Многочлен

[Navid]

Позволять многочлены $P(x,y,z)$ where for all $ x,y,z$ we have:
$ P(x+y, y+z, z+x)= 2P(x,y,z) $
Докажи это $ P(x,y,z)=(x+y+z)Q( (x-y) ^2 , (y-z) ^2 , (z-x)^2) $ где $Q$ симметрический полином

[Lia]

Navid
Please, provide an English version of your post.

[Navid]

Let $P(x,y,z)$ be a polynomial in three variables, such that :
$ P(x+y, y+z, z+x)= 2P(x,y,z) $
Prove that there exist a Symmetric Polynomial $Q$ such that: $ P(x,y,z)=(x+y+z)Q( (x-y) ^2 , (y-z) ^2 , (z-x)^2) $ .

[DeBill]

Navid
The assertion is wrong.
Let $x+y+z = 0$. Using the functional equation 6 times we get $2^6\cdot P(x,y,z)= P(x,y,z)$, so $P(x,y,z) = 0$.
Hence $P(x,y,z) = (x+y+z)\cdot F(x,y,z)$ for some polynomial $F$.
Then $F(x+y,y+z,z+x) = F(x,y,z)$ for all $x,y,z$. (1)
Let $F(S+b-c,S+c-a,S+a-b) =f(S,a,b,c)$. By the same way we get
$f(64S,a,b,c) = f(S,a,b,c)$, so $f$ does not depend on $S$, $f=f(a,b,c)$.
We have from (1):
$f(a,b,c) = f(-c,-a,-b)$ (2).
Using (2) we obtained $f(a,b,c) = f(b,c,a) = f(c,a,b)$ and $f(a,b,c) = f(-a,-b,-c)$
So, $f$ is "almost symmetric" and "almost even"....
Substituting $S=\frac{x+y+z}{3}, a=\frac{z-y}{3}, b=\frac{x-z}{3},c = \frac{y-x}{3}$ we get $F(x,y,z)=f(a,b,c) $
Example: $P(x,y,z)= (x+y+z)\cdot ((x-y)^4\cdot (y-z)^2 +(y-z)^4\cdot(z-x)^2 +(z-x)^4\cdot(x-y)^2)$.

[Navid]

DeBill в сообщении #1139466 писал(а):

Navid
The assertion is wrong.
Let $x+y+z = 0$. Using the functional equation 6 times we get $2^6\cdot P(x,y,z)= P(x,y,z)$, so $P(x,y,z) = 0$.
Hence $P(x,y,z) = (x+y+z)\cdot F(x,y,z)$ for some polynomial $F$.
Then $F(x+y,y+z,z+x) = F(x,y,z)$ for all $x,y,z$. (1)
Let $F(S+b-c,S+c-a,S+a-b) =f(S,a,b,c)$. By the same way we get
$f(64S,a,b,c) = f(S,a,b,c)$, so $f$ does not depend on $S$, $f=f(a,b,c)$.
We have from (1):
$f(a,b,c) = f(-c,-a,-b)$ (2).
Using (2) we obtained $f(a,b,c) = f(b,c,a) = f(c,a,b)$ and $f(a,b,c) = f(-a,-b,-c)$
So, $f$ is "almost symmetric" and "almost even"....
Substituting $S=\frac{x+y+z}{3}, a=\frac{z-y}{3}, b=\frac{x-z}{3},c = \frac{y-x}{3}$ we get $F(x,y,z)=f(a,b,c) $
Example: $P(x,y,z)= (x+y+z)\cdot ((x-y)^4\cdot (y-z)^2 +(y-z)^4\cdot(z-x)^2 +(z-x)^4\cdot(x-y)^2)$.

Thank you very much!, I think we must change the Condition of The Statement for polynomial $Q$, By this manner:
where $Q$- arbitrary polynomial, unchanged under cyclic permutations of variables.

[DeBill]

Navid
and "almost even"...That is, with all monomials of even degree.

[Navid]

DeBill в сообщении #1139495 писал(а):

Thank you!