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Let $A$ be a $(r \times r)$-matrix. From the equation$ \det\left(1+A\right)=\sum_{0\leq j \leq r} {r \choose j} H_j (A) $ where $H_j (A)$ are homogenous polynomials of order $j$ in the entries of the matrix $A$. We define the the polarization $P_l$ of $H_l$,to be a $l$-multilinear, symmetric polynomial, satisfying $P_l(A, \cdots , A) =H_l (A)$, is there a general formula to write $P_l(A_1, ... , A_l)$ in terms of the $H_l$?

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    @joriki I edited and defined $r$ and $H_j$. I added combinatorics because of how the formula should look like.2012-09-21

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I think I have the answer: $ P_l(A_1,\cdots , A_l)=\frac{(-1)^l}{l!}\sum_{j=1}^l \sum_{i_j < \cdots < i_j} (-1)^j H_l (A_{i_1} + \cdots + A_{i_j}). $