mainly deal with fields of characteristic $\not =2$. Results for fields of characteristic $2$ are also given later in this paper. Now, we define formally [`IP1S`]{}:

Let $\big( {{\mathbf}{f}}=(f_1,\ldots,f_m),{{\mathbf}{g}}=(g_1,\ldots,g_m)\big) \in
  {{\mathbb}{K}}[x_1,\ldots,x_n]^m \times {{\mathbb}{K}}[x_1,\ldots,x_n]^m$. We shall say that ${{\mathbf}{f}}$ and ${{\mathbf}{g}}$ are equivalent, denoted ${{\mathbf}{f}}\sim {{\mathbf}{g}}$, if $\exists$ $A
  \in \operatorname{GL}_n({{\mathbb}{K}})$ such that: $${{\mathbf}{g}}({{\mathbf}{x}})={{\mathbf}{f}}(A \cdot {{\mathbf}{x}}).$$ [`IP1S`]{}is then the problem of finding – if any – $A \in \operatorname{GL}_n({{\mathbb}{K}})$ that makes ${{\mathbf}{g}}$ equivalent to ${{\mathbf}{f}}$ (*i.e.* $A \in \operatorname{GL}_n({{\mathbb}{K}})$ such that ${{\mathbf}{g}}({{\mathbf}{x}})={{\mathbf}{f}}(A \cdot {{\mathbf}{x}})$).

In such a case, we present a randomized polynomial-time algorithm for solving [`IP1S`]{}with quadratic polynomials. To do so, we show that such a problem can be reduced to the variant of a classical problem of representation theory over finite dimensional algebras. In our setting we need, as in the case $m=1$, to provide a canonical form of the problem.

### Canonical Form of [`IP1S`]{} {#canonical-form-of-ip1s .unnumbered}

Let $\big( {{\mathbf}{f}}=(f_1,\ldots,f_m),{{\mathbf}{g}}=(g_1,\ldots,g_m)\big) \in
{{\mathbb}{K}}[x_1,\ldots,x_n]^m \times {{\mathbb}{K}}[x_1,\ldots,x_n]^m$ be homogeneous quadratic polynomials. 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 $g_1,\ldots,g_m$). Recall that the Hessian matrix associated to a $f_i$ is defined as $H_i=\left(\frac{\partial^2
    f_i}{\partial x_k \partial x_{\ell}}\right)_{k,\ell}\in
\mathbb{K}^{n\times n}$. Consequently, [`IP1S`]{}for quadratic forms is 