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The signature(s) of the curvature of the zero set V of a free(non-commutative) polynomial is defined as the number of positive and negativeeigenvalues of the non-commutative second fundamental form on V determined byp. With some natural hypotheses, the degree of p is bounded in terms of thesignature. In particular, if one of the signatures is zero, then the degree ofp is at most two.

Non-commutative varieties with curvature having bounded signature