# Time complexity of Gaussian elimination over polynomial ring

I have a $t \times l$-polynomial matrix $A$ over $\mathbb{F}_q[x]$. The entries of $A$ are of degree $\le m$. I want to reduce $A$ to upper-triangular form by Gaussian elimination in case of using the Euclidean division instead of the exact one. What the worst-case computational (time) complexity it will take?

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