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Silk Road
2023 Silk Road
4
4
Part of
2023 Silk Road
Problems
(1)
polynomials of three variables with rational coefficients
Source: SRMC 2023, P4
12/28/2023
Let
M
=
Q
[
x
,
y
,
z
]
\mathcal{M}=\mathbb{Q}[x,y,z]
M
=
Q
[
x
,
y
,
z
]
be the set of three-variable polynomials with rational coefficients. Prove that for any non-zero polynomial
P
∈
M
P\in \mathcal{M}
P
∈
M
there exists non-zero polynomials
Q
,
R
∈
M
Q,R\in \mathcal{M}
Q
,
R
∈
M
such that
R
(
x
2
y
,
y
2
z
,
z
2
x
)
=
P
(
x
,
y
,
z
)
Q
(
x
,
y
,
z
)
.
R(x^2y,y^2z,z^2x) = P(x,y,z)Q(x,y,z).
R
(
x
2
y
,
y
2
z
,
z
2
x
)
=
P
(
x
,
y
,
z
)
Q
(
x
,
y
,
z
)
.
algebra
polynomial
Multivariable