Global structure behind pointwise equivalences of noncommutative polynomials

TL;DR

The paper analyzes how local equivalences of noncommutative polynomials, viewed through evaluations on matrix tuples, align with global ring-theoretic relations in the free algebra ${\mathbbm k}<\underline x>$. It proves that rank-equivalence corresponds to stable association, isospectrality corresponds to intertwinement (via elementary intertwined chains and their transitive closure), and pointwise similarity corresponds to exact equality, establishing a clean global structure behind pointwise NC polynomial equivalences. It then develops tools connecting rank-equivalence to joint similarity, explores how isospectrality behaves under polynomial composition, and characterizes norm- and singular-value equivalences over $\mathbb{C}$ as unimodular scalings. These results illuminate how algebraic factorization, representation theory, and NC function perspectives converge to explain when two NC polynomials share local evaluation behavior across all matrix sizes.

Abstract

This paper investigates the interplay between local and global equivalences on noncommutative polynomials, the elements of the free algebra. When the latter are viewed as functions in several matrix variables, a local equivalence of noncommutative polynomials refers to their values sharing a common feature point-wise on matrix tuples of all dimensions, such as rank-equivalence (values have the same ranks), isospectrality (values have the same spectrum), and pointwise similarity (values are similar). On the other hand, a global equivalence refers to a ring-theoretic relation within the free algebra, such as stable association or (elementary) intertwinedness. This paper identifies the most ubiquitous pairs of local and global equivalences. Namely, rank-equivalence coincides with stable association, isospectrality coincides with both intertwinedness and transitive closure of elementary intertwinedness, and pointwise similarity coincides with equality. Using these characterizations, further results on spectral radii and norms of values of noncommutative polynomials are derived.

Theorems & Definitions (58)

• Example 1.1
• Theorem 1.2
• Theorem 2.1
• Theorem 2.2
• Proposition 2.3
• proof
• Lemma 2.4
• proof
• Lemma 3.1
• proof