Symmetric Sums Thứ k tiểu đối xứng chức năng của n biến x1, x2,. . . , Xn là các đa thức định nghĩa ek công thức (x1, x2,., Xn)Các chức năng này được sử dụng trong hầu như tất cả các khoa học toán học, nhưng họ rút ra nhiều tầm quan trọng của kết nối mà họ cung cấp từ các hệ số của một đa thức và chức năng của nguồn gốc của nó. | 12 Symmetric Sums The fcth elementary symmetric function of the n variables x1 x2 . xn is the polynomial defined the formula efc x X2 . Xn Xil Xi2 Xik . 1 ii i2 --- ik n The first three of these polynomials are simply eo xi X2 . . . Xn 1 e1 x1 X2 . . . Xn X1 X2 . Xn and e2 xi X2 . xn yt XjXk while the nth elementary symmetric function is simply the full product en xi X2 . . . Xn X1X2 Xn. These functions are used in virtually every part of the mathematical sciences yet they draw much of their importance from the connection they provide between the coefficients of a polynomial and functions of its roots. To be explicit if the polynomial P t is written as the product P t t X1 t x2 t xn then it also has the representation P t tn e1 x tn 1 1 k ek x tn k 1 nen x where for brevity we have written ek x in place of ek x1 x2 . xn . The Classical Inequalities of Newton and Maclaurin The elementary polynomials have many connections with the theory of inequalities. Two of the most famous of these date back to the great Isaac Newton 1642-1727 and the Scottish prodigy Colin Maclaurin 178 Symmetric Sums 179 1696-1746 . Their namesake inequalities are best expressed in terms of the averages - . A _ ek x1 x2 . xn Ek x Ek x1 x2 . . . xn n which bring us to our first challenge problem. Problem Inequalities of Newton and Maclaurin Show that for all x G Rn one has Newton s inequalities Ek-1 x Ek 1 x E2 x for 0 k n and check that they imply Maclaurin s inequalities which assert that En n x En- n-1 x E2 x 1 2 Ei x for all x xi X2 . . xn such that Xk 0 for all 1 k n. Orientation and the AM-GM Connection If we take n 3 and set x x y z then Maclaurin s inequalities simply say xyz 1 3 i xy xz yz 1 2 x y z I 3 3 which is a sly refinement of the AM-GM inequality. In the general case Maclaurin s inequalities insert a whole line of ever increasing expressions between the geometric mean x1x2 xn 1 n and the arithmetic mean xi x2 xn n. From Newton to Maclaurin By Geometry .
