Explain the terms : homogeneous , symmetric , anti-symmetric , cyclic with respect to polynomials. In answers to some of my previous questions , a lot of people used the terms homogeneous polynomial ( in a,b,c ) (under permutations of variables ) , cyclic polynomial ( in a,b,c)  (under permutations of variables ) , anti-symmetric and symmetric polynomial . Please explain their meaning .
Also in answering my previous question , Barry Cipra said  , ( link )
" The expression
$P(a,b,c)=(a+b+c)(−a+b+c)(a−b+c)(a+b−c)$
is clearly a homogeneous polynomial of degree 4, symmetric in its three variables. It's also clear that the coefficient of $a^4$ (hence also $b^4$ and $c^4$) is −1. Moreover,
$P(−a,b,c)=P(a,−b,c)=P(a,b,−c)=P(a,b,c)$
which implies P has no terms with any variable taken to an odd degree."
I don't understand why P has no variable with odd degree. So please explain that too.
( For context , see my previous questions :
Simplify : $( \sqrt 5 + \sqrt6 + \sqrt7)(− \sqrt5 + \sqrt6 + \sqrt7)(\sqrt5 − \sqrt6 + \sqrt7)(\sqrt5 + \sqrt6 − \sqrt7) $
Factorize: $a^2(b − c)^3 + b^2(c − a)^3 + c^2(a − b)^3$
Factorise: $2a^4 + a^2b^2 + ab^3 + b^4$ )
 A: Homogeneous
A function $f$ of $n$ variables is homogeneous of degree $k$ if
$$
f(tx_1, \dots, tx_n) = t^k f(x_1, \dots, x_n).
$$
For example:
$$
 x^2 + y^2,\quad xyz,\quad x^4 - 5x^3y,\quad \cos \frac{x}{y}.
$$
Note that the total degree of each term of the polynomials are the same. For $x^4-5x^3y$, the total degree of the first term is $4 + 0 = 4$, and for the second term it's $3 + 1 = 4$.
Euler showed that this implies for differentiable $f$:
$$
x_1 \frac{\partial f}{\partial x_1} + \cdots + x_n \frac{\partial f}{\partial x_n} = k f.
$$
Symmetric
A function f is symmetric if
$$
f(\dots, x_i, \dots, x_j, \dots) = f(\dots, x_j, \dots, x_i, \dots)\quad\text{for all }i, j.
$$
For example, 
$$
x + y,\quad xyz,\quad xy + xz + yz,\quad x^8 + y^8 + z^8 + w^8.
$$
If a symmetric polynomial has a monomial $x^3 y^2 z$, it also has a monomial $x y^3 z^2$, for example.  Any symmetric polynomial is a polynomial in the elementary symmetric functions:
$$
\begin{align}
e_0(x_1, \dots, x_n) &= 1 \\
e_1(x_1, \dots, x_n) &=\sum_{1\le i \le n} x_i \\
e_2(x_1, \dots, x_n) &=\sum_{1\le i < j \le n} x_i x_j\\
\dots\\
e_n(x_1, \dots, x_n) &=  x_1 x_2 \cdots x_n.
\end{align}
$$
Anti-symmetric
A function f is anti-symmetric if
$$
f(\dots, x_i, \dots, x_j, \dots) = - f(\dots, x_j, \dots, x_i, \dots)\quad\text{for all }i, j.
$$
For example, 
$$
(x-y) (x-z) (y-z)
$$
What happens to this if you swap $x$ and $y$?
$$
(y-x)(y-z)(x-z) = -(x-y) (x-z) (y-z)
$$
Swapping $x$ and $z$, or swapping $y$ and $z$ does the same.
Cyclic
A polynomial is cyclic if
$$
f(x_1, \dots, x_n) = f(x_2, \ldots, x_n, x_1)
$$
This implies that 
$$
f(x_1, \dots, x_n) = f(x_{k+1}, \ldots, x_n, x_1, \dots, x_k)\quad\mbox{for any } k.
$$
For example, $x^3y + y^3z + z^3x$.  If you replace $(x, y, z)$ with $(y, z, x)$, you get $y^3z + z^3x + x^3 y$, which equals the original.
