only essential variables.

\[lm:rank\] Let $({{\mathbf}{f}},{{\mathbf}{g}})\in{{\mathbb}{K}}[x_1,\ldots,x_n]^m \times {{\mathbb}{K}}[x_1,\ldots,x_n]^m $ be two sets of quadratic polynomials. If ${{\mathbf}{f}}\sim {{\mathbf}{g}}$, then their numbers of essential variables must be the same. Let $s$ be the number of essential variables of ${{\mathbf}{f}}$. Finally, let $(\tilde{{{\mathbf}{f}}},\tilde{{{\mathbf}{g}}})\in{{\mathbb}{K}}[y_1,\ldots,y_s]^m \times
  {{\mathbb}{K}}[y_1,\ldots,y_s]^m$ be such that: $${{\mathbf}{f}}=\tilde{{{\mathbf}{f}}}(\ell_1,\ldots,\ell_{s}) \mbox{ and }
  {{\mathbf}{g}}=\tilde{{{\mathbf}{g}}}(\ell'_1,\ldots,\ell'_{s}),$$ with $\ell_1,\ldots,\ell_{s}$ (resp. $\ell'_1,\ldots,\ell'_{s}$) linear forms in ${{\mathbf}{x}}$ of rank $s$ and $\tilde{{{\mathbf}{f}}},
  \tilde{{{\mathbf}{g}}}\in{{\mathbb}{K}}[y_1,\ldots,y_{s}]^m$. It holds that: $${{\mathbf}{f}}\sim {{\mathbf}{g}}\iff \tilde{{{\mathbf}{f}}} \sim \tilde{{{\mathbf}{g}}}.$$

Let $H_1,\ldots,H_m$ be the Hessian matrices of $f_1,\ldots,f_m$ (resp. $H^{\prime}_1,\ldots,H^{\prime}_m$ be the Hessian matrices of $g_1,\ldots,g_m$). Similarly, we define the Hessian matrices $\tilde{H}_1,\ldots,\tilde{H}_m$ (resp. $\tilde{H}^{\prime}_1,\ldots,\tilde{H}^{\prime}_m$) of $\tilde{f}_1,\ldots,\tilde{f}_m$ (resp. $\tilde{g}_1,\ldots,\tilde{g}_m$). Let also $M$ and $N$ be matrices in $\operatorname{GL}_n({{\mathbb}{K}})$ such that $H_i=M^{\operatorname{T}}{\left(\begin{smallmatrix} \tilde{H}_i & {\mathbf}{0}\\{\mathbf}{0}
      &{\mathbf}{0}\end{smallmatrix}\right)}M$ and $H_i'=N^{\operatorname{T}}{\left(\begin{smallmatrix}
      \tilde{H}_i' &{\mathbf}{0}\\{\mathbf}{0} &{\mathbf}{0}\end{smallmatrix}\right)}N$ for all $i,1\leq i\leq
  m$. There exist such $M$ and $N$, as ${{\mathbf}{f}}$ and ${{\mathbf}{g}}$ have essentially $s$ variables. Up to re-indexing the rows and columns of $H_i$ and $H_i'$, so that they remain symmetric, one can always choose $M$ and $N$ such that $M={\left(\begin{smallmatrix}M_1&M_2\\{\mathbf}{0}&{{\mathrm}{Id}}\end{smallmatrix}\right)}$ and $N={\left(\begin{smallmatrix}N_1&N_2\\{\mathbf}{0}&{{\mathrm}{Id}}\end{smallmatrix}\right)}$, with $M_1,N_1\in\operatorname{GL}_s({{\mathbb}{K}})$.

If $\tilde{{{\mathbf}{f}}} \sim \tilde{{{\mathbf}{g}}}$, $\exists \, \tilde{A}\in\operatorname{GL}_s({{\mathbb}{K}})$ such that $A^{\operatorname{T}}\tilde{H}_i\tilde{A}=\tilde{H}_i'$, for all $i,1\leq i\leq m$. Then, for all $B\in{{\mathbb}{K}}^{(n-s)\times s}$ and $C\in\operatorname{GL}_{n-s}({{\mathbb}{K}})$: $$\begin{aligned}
    {\left(\begin{smallmatrix}\tilde{A}^{\operatorname{T}} &B^{\operatorname{T}}\\ {\mathbf}{0}
        &C^{\operatorname{T}}\end{smallmatrix}\right)}
  {\left(\begin{smallmatrix}\tilde{H}_i\vphantom{^{\operatorname{T}}}&{\mathbf}{0}\\{\mathbf}{0} 