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TR11-088 | 7th June 2011 17:35
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#### How much commutativity is needed to prove polynomial identities?

**Abstract:**
Let $f$ be a non-commutative polynomial such that $f=0$ if we assume that the variables in $f$ commute. Let $Q(f)$ be the smallest $k$ such that there exist polynomials $g_1,g_1', g_2, g_2',\dots, g_k, g_k' $ with \[f\in I([g_1,g_1'], [g_2, g_2'],\dots, [g_k, g_k'] )\,,\]

where $[g,h]=gh-hg$. Then $Q(f)\leq {n\choose 2}$, where $n$ is the number of variables of $f$. We show that there exists a polynomial $f$ with $Q(f)=\Omega(n^2)$. We pose the problem of constructing such an $f$ explicitly, pointing out that the solution may have applications to complexity of proofs.