Polynomial Forms

# Polynomial Forms

 Definition: Let $R$ be a ring. A polynomial $F \in R[x_1, x_2, ..., x_n]$ is said to be a Form of Degree $d$ if every term in $F$ is of degree $d$.

For example, the following polynomials are forms of degree $2$, $3$, and $4$ respectively:

(1)
\begin{align} \quad F(x, y, z) = x^2 + xy \quad , \quad G(x, y, z) = xyz + y^2z + z^3 \quad , \quad H(x, y, z) = x^4 + y^4 + z^4 \end{align}
 Definition: Let $R$ be an integral domain and let $F \in R[x_1, x_2, ..., x_n]$ be a form. The Dehomogenization of $F$ is a polynomial $F_* \in R[x_1, x_2, ..., x_{n-1}]$ defined by $F_*(x_1, x_2, ..., x_{n-1}) = F(x_1, x_2, ..., x_{n-1}, 1)$.

For example, if $F(x, y, z) = x^3 + xyz + y^2z$, then we dehomogenize $F$ to get:

(2)
\begin{align} \quad F_*(x, y, z) = F(x, y) = x^3 + xy + y^2 \end{align}

Note that $F_*$ is not a form in general.

 Definition: Let $R$ be an integral domain and let $F \in R[x_1, x_2, ..., x_n]$ be a polynomial of degree $d$ where we write $F = \sum_{k=0}^{d} F_k$ where each $F_k$ is the sum of all terms in $F$ of degree $k$ (so that each $F_k$ is a form of degree $k$). The Homogenization of $F$ is a polynomial $F^* \in R[x_1, x_2, ..., x_{n+1}$ defined by $F^*(x_1, x_2, ..., x_n, x_{n+1}) = \sum_{k=0}^{d} x_{n+1}^{d - k}F_k(x_1, x_2, ..., x_n)$.

For example, if $F(x, y, z) = x + y + xy + z^2 + xyz$, then:

(3)
\begin{align} \quad F_0(x, y, z) &= 0 \\ \quad F_1(x, y, z) &= x + y \\ \quad F_2(x, y, z) &= xy + z^2 \\ \quad F_3(x, y, z) &= xyz \end{align}

So we homogenize $F$ to get:

(4)
\begin{align} \quad F^*(x, y, z, w) = w^2F_1(x, y, z) + wF_2(x, y, z) + F_3(x, y, z) = w^2x + w^2y + wxy + wz^2 + xyz \end{align}

Note that if $F$ is a polynomial of degree $d$ then the homogenization $F^*$ is a form of degree $d$.