the affine case; typically by considering the homogeneous components, see [@FaugerePerret06]. However, homogenizing the instances allows us to make the proofs simpler and cleaner. As such, we focus our attention to solve [`IP1S`]{}for quadratic homogeneous forms.

When $m=1$, the [`IP1S`]{}problem can be easily solved by computing a reduced form of the input quadratic forms. In [@DBLP:conf/pkc/BouillaguetFFP11], the authors present an efficient heuristic algorithm for solving [`IP1S`]{}on quadratic instances. However, the algorithm requires to compute a Gröbner basis. So, its complexity could be exponential in the worst case. More recently, [@MacariotRatPG2013] proposed a polynomial-time algorithm for solving [`IP1S`]{}on quadratic instances with $m=2$ over fields of any characteristic. We consider here arbitrary $m>1$.

In computer algebra, a fundamental and related problem is the simplification of a homogeneous polynomial system ${{\mathbf}{f}}\in{{\mathbb}{K}}[{{\mathbf}{x}}]^m$. That is, compute $A\in\operatorname{GL}_n({{\mathbb}{K}})$ such that ${{\mathbf}{g}}({{\mathbf}{x}})={{\mathbf}{f}}(A\cdot{{\mathbf}{x}})$ is easier to solve. In this setting, <span style="font-variant:small-caps;">Ridge</span> algorithm (see [@BerthomieuHM2010; @Hironaka1970; @Giraud1972]) and <span style="font-variant:small-caps;">MinVar</span> algorithm (see [@Carlini05reducingthe; @Kayal2011]) reduce to the best the number of variables of the system. More generally, for a given homogeneous polynomial system ${{\mathbf}{f}}$, the *Functional Decomposition Problem* is the problem of computing $\mathbf{h}=(h_1,\ldots,h_s)$ homogeneous and ${{\mathbf}{g}}$ such that ${{\mathbf}{f}}({{\mathbf}{x}})={{\mathbf}{g}}(\mathbf{h}({{\mathbf}{x}}))$.

To simplify the presentation in this introduction, we 