The standard polynomial of degree n is in non-commuting variables x1, ..., xn, where the sum is taken over all n!
The Amitsur–Levitzki theorem states that for n × n matrices A1, ..., A2n whose entries are taken from a commutative ring then Amitsur and Levitzki (1950) gave the first proof.
Swan showed that this was the case provided the number of edges in the graph is at least 2n, thus proving the Amitsur–Levitzki theorem.
Rosset (1976) gave a short proof using the exterior algebra of a vector space of dimension 2n.
Procesi (2015) gave another proof, showing that the Amitsur–Levitzki theorem is the Cayley–Hamilton identity for the generic Grassman matrix.