variables $x_{s+1}\ldots,x_n$ will be called *redundant* variables. If $\operatorname{char}{{\mathbb}{K}}=0$ or $\operatorname{char}{{\mathbb}{K}}>\deg f$, and $f$ has $s$ essential variables, then we can compute in randomized polynomial time $M \in \operatorname{GL}_n({{\mathbb}{K}})$ such that $f(M\,{{\mathbf}{x}})$ depends only on the first $s$ variables.

For a quadratic form, $s$ is simply the rank of the associated Hessian matrix. As such, for $m=1$, a quadratic instance is regular if and only if the associated Hessian matrix is invertible. For a set of equations, we extend the notion of essential variables as follows.

\[def:ess\] The number of *essential variables* of ${{\mathbf}{f}}=(f_1,\ldots,f_m)\in{{\mathbb}{K}}[x_1,\ldots,x_n]^m$ is the smallest $s$ such that ${{\mathbf}{f}}$ can be decomposed as: $${{\mathbf}{f}}=\tilde{{{\mathbf}{f}}}(\ell_1,\ldots,\ell_{s})$$ with $\ell_1,\ldots,\ell_{s}$ being linear forms in $x_1,\ldots,x_n$ of rank $s$ and $\tilde{{{\mathbf}{f}}}\in{{\mathbb}{K}}[y_1,\ldots,y_{s}]^m$.

The linear forms $\ell_1,\ldots,\ell_s$ can be easily computed thanks to Proposition \[prop:carl\] when the characteristic of ${{\mathbb}{K}}$ is zero or greater than the degrees of $f_1,\ldots,f_m$. In characteristic $2$, when ${{\mathbb}{K}}$ is perfect (which is always true if ${{\mathbb}{K}}$ is finite for instance) the linear forms can also be recovered in polynomial time (see [@BerthomieuHM2010; @Giraud1972; @Hironaka1970] for instance). Below, we show that we can restrict our attention to only essential variables. Namely, solving [`IP1S`]{}on $({{\mathbf}{f}},{{\mathbf}{g}})$ reduces to solving [`IP1S`]{}on instances having 