the square root in a certain representation in a matrix algebra. While computing square roots of matrices can be done efficiently using numerical methods, it seems difficult to control the bit complexity of such methods. However, we present exact and polynomial-time algorithms for computing a representation of the square root of a matrix in $\mathbb{K}^{n \times n}$, for various fields (including finite fields), as a product of two matrices. Each coefficient of these matrices lie in an extension field of $\mathbb{K}$ of polynomial degree. We then consider \#`IP1S`, the counting version of `IP1S` for quadratic instances. In particular, we provide a (complete) characterization of the automorphism group of homogeneous quadratic polynomials. Finally, we also consider the more general *Isomorphism of Polynomials* (`IP`) problem where we allow an invertible linear transformation on the variables *and* on the set of polynomials. A randomized polynomial-time algorithm for solving `IP` when $\mathbf{f}=(x_1^d,\ldots,x_n^d)$ is presented. From an algorithmic point of view, the problem boils down to factoring the determinant of a linear matrix (*i.e.* a matrix whose components are linear polynomials). This extends to `IP` a result of Kayal obtained for `PolyProj`.
address:
- 'Sorbonne Universités, <span style="font-variant:small-caps;">UPMC</span> Univ Paris 06, Équipe <span style="font-variant:small-caps;">PolSys</span>, <span 