关于多项式的Hadamard幂的一个猜想
On a conjecture of Hadamard powers of polynomials
设f(x)=∑ i=0^n aix^i是一个n次正系数多项式,它的p次Hadamard幂是多项式f^[p](x)=∑ i=0^n a i^px^i.一个长期没解决的猜想是：设p>1,若多项式f(x)仅具实零点,则其p次Hadamard幂f^[p]}(x)亦然。本文证明此猜想当n=3时成立,并举例说明当n=4时不成立。本文也证明存在正数Pn,使得当p>Pn,若$n$次正系数多项式f(x)仅具实零点,则其p次Hadamard幂f^[p](x)亦然。
Let f(x)=∑ i=0^n aix^i be a polynomial with positive coefficients and p>0.The pth Hadamard power of f(x) is the polynomial f^[p](x)=∑ i=0^n ai^px^i.It is conjectured that if f(x) has only real zeros, then so does f^[p](x) for p≥1.We verify the conjecture when n=3 and give a counterexample when n=4.We also show that there exists a positive number Pn such thatif f(x) has only real zeros, then so does f^[p](x) for p>Pn.
王毅、张滨
数学
组合数学实零点多项式Hadamard幂
ombinatoricsPolynomial with only real zerosHadamard power
王毅,张滨.关于多项式的Hadamard幂的一个猜想[EB/OL].(2013-09-06)[2025-08-16].http://www.paper.edu.cn/releasepaper/content/201309-83.点此复制
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