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(This question was originally asked at Math.SE, where it didn't receive any answers.)

Let $A(x_1, \ldots, x_m)$ be a $n$ x $n$ matrix whose entries are polynomials on real variables $x_1, \ldots, x_m$, with real coefficients and with degree up to $d$, that is,

$$ A(x_1, \ldots, x_m) = \begin{bmatrix} p_{11}(x_1, \ldots, x_m) & \cdots & p_{1n}(x_1, \ldots, x_m) \\ \vdots & \ddots & \vdots \\ p_{n1}(x_1, \ldots, x_m) & \dots & p_{nn}(x_1, \ldots, x_m) \end{bmatrix}, $$

with $p_{ij}$ multivariate polynomials and

$$ \max_{i=1,\ldots,n; j=1,\ldots,n} \deg p_{ij} = d. $$

What is a tight upper bound (possibly in function of $m$,$n$ and $d$) for the degree of the entries of the adjugate matrix of $A(x_1, \ldots, x_m)$?

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    $\begingroup$ The entries of the adjugate matrix are polynomials of degree $n-1$ in the entries of $A$ and since the entries of $A$ are polynomials of degree $d$ we get that the entries of the adjugate matrix are polynomials of degree $(n-1)\cdot d$. Equality can be achieved if $A$ is a diagonal matrix with entries $x_1^d$. Do I miss something? $\endgroup$
    – user35593
    Commented Jul 7, 2016 at 8:15
  • $\begingroup$ @user35593 You're right, but I believe it should be $n \cdot d$ instead of $(n-1) \cdot d$. I was thinking that we could do better than $n \cdot d$... Thanks for the answer! :) $\endgroup$
    – Tadashi
    Commented Jul 8, 2016 at 2:02

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