Suppose the following real polynomial of $n$ variables $$f(X_1,X_2,\cdots,X_n)=\sum_{I=(i_1,i_2,\cdots,i_n)}a_IX_1^{i_1}X_2^{i_2}\cdots X_n^{i_n}$$ is easy or familiar to us. However, I need to deal with the following real polynomial: $$F(X_1,X_2,\cdots,X_n)=\sum_{I=(i_1,i_2,\cdots,i_n)}sign(a_I)a_I^2X_1^{i_1}X_2^{i_2}\cdots X_n^{i_n},$$ where $\begin{eqnarray}sign(a_I)= \begin{cases} 1,&a_I>0\cr 0, &a_I=0 \cr -1, &a_I<0\end{cases} \end{eqnarray}.$

I want to ask if there is any method or skill to associate $F(X_1,X_2,\cdots,X_n)$ with $f(X_1,X_2,\cdots,X_n)$?