Newton inequality

Newton's inequality

brief introduction

In the field of Mathematics , Newton's inequality is based on Isaac · Newton's name . hypothesis Is the set of real Numbers , Make Express Upper Fundamental symmetric polynomials of order . So the basic symmetric mean Satisfy the inequality

If and only if all Take the equal sign when equal .

prove

A concise proof is the use of Rolle's theorem in mathematical analysis . Equipped with A real number ： . Constructed to A polynomial of roots ：

This polynomial can be written as ：

First of all, prove that ： There is another group A real number ： , Make their basic symmetric mean It happens to be the original The first of the basic symmetric mean values of real numbers n-1 individual ： .

The specific method is to examine polynomials Derivative polynomial of . According to Rolle's Theorem , If two real numbers and inequality , Then there must be a number between them bring . And if the Is a polynomial One of the Second root , So it's also Of Secondary root . therefore , There must be A real root . Let these real roots be equal to , that ：

At the same time ：

Compare the coefficients on both sides , You can get ：

However, among the combinatorial numbers ：

So the equation becomes ：

So we found A real number “ Instead of ” The original A real number , Make the front of the basic symmetric mean None of them will change . It looks like , For arbitrary , After several transformations , It can be transformed into A real number , Make the basic symmetric mean Become the most “ Pull over ” The one of . actually , The above conversion instructions ： Just prove

This one will do .

The following proves this . First , If One of them is 0, So the left side of the inequality , So the left is equal to 0, Obviously smaller than the right . And if the None of them is 0 Words , So since this inequality is homogeneous , So let's assume that . In this case , Inequality becomes ：

That is to say

The final inequality is the mean square inequality , It must be true . So the inequality is proved .

And The relationship between

Another method of proof involves a conclusion in higher mathematics as a lemma ： If for a homogeneous polynomial with respect to two arguments

There are real numbers , Properly There will be , So the partial derivatives of this polynomial of any degree （ It's still a homogeneous polynomial ） The equation formed by ：

Will also meet this condition ： There are real numbers , Properly There will be

The specific proof is to consider the polynomial used in the proof in the previous section ：

Rewrite it as a polynomial with respect to two arguments ：

This polynomial satisfies the condition of lemma , So just consider a particular partial derivative equation ：

This equation can be written as

According to lemma , Corresponding to quadratic equation There are two real roots . So the discriminant of this equation is greater than or equal to zero , in other words ：

It can be seen from this proof that , Newton's inequality is also a discriminant condition corresponding to a quadratic equation , Like the Cauchy inequality . Using the properties of discriminant , A series of inequalities similar to Newton's inequality can be obtained .

history

This inequality was first used by Newton as a method to estimate the number of imaginary roots of polynomials with real coefficients . Newton in his book 《 generalized Arithmetic 》（Arithmetica Universalis） In the second section of the second chapter of the first chapter, an assertion is made without any proof ： polynomial

The number of virtual roots in is equal to the inequality The number that does not hold in . let me put it another way , If a polynomial has only real roots and no imaginary roots , So the above polynomials for all k All should be established .1729 year , McLaughlin has given a direct proof , But a satisfactory solution to this problem will have to wait until 1865 year , Sylvester proved a more general result .