四次方程实根的分类及其勾股
Classification of the Real Roots of the Quartic Equation and their Pythagorean Tunes
摘要：
给出了一个两层分析的位置的实根的一般四次方程$x^4 + ax^3 + bx^2 + cx + d = 0$的实系数和分类的根$a$，$b$，$c$，和$d$，而不使用任何数值逼近。与一般的四次曲线相关联的是一些辅助二次方程（预解二次方程），其根允许这种系统化以及确定四次曲线的各个根的界限。在许多情况下，根的孤立区间被发现。第二层的分析使用两个辅助三次方程（辅助三次方程）和解决这些，连同一些预解的二次方程，允许的一般四次的根的完整分类，也确定每个根的孤立区间。这些隔离区间涉及的四次（除其他外）的驻点，并通过解决一些预解二次方程，隔离区间的四次的驻点也被确定。对每种可能的情况都进行了仔细的研究，并以详细的图表说明了其具体特征，基于求解三次方程的分析和仅基于求解二次方程的分析。由于四次方程的根的分析是通过研究“次四次”$x^4 + ax^3 + bx^2$与一组合适的平行线的交点来完成的，因此可以在这些交点和一组平行线以及音符和代表不同音高的五线谱之间找到美丽的毕达哥拉斯类比：四次方程的每种特殊情况都有其自己的短调。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
