广义多项式的Descartes符号法则及其在降维方法中的应用

Descartes' Law of Signs for Generalized Polynomials and Its Application in the Method of Descent

一般幂函数的实线性组合称为广义多项式.本文证明了广义多项式的正根数目不超过其系数组的变号数,即广义多项式上的Descartes符号法则成立.应用此结果发展了对称函数不等式证明的降维方法,用发展后的降维方法处理幂平均问题,得到了更优结果.

The real linear combination of power functions is called a generalized polynomial. In this paper, we prove that the Descartes＇ law of signs for generalized polynomials holds. That is to say the number of positive roots of a generalized polynomial is not more than the number of the sign changes of its coefficients. Then, using this law, we generalize the method of descent for proving the proposition of symmetric inequalities. Finally, the generalized method of descent is applied to the problems of power mean, which is more efficient and effective.