Let $A,B,C \in k[x]^{n \times n}$ be polynomial square matrices with $\deg(A),\deg(C) \leq d$ where we define the degree of a matrix to be the maximum of the degrees of its entries. Moreover, assume that all involved matrices have full rank $n$ over $k(x)$ and, moreover, that $B$ is unimodular over $k[x]$, that is $\deg \det B = 0$.
Does $AB = C$ imply $\deg(B) \leq d$ or is it possible that $B$ has degree indpendent of $d$?
In general, $B$ has degree independent of $d$. For example, consider $$ A = \pmatrix{1&-x\\ &1 & -x\\ && \ddots & \ddots\\ &&&&-x\\&&&&1}, \quad C = I,\\ B = \pmatrix{ 1&x&x^2 & \cdots & x^{n-1}\\ &1&x&\ddots & \vdots\\ &&\ddots & \ddots \\ &&& 1 & x\\ &&&&1}. $$