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Can we perform integer division with a polynomial size arithmetic circuit over $\mathbb{Q}$ that takes as input the numerator and denominator?

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By arithmetic circuit do you mean a circuit composed of addition and multiplication gates? –  Kaveh Apr 11 at 7:32

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See Eric Allender's survey of "recent" breakthroughs (circa more than a decade ago) in the complexity of division. The bottom line is that

Division is complete for DLOGTIME-uniform $\mathsf{TC}^0$

so in particular there is a poly-sized circuit (of constant depth) that takes $x$ and $y$ as input and produces $x/y$. Notice that this circuit doesn't even need to be arithmetic (it only needs MAJORITY gates in addition to boolean gates)

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