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Let there be a polynomial of degree
n
≥
1
{\displaystyle n\geq 1}
P
(
z
)
=
a
n
z
n
+
a
n
−
1
z
n
−
1
+
⋯
+
a
1
z
+
a
0
∈
C
[
z
]
{\displaystyle P(z)=a_{n}z^{n}+a_{n-1}z^{n-1}+\cdots +a_{1}z+a_{0}\in \mathbb {C} [z]}
By the Fundamental Theorem of Algebra , it has
n
{\displaystyle n}
complex roots (with multiplicity). Then we can write:
P
(
z
)
=
a
n
(
z
−
z
1
)
(
z
−
z
2
)
⋯
(
z
−
z
n
)
{\displaystyle P(z)=a_{n}(z-z_{1})(z-z_{2})\cdots (z-z_{n})}
As we know, Vieta's formulae link between the coefficients and roots of a polynomial:
{
z
1
+
z
2
+
⋯
+
z
n
=
−
a
n
−
1
a
n
(
z
1
z
2
+
z
1
z
3
+
⋯
+
z
1
z
n
)
+
(
z
2
z
3
+
z
2
z
4
+
⋯
+
z
2
z
n
)
+
⋯
+
z
n
−
1
z
n
=
a
n
−
2
a
n
(
z
1
z
2
z
3
+
⋯
+
z
1
z
2
z
n
)
+
(
z
1
z
3
z
4
+
⋯
+
z
1
z
3
z
n
)
+
⋯
+
(
z
2
z
3
z
4
+
⋯
+
z
2
z
n
−
1
z
n
)
+
⋯
+
z
n
−
2
z
n
−
1
z
n
=
−
a
n
−
3
a
n
⋮
z
1
⋯
z
n
=
(
−
1
)
n
a
0
a
n
{\displaystyle {\begin{cases}z_{1}+z_{2}+\cdots +z_{n}=-{\dfrac {a_{n-1}}{a_{n}}}\\(z_{1}z_{2}+z_{1}z_{3}+\cdots +z_{1}z_{n})+(z_{2}z_{3}+z_{2}z_{4}+\cdots +z_{2}z_{n})+\cdots +z_{n-1}z_{n}={\dfrac {a_{n-2}}{a_{n}}}\\(z_{1}z_{2}z_{3}+\cdots +z_{1}z_{2}z_{n})+(z_{1}z_{3}z_{4}+\cdots +z_{1}z_{3}z_{n})+\cdots +(z_{2}z_{3}z_{4}+\cdots +z_{2}z_{n-1}z_{n})+\cdots +z_{n-2}z_{n-1}z_{n}=-{\dfrac {a_{n-3}}{a_{n}}}\\\vdots \\z_{1}\cdots z_{n}=(-1)^{n}{\dfrac {a_{0}}{a_{n}}}\end{cases}}}
As we can see, these sums are symmetric polynomial, and are called elementary symmetric polynomials .
The elementary symmetric polynomials in variables
X
1
,
…
,
X
n
{\displaystyle X_{1},\ldots ,X_{n}}
, are defined as such:
E
1
(
X
→
n
)
=
∑
1
≤
i
≤
n
X
i
E
2
(
X
→
n
)
=
∑
1
≤
i
1
<
i
2
≤
n
X
i
1
X
i
2
⋮
E
k
(
X
→
n
)
=
∑
1
≤
i
1
<
⋯
<
i
k
≤
n
X
i
1
⋯
X
i
k
⋮
E
n
(
X
→
n
)
=
X
1
⋯
X
n
{\displaystyle {\begin{aligned}E_{1}({\vec {X}}{}^{n})&=\sum _{1\leq i\leq n}X_{i}\\[5pt]E_{2}({\vec {X}}{}^{n})&=\sum _{1\leq i_{1}<i_{2}\leq n}\!\!\!\!X_{i_{1}}X_{i_{2}}\\&\,\,\,\vdots \\[5pt]E_{k}({\vec {X}}{}^{n})&=\sum _{1\leq i_{1}<\cdots <i_{k}\leq n}\!\!\!\!\!\!X_{i_{1}}\!\cdots X_{i_{k}}\\&\,\,\,\vdots \\[5pt]E_{n}({\vec {X}}{}^{n})&=X_{1}\!\cdots X_{n}\end{aligned}}}