多项式方程

This paper reviews the historical development of the algebraic solutions of polynomial equations, introduces the life and scientific achievements of Cardano together with the historical background of Ars Magna. Based on this, the paper makes an elaborate study on Ars Magna.

本文首先回顾了多项式方程代数解法的发展过程，介绍了卡尔达诺的生平、科学与数学成就以及《大术》的历史背景，然后在此基础上详细研究了《大术》各章的内容。

Cardano's Ars Magna is a landmark in the history of mathematics. It started the theoretical research on algebraic equations, solved the polynomial equations of third-degree and forth-degree in a general and systematical way, and studied the imaginary number as well as its operations for the first time.

卡尔达诺的《大术》在数学史上具有重要的地位，它开创了代数方程的理论研究，首次系统地给出了三、四次多项式方程的一般解法，并且最早讨论了虚数及其运算。

It showed that the inverse circular function equation had the best fitting to the process of yellow soil releasing cadmium at conditions with different acid rain. Besides, polynomial equations and two-constant equations can also describe this process properly.

结果表明，反三角函数方程对多种酸雨条件下黄壤吸附态镉释放过程的拟合度最佳，此外多项式方程、双常数方程也能较好地描述这一过程。

Dixon method was adopted to construct a 22×22 Dixon matrix and then the greatest common divisor of two columns was extracted.

结合矢量法和复数法建立了4个几何约束方程式，使用Dixon结式构造22 22的Dixon矩阵，对两列提取公因式后展开矩阵的行列式得到一元64次多项式方程，回代过程中去掉6组增根后得到58组解。

The direct displacement problem can be changed into solving a polynomial equation called the equivalent polynomial equation.

研究表明其正解问题最后可转化为一个高次多项式方程求解问题，称此方程为正解的等价多项式方程。

Extraneous roots are found during the process to obtain the other 3 variables by using Euclidean algorithm.

首先使用结式对四个多项式方程直接消元，得到一个一元52次的多项式方程，用辗转相除法求其他三个变量的过程中发现了6个增根。

Dixon method was adopted to construct a 22×22 Dixon matrix and then the greatest common divisor of two columns was extracted. After expanding the determinant of the matrix, a 64 degree univariate polynomial equation was obtained and 6 extraneous roots were canceled in the process of solving the other 3 variables.

结合矢量法和复数法建立了4个几何约束方程式，使用Dixon结式构造22×22的Dixon矩阵，对两列提取公因式后展开矩阵的行列式得到一元64次多项式方程，回代过程中去掉6组增根后得到58组解。

Firstly, four geometric loop equations are set up by using vector method in complex number fields. Secondly, three constraint equations are used to construct the Dixon resultants, which is a 6×6 matrix and contain two variables to be eliminated. Extract the greatest common divisor of two rows and two columns of Dixon matrix and compute its determinant to obtain a new equation. This equation together with the forth constraint equation can be used to construct a Sylvester resultant.

首先使用矢量法和复数法建立4个几何约束方程式；再使用Dixon结式法对3个方程式构造一个含有2个变元的6×6 Dixon矩阵，提取其中2行列元素的公因式，将新矩阵的行列式展开后得到二元高次多项式方程，该方程与剩下一个方程使用Sylvester结式消去一变元，得到一元高次方程。

The transcendental characteristic equation is converted into a polynomial equation at the imaginary axis without any substitutions. This method needs less computation than the CTCR method.

该方法将含有指数项的特征方程在虚轴上转化为多项式方程进行研究，无需任何中间变量代换，处理方法简单，计算量小。

Based on the discrete-time difference equation of the plant,an explicit solution of Diophantine matrix polynomial equations in multivariable generalized predictive control is derived without appealing to Diophantine matrix polynomial equations recursions or iterations,which makes the applications of generalized predictive control much more convenient.

直接利用被控对象的离散差分方程推导出多变量广义预测控制中Diophantine矩阵多项式方程的显式解，从而避免了其递推求解或迭代求解，使广义预测控制的应用更加方便。

As she looked at Warrington's manly face, and dark, melancholy eyes, she had settled in her mind that he must have been the victim of an unhappy attachment.

每逢看到沃林顿那刚毅的脸，那乌黑、忧郁的眼睛，她便会相信，他一定作过不幸的爱情的受害者。

Maybe they'll disappear into a pothole.

也许他们将在壶穴里消失