systems are not equivalent or a canonical form $${\left(\big(\sum_{i=1}^{n}d_ix_{i}^{2},f_2,\ldots,f_m\big),
    \big(\sum_{i=1}^{n}d_ix_{i}^{2},g_2,\ldots,g_m\big)\right)},$$ where the $d_i$ are equal to $1$ or a nonsquare in ${{\mathbb}{K}}$, $f_i$ and $g_i$ are *nondegenerate* homogeneous quadratic polynomials in ${{\mathbb}{K}}[x_1,\ldots,x_n]$. Any solution on ${{\mathbb}{K}}$ on the canonical form can be efficiently mapped to a solution of the initial instance (and conversely).

Let us note that over the rationals, computing the exact same sum of squares for the first quadratic forms of each set is difficult, see [@Saxena2006 Chapter 3], [@Wallenborn2013 Chapter 1]. As such, one could only assume that the first quadratic form of the second set is $\sum_{i=1}^nd_i'x_i^2$. This does not fundamentally change the algorithms presented in this paper, beside some matrices denoted by $D$ which could be changed into $D'=\operatorname{Diag}(d_1',\ldots,d_n')$.

Note that the success probability of the algorithms presented here will depend on the size of the field. If one looks for $A\in{{\mathbb}{L}}^{n\times n}$ with ${{\mathbb}{L}}$ an extension of ${{\mathbb}{K}}$, one can amplify the success probability over a small field by using the fact that matrices are conjugate over ${{\mathbb}{K}}$ if and only if they are conjugate over an algebraic extension ${{\mathbb}{L}}$ (see [@Pazzis2010]). Thus, one can search linear change of variables with coefficients in some 