NON-COMMUTATIVE VARIETIES WITH CURVATURE HAVING BOUNDED SIGNATURE

A natural notion for the signature C-+/-(V(p)) of the curvature of the zero set V(p) of a non-commutative polynomial p is introduced. The main result of this paper is the bound deg p <= 2C(+/-)(V(p)) + 2. It is obtained under some irreducibility and nonsingularity conditions, and shows that the signature of the curvature of the zero set of p dominates its degree. The condition C+(V(p)) = 0 means that the non-commutative variety V(p) has positive curvature. In this case, the preceding inequality implies that the degree of p is at most two. Non-commutative varieties V(p) with positive curvature were introduced in (Indiana Univ. Math. J. 56 (2007) 1189-1231). There a slightly weaker irreducibility hypothesis plus a number of additional hypotheses yielded a weaker result on p. The approach here is quite different; it is cleaner, and allows for the treatment of arbitrary signatures. In (J. Anal. Math. 108 (2009) 19-59), the degree of a non-commutative polynomial p was bounded by twice the signature of its Hessian plus two. In this paper, we introduce a modified version of this non-commutative Hessian of p which turns out to be very appropriate for analyzing the variety V(p).