equivalent to finding $A \in \operatorname{GL}_n({{\mathbb}{K}})$ such that: $$\label{eq:orig}
  H_i^{\prime}=A^{\operatorname{T}}\cdot H_i \cdot A,   \text{ for all } i, 1 \leq i \leq m.$$ Assuming $H_j$ is invertible, and thus so is $H_j'$, one has $H_j'^{-1}=A^{-1}H_j^{-1}A^{-\operatorname{T}}$. Combining this with equation  yields $H_j'^{-1}H_i'=A^{-1}\cdot
H_j^{-1}H_i\cdot A$. If none of the $H_i$’s is invertible, then we look for an invertible linear combination thereof. For this reason, we assume all along this paper:

 \[as:regular\] Let ${{\mathbf}{f}}=(f_1,\ldots,f_m)\in{{\mathbb}{K}}[x_1,\ldots,x_n]$. We assume that a linear combination over ${{\mathbb}{K}}$ of the quadratic forms $f_1,\ldots,f_m$ is not degenerate[^1]. In particular, we assume that $|{{\mathbb}{K}}| > n$.

Taking as variables the entries of $A$, we can see that  naturally yields a nonlinear system of equations. However, we show that one can essentially linearize equations . To this end, we prove in Section \[s:normalization\] that under Assumption \[as:regular\] any quadratic homogeneous instance [`IP1S`]{}can be reduced, under a randomized process, to a canonical form on which – in particular – all the quadratic forms are nondegenerate. We shall call these instances *regular*. More precisely:

\[th:normal\] Let ${{\mathbb}{K}}$ be a field of $\operatorname{char}{{\mathbb}{K}}\neq 2$. There exists a randomized polynomial-time algorithm which given a regular quadratic homogeneous instance of [`IP1S`]{}returns “[<span style="font-variant:small-caps;">NoSolution</span>]{}” only if the two 