There are in-depth studies on solving multivariate higher-order equations, higher-order arithmetic progression summation, and higher-order interpolation methods. He is the author of three volumes: Arithmetic Enlightenment (1299) and Siyuan Jade Mirror (1303). In the latter, he discussed the solution, expression, operation and elimination of related polynomials of high-order four-element simultaneous equations.
It is another outstanding creation of mathematicians in Song and Yuan Dynasties to extend celestial sphere to higher-order simultaneous equations of binary, ternary and quaternary. What has been handed down to this day is Zhu Shijie's Meet with Siyuan, which systematically discusses this outstanding creation. Meeting in Philip Burkart was written on 1303. The book has 3 volumes, 24 subjects and 288 questions. This paper mainly discusses the solution of higher-order equation (which is also Zhu Shijie's greatest contribution), higher-order arithmetic progression summation and higher-order interpolation method. It is an important masterpiece that has been circulated so far and systematically discusses the four elements.
On the basis of the theory of heaven, Zhu Shijie established the theory of four-dimensional higher-order equation, and he put the constant term in the center (that is, "Tai"), and then "set Tianyuan in the bottom, the ground in the left, people in the right and things in the top". The four elements of "heaven, earth, people and things" represent the unknown (that is, equivalent to today's). If modern X, Y, Z and W are used to represent heaven, earth, people and things, then we can express Zhu Shijie's high-order multivariate equation. The equations expressed by the above two graphs "Quadratic Primary Support Formula" and "Quadratic Secondary Support Formula" are: x+y+z+w=0.
After listing the quaternary high-order equations in the above way, solve the equations simultaneously. The method is to solve the equations by elimination method. First, one variable is unknown, and polynomials composed of other elements are used as the coefficients of this unknown. Then the quartic equation with four variables is eliminated to form the quartic equation with three variables, then the univariate is changed into the quartic equation with two variables, and then the univariate is eliminated to obtain the Tianyuan open module with only one variable. Then find the positive root by multiplying the root. This is an important development of linear method for solving problems. In the west, the systematic study of multivariate equations will wait until16th century. Higher-order arithmetic progression summation and higher-order interpolation are also important contents that Siyuan meets. In many summation problems, formulas can be derived from a series of triangular superposition formulas. Zhu Shijie gave the formula when p = 1, 2, ... 6 in the above formula. In addition, there are other higher-order arithmetic progression summation formulas. In terms of recruitment methods, Zhu Shijie has given the formula of recruitment, which is more than 400 years earlier than the West.
Sutton, a famous American historian of science, commented: "Zhu Shijie was an outstanding mathematician in his time and throughout the ages", and "Meeting Philip Burkart" is "the most important mathematical work of China and one of the most outstanding mathematical works in the whole Middle Ages." Zhu Shijie is not only an outstanding mathematician, but also a mathematics educator. He has traveled all over the world and taught students for more than 20 years. And personally wrote an introductory book on mathematics called "Arithmetic Enlightenment". Zhu Shijie put forward the method of solving Pythagorean formula with known chord sum and string sum in the volume of "Arithmetic Enlightenment", which supplemented the deficiency of "Nine Chapters Arithmetic".