某个6次多项式满足s1=s3=0,求证:s7/7=(s5/5)(s2/2)其中si为幂和对称多项式

Question

某个6次多项式满足s1=s3=0,求证:s7/7=(s5/5)(s2/2)
其中si为幂和对称多项式

Answer 1

设ti为i次初等对称多项式.
有牛顿恒等式:
0 = s1 = t1,故t1 = 0.
s2 = s1·t1-2·t2 = -2·t2.
0 = s3 = s2·t1-s1·t2+3·t3 = 3·t3,故t3 = 0.
s5 = s4·t1-s3·t2+s2·t3-s1·t4+5·t5 = 5·t5.
s7 = s6·t1-s5·t2+s4·t3-s3·t4+s2·t5-s1·t6 = -s5·t2+s2·t5 = -7·t2·t5.
于是s7/7 = -t2·t5 = (s2/2)(s5/5).