Every System is equivalent to a System Of Quadrics

1. Show that solving every (0-dimensional) system of equations $F(\x)=0$
is equivalent to solving a system of quadratic equations $G(\y)=0$ in the following sense: there is a map $\phi: k[\y]=k[y_1,\ldots,y_m]\to k[\x]=k[x_1,\ldots,x_n]$ such that $\phi$ is an isomorphism of $\V(F)$ and $\V(G)$ and the latter is defined by

(1)
\begin{align} G(\y) = F(\phi(\y)) \cup \{y_i-\phi(y_i) : i\in[m]\}, \end{align}

which are polynomials of degree $2$.

2. Construct such $\phi$ for

(2)
\begin{align} F = \{ & 4 x_2 x_3^{8}-5 x_1 x_3^{3}-3 x_1^{2} x_2+x_1 x_2^{2}-8,\\ & x_3^{9}-3 x_1 x_2 x_3^{5}+x_1 x_3^{3}-7 x_1^{2} x_2-2 x_1 x_2^{2}-x_2-1,\\ & 2 x_1 x_2 x_3^{9}+5 x_1 x_3^{9}+5 x_2 x_3^{8}-x_1^{2} x_2-4 x_1 x_2^{2}+5 x_3+1 \} \end{align}

3. (!!!experiment!!!) Try computing the Groebner basis of $\<G\>$, eliminate all but one variables, compute numerically the points of $\V(G)$. How does the performance of these compare to that of the same procedures executed for $\<F\>$ and $\V(F)$?

4. What is the Waring rank of a (quadratic) homogeneous polynomial? Prove that any projective variety (in $\P^n$) is isomorphic to a projective variety (embedded in $\P^N$ for some $N$) cut out by a linear polynomial and quadtratic polynomials of rank at most 4.