Non-commutative varieties with curvature having bounded signature

TL;DR

This paper investigates the relationship between the curvature signature of zero sets of free non-commutative polynomials and their degree, establishing bounds based on the signature, especially when one signature component is zero.

Contribution

It introduces a bound on the degree of non-commutative polynomials based on the curvature signature of their zero sets, extending understanding of geometric properties in free algebra.

Findings

Degree of p is bounded by the signature of curvature.

If one signature component is zero, the degree of p is at most two.

Curvature signature imposes algebraic restrictions on polynomial degree.

Abstract

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 negative eigenvalues of the non-commutative second fundamental form on V determined by p. With some natural hypotheses, the degree of p is bounded in terms of the signature. In particular, if one of the signatures is zero, then the degree of p is at most two.