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Nature of the Roots of Real Polynomial Equations
E 2055 [1968, 188]. Proposed by M. F. Capobianco, St. John's University
Consider a real polynomial equation of degree $n$. Attention is paid to whether the roots are real and unequal, real and equal (in various combinations), or simple or multiple complex conjugates. If $n=2$ there are but three possibilities, namely, all roots real and equal, all roots real and unequal, and all roots complex. If $n=3$, there are four possibilities: three equal, two equal, three unequal, two complex. For $n=4$, there are nine possibilities. How many possibilities are there for general $n$ ?
Solution. Denoting by $m_i, i=1,2, \cdots, r$, the multiplicities of the conjugate pairs of imaginary roots and by $n_i, i=1,2, \cdots, s$, the multiplicities of the real roots, we seek the number of solutions of
\[
2 \sum_{i=1}^r m_i+\sum_{i=1}^s n_i=n
\]
subject to the restrictions $m_1 \leqq m_2 \leqq \cdots \leqq m_r, n_1 \leqq n_2 \leqq \cdots \leqq n_s$. This is given by the convolution
\[
\mu_n=\sum_{k=0}^{[n / 2]} P(k) P(n-2 k)
\]
where $P(k)$ is the number of unrestricted partitions of $k$, with the well-known (see, for example, Riordan, An Introduction to Combinatorial Analysis, Wiley, New York, 1958, p. 111) generating function
\[
\begin{aligned}f(t)=\sum_{k=0}^{\infty} P(k) t^k&=\prod_{k=0}^{\infty}\left(1-t^k\right)^{-1}\\
&= 1+t+3 t^2+4 t^3+9 t^4+12 t^5+23 t^6+31 t^7+54 t^8+73 t^9+118 t^{10}+\cdots
\end{aligned}\] |
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ljh25252
发表于 2024-12-29 05:25
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