making the hardness of the irregular case intriguing and then an interesting open question.

Special case of [`IP`]{} {#special-case-of-ip .unnumbered}
------------------------

In our quest of finding instances of [`IP`]{}solvable in polynomial-time, we take a first step in Section \[s:ip\]. We consider [`IP`]{}for a specific set of polynomials with $m=n$. In the aforementioned Section \[s:ip\], we prove the following:

\[th:ip\] Let ${{\mathbf}{g}}=(g_1,\ldots,g_n) \in {{\mathbb}{K}}[x_1,\ldots,x_n]^n$ be given in dense representation, and ${{\mathbf}{f}}={{\mathrm}{{\mathbf}{POW}}}_{n,d}=(x_1^d,\ldots,x_n^d) \in {{\mathbb}{K}}[x_1,\ldots,x_n]^n$ for some $d>0$. Whenever $\operatorname{char}{{\mathbb}{K}}=0$, let $e=d$ and $\tilde{{{\mathbf}{g}}}={{\mathbf}{g}}$. Otherwise, let $p=\operatorname{char}{{\mathbb}{K}}$, let $e$ and $r$ be integers such that $d=p^r\,e$, $p$ and $e$ coprime, and let $\tilde{{{\mathbf}{g}}}\in{{\mathbb}{K}}[{{\mathbf}{x}}]^n$ be such that ${{\mathbf}{g}}({{\mathbf}{x}})=\tilde{{{\mathbf}{g}}}({{\mathbf}{x}}^{p^r})$. Let $L$ be a polynomial size of an arithmetic circuit to evaluate the determinant of the Jacobian matrix of $\tilde{{{\mathbf}{g}}}$. If the size of ${{\mathbb}{K}}$ is at least $12\,\max
  (2^{L+2},e\,(n-1)\,2^{e\,(n-1)}+e^3\,(n-1)^3,2\,(e\,(n-1)+1)^4)$, then there is a randomized polynomial-time algorithm which recovers – if any – $(A,B)\in\operatorname{GL}_n({{\mathbb}{K}})\times\operatorname{GL}_n({{\mathbb}{K}})$ such that: $${{\mathbf}{g}}=B\cdot{{\mathrm}{{\mathbf}{POW}}}_{n,d}(A\cdot{{\mathbf}{x}}).$$

This extends a similar result of [@Kayal2011 Section 5] who considered [`PolyEquiv`]{}for a sum of $d$-power polynomials. We show that solving [`IP`]{}for ${{\mathrm}{{\mathbf}{POW}}}_{n,d}$ reduces to factoring the determinant of a Jacobian matrix (in [@Kayal2011], the Hessian matrix is considered). This illustrates, how powerful partial derivatives can be in equivalence 