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2014 BMT Spring
P2
BMT 2014 Spring - Analysis P2
BMT 2014 Spring - Analysis P2
Source:
January 6, 2022
Polynomials
Problem Statement
Define
η
(
f
)
\eta(f)
η
(
f
)
to be the number of roots that are repeated of the complex-valued polynomial
f
f
f
(e.g.
η
(
(
x
−
1
)
3
⋅
(
x
+
1
)
4
)
=
5
\eta((x-1)^3\cdot(x+1)^4)=5
η
((
x
−
1
)
3
⋅
(
x
+
1
)
4
)
=
5
). Prove that for nonconstant, relatively prime
f
,
g
∈
C
[
x
]
f,g\in\mathbb C[x]
f
,
g
∈
C
[
x
]
,
η
(
f
)
+
η
(
g
)
+
η
(
f
+
g
)
<
deg
f
+
deg
g
\eta(f)+\eta(g)+\eta(f+g)<\deg f+\deg g
η
(
f
)
+
η
(
g
)
+
η
(
f
+
g
)
<
de
g
f
+
de
g
g
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