&{\mathbf}{0}\vphantom{^{\operatorname{T}}}
    \end{smallmatrix}\right)}
  {\left(\begin{smallmatrix}\tilde{A}\vphantom{^{\operatorname{T}}} &{\mathbf}{0}\\ B &C\vphantom{^{\operatorname{T}}}
    \end{smallmatrix}\right)}
  &=
  {\left(\begin{smallmatrix}\tilde{H}_i^{\prime}\vphantom{^{\operatorname{T}}}&{\mathbf}{0}\\{\mathbf}{0}
      &{\mathbf}{0}\vphantom{^{\operatorname{T}}}
    \end{smallmatrix}\right)},\\
  N^{\operatorname{T}}{\left(\begin{smallmatrix}\tilde{A}^{\operatorname{T}} &B^{\operatorname{T}}\\ {\mathbf}{0}
      &C^{\operatorname{T}}\end{smallmatrix}\right)} M^{-\operatorname{T}}H_i\,M^{-1}
  {\left(\begin{smallmatrix}\tilde{A}\vphantom{^{\operatorname{T}}} &{\mathbf}{0}\\ B
      &C\vphantom{^{\operatorname{T}}}
    \end{smallmatrix}\right)}N &=H_i^{\prime}.\end{aligned}$$ Therefore, ${{\mathbf}{f}}$ and ${{\mathbf}{g}}$ are equivalent.

Conversely, we assume now that ${{\mathbf}{f}}\sim {{\mathbf}{g}}$, *i.e.* there exists $A\in\operatorname{GL}_n({{\mathbb}{K}})$ such that $A^{\operatorname{T}}\cdot H_i \cdot A=H^{\prime}_i$, for all $i,1 \leq i \leq m$. This implies that: $$N^{-\operatorname{T}}A^{\operatorname{T}}M^{\operatorname{T}}{\left(\begin{smallmatrix}\tilde{H}_i &{\mathbf}{0}\\{\mathbf}{0} &{\mathbf}{0}
  \end{smallmatrix}\right)}M\,A\,N^{-1}={\left(\begin{smallmatrix}\tilde{H}^{\prime}_i
    &{\mathbf}{0}\\{\mathbf}{0} &{\mathbf}{0}
  \end{smallmatrix}\right)}, \forall\, i,1 \leq i \leq m.$$ We then define $\tilde{A}=((MAN^{-1})_{i,j})_{1\leq i,j\leq s}$, so that $\tilde{{{\mathbf}{f}}}(\tilde{A}{{\mathbf}{x}})=\tilde{{{\mathbf}{g}}}({{\mathbf}{x}})$. As ${{\mathbf}{g}}$ has $s$ essential variables, then $\operatorname{rank}\tilde{A}$ cannot be smaller than $s$, hence $\tilde{A} \in \operatorname{GL}_s({{\mathbb}{K}})$. We then get $\tilde{A}^{\operatorname{T}}\tilde{H}_i\tilde{A}=\tilde{H}_i'$ for all $i,1 \leq i
\leq m$, *i.e.* $\tilde{{{\mathbf}{f}}} \sim \tilde{{{\mathbf}{g}}}$.

According to Proposition \[lm:rank\], there is an efficient reduction mapping an instance $({{\mathbf}{f}},{{\mathbf}{g}})$ of [`IP1S`]{}to an instance $(\tilde{{{\mathbf}{f}}},\tilde{{{\mathbf}{g}}})$ of [`IP1S`]{}having only essential variables. From now on, we will then assume that we consider instances of [`IP1S`]{}with $n$ essential variables for both ${{\mathbf}{f}}$ and ${{\mathbf}{g}}$.

#### Canonical Form {#par:identity}

We now assume that $\operatorname{char}{{\mathbb}{K}}\neq 2$.

Let ${{\mathbf}{f}}=(f_1,\ldots,f_m)\in{{\mathbb}{K}}[x_1,\ldots,x_n]^m$ be quadratic homogeneous forms with Hessian matrices $H_1,\ldots,H_m$. We shall say that ${{\mathbf}{f}}$ is *regular* if its number of essential variables is $n$ and if $\exists\,\lambda_1,\dots,\lambda_m\in{{\mathbb}{K}}$ such that $\det{\left(\sum_{i=1}^m\lambda_i\,H_i\right)}\neq 0$.

Our algorithm requires that amongst 