$\def\<{\langle} \def\>{\rangle} \newcommand{\p}{\partial} \newcommand{\x}{{\boldsymbol{x}}} \newcommand{\y}{{\boldsymbol{y}}} \newcommand{\A}{{\mathbb{A}}} \renewcommand{\P}{{\mathbb{P}}} \newcommand{\V}{{\mathbb{V}}} \newcommand{\G} {{\mathbb{G}}} \newcommand{\Gr}{{\operatorname{Gr}}} \def\Grob{Gr\"obner} \newcommand{\bC}{{\mathbb{C}}} \newcommand{\bR}{{\mathbb{R}}} \newcommand{\bN}{{\mathbb{N}}} \newcommand{\bQ}{{\mathbb{Q}}} \newcommand{\bH}{{\mathbb{H}}} \newcommand{\MM}{{\bf{M2}}}$

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

which are polynomials of degree $2$.

2. Construct such $\phi$ for

(2)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.