsidiary cubic equations (auxiliary cubic equations) and solving these, together with some of the resolvent quadratic equations, allows the full classification of the roots of the general quartic and also the determination of the isolation interval of each root. These isolation intervals involve the stationary points of the quartic (among others) and, by solving some of the resolvent quadratic equations, the isolation intervals of the stationary points of the quartic are also determined. Each possible case has been carefully studied and illustrated with a detailed figure containing a description of its specific characteristics, analysis based on solving cubic equations and analysis based on solving quadratic equations only. As the analysis of the roots of the quartic equation is done by studying the intersection points of the "sub-quartic" $x^4 + ax^3 + bx^2$ with a set of suitable parallel lines, a beautiful Pythagorean analogy can be found between these intersection points and the set of parallel lines on one hand and the musical notes and the staves representing different musical pitches on the other: each particular case of the quartic equation has its own short tune.
