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Vojtěch Jarník IMC
1995 VJIMC
Problem 2
polynomial in z+1/z
polynomial in z+1/z
Source: VJIMC 1995 2.2
November 25, 2021
algebra
polynomial
Problem Statement
Let
f
=
f
0
+
f
1
z
+
f
2
z
2
+
…
+
f
2
n
z
2
n
f=f_0+f_1z+f_2z^2+\ldots+f_{2n}z^{2n}
f
=
f
0
+
f
1
z
+
f
2
z
2
+
…
+
f
2
n
z
2
n
and
f
k
=
f
2
n
−
k
f_k=f_{2n-k}
f
k
=
f
2
n
−
k
for each
k
k
k
. Prove that
f
(
z
)
=
z
n
g
(
z
+
z
−
1
)
f(z)=z^ng(z+z^{-1})
f
(
z
)
=
z
n
g
(
z
+
z
−
1
)
, where
g
g
g
is a polynomial of degree
n
n
n
.
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