Let $\widehat { \mu } = \left( \mu _ { 1 } > \cdots > \mu _ { n } \right) \in \mathbb { R } ^ { n }$. We are given integers $m _ { k } \geqslant 1$ for $k = 1 , \ldots , n$. We set $$Q _ { 1 } = \prod _ { k = 1 } ^ { n } \left( X - \mu _ { k } \right) ^ { m _ { k } } \quad \text { and } \quad P _ { j } = \frac { Q _ { 1 } } { X - \mu _ { j } } .$$ Let $\left( a , \alpha _ { 1 } , \alpha _ { 2 } , \ldots , \alpha _ { n } \right) \in \mathbb { R } ^ { n + 1 }$ and let $P \in \mathbb { R } [ X ]$ be defined by the formula $$P = ( X - a ) Q _ { 1 } - \sum _ { j = 1 } ^ { n } \alpha _ { j } P _ { j }$$ (a) Give an expression of $P \wedge Q _ { 1 }$ in terms of the $\mu _ { j }$, the $m _ { j }$ and the set $J$ of indices for which $\alpha _ { j } = 0$. (b) Assume that the numbers $\alpha _ { 1 } , \ldots , \alpha _ { n }$ are non-negative. Show that all roots of $P$ are real.
Let $\widehat { \mu } = \left( \mu _ { 1 } > \cdots > \mu _ { n } \right) \in \mathbb { R } ^ { n }$. We are given integers $m _ { k } \geqslant 1$ for $k = 1 , \ldots , n$. We set
$$Q _ { 1 } = \prod _ { k = 1 } ^ { n } \left( X - \mu _ { k } \right) ^ { m _ { k } } \quad \text { and } \quad P _ { j } = \frac { Q _ { 1 } } { X - \mu _ { j } } .$$
Let $\left( a , \alpha _ { 1 } , \alpha _ { 2 } , \ldots , \alpha _ { n } \right) \in \mathbb { R } ^ { n + 1 }$ and let $P \in \mathbb { R } [ X ]$ be defined by the formula
$$P = ( X - a ) Q _ { 1 } - \sum _ { j = 1 } ^ { n } \alpha _ { j } P _ { j }$$
(a) Give an expression of $P \wedge Q _ { 1 }$ in terms of the $\mu _ { j }$, the $m _ { j }$ and the set $J$ of indices for which $\alpha _ { j } = 0$.\\
(b) Assume that the numbers $\alpha _ { 1 } , \ldots , \alpha _ { n }$ are non-negative. Show that all roots of $P$ are real.