| Etienne Bezout was a French mathematician in the18th century. He was much occupied with his teaching duties and official business as public officers, therefore he could devote to research within limited time, and he chose theory of equations for study, which is a branch of algebra he loved. Bezout's polynomial multiplier method for resolving equations was the most widely recognized elimination method, and it became the theoretical basis of using the polynomial multiplier in modern polynomial optimization. The Bezout theorem gotten in solving the degree of resultant is his grea-test achievements, which is crucial to the study of the intersection of manifolds in al-gebraic geometry and used in the algebraic geometry widely. What's more, Bezout 's theory of resultant inspired the studies of the modern elimination theory, including Lagrange and Cauchy's refinements of elimination procedure and Sylvester's work on resultants and inertia forms.Basing on the reviews the development process of western elimination method in the18th century, it is, then, introduced that life of Bezout, his mathematical achie-vements and their historical background. And the main mathematical contents in his theory are studied in detail, including the polynomial multiplier method, the derivati-on of Bezout theorem and his work on the degree of resultant. Finally, it is discussed that the influence of Bezout's resultant theory on geometry, and it is also analyzed that development of Bezout theorem and its application.On the basis of reading original sources and investigating previous researching literatures on this subjects, the major contributions made in the paper are the follows.Firstly, it is analyzed that Bezout's elimination method, and its main idea is the polynomial multiplier. It is also discussed that the mathematical thought and charac-teristic of his two kinds of polynomial multiplier method, and illustrated the motivation for Bezout to found this two methods.Secondly, it is combed that the development process of western elimination theory. It had first appeared in Newton's Arithmetica universalis in1707, in which there are a few rules for elimination in special cases. Then Euler took the next important step. Euler's early work on elimination was motivated instead by his investigations into intersections of curves, and it is the same as Cramer who got similar results around the same time. Meanwhile, Bezout had taken up Newton's rules together with the early works of Euler and Cramer, extending the elimination theory to more than two equations of more than two unknowns.Thirdly, it's studied of Bezout's work on the degree of resultants, and to discuss the reason for him and Euler to get different conclusions. Bezout was the first to found that arriving at equations of higher degree was not just a quirk of any particular method but a deep-rooted problem:that the degree of a resolvent would in general always be higher than that of the original equations. He pointed out the reason for Euler and other people failed to the elimination, and improved the elimination method successfully.Fourthly, it is discussed that the specific application and example of Bezout theorem in the algebraic geometry, and the weak Bezout theorem and strong Bezout theorem. And it is analyzed that the classical and non-classical cases of Bezout theorem in the algebraically closed field, interspersed about the historical background of these cases.Finally, it is discussed that the influence of Bezout's resultant theory on geometry, and to elaborate some geometrical theory arising from this theory. Then, It is tentatively discussed in brief that the possibility of the development of the theorem on the degree of the resultant. At the same time it is discussed that theories reflecting the value of Bezout theorem in geometry. |
