Non-commutative standard polynomials applied to matrices

Denis Serre (1)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

The Amitsur–Levitski Theorem tells us that the standard polynomial in 2n non-commuting indeterminates vanishes identically over the matrix algebra M n (K). For K = R or C and 2 ≤ r ≤ 2n − 1, we investigate how big S r (A 1 ,. .. , A r) can be when A 1 ,. .. , A r belong to the unit ball. We privilegiate the Frobenius norm, for which the case r = 2 was solved recently by several authors. Our main result is a closed formula for the expectation of the square norm. We also describe the image of the unit ball when r = 2 or 3 and n = 2. MSC classification : 15A24, 15A27, 15A60

Domaines: Equations aux dérivées partielles [math.AP]; Mathématiques générales [math.GM]; Mécanique des fluides [physics.class-ph]

Denis Serre. Non-commutative standard polynomials applied to matrices. Linear Algebra and its Applications, 2016, 490, pp.202 - 223. ⟨10.1016/j.laa.2015.11.003⟩. ⟨ensl-01402364⟩