Central polynomials of minimal degree for matrices

Summary · qwen2.5:32b

Formanek's conjecture on the minimal degree of central polynomials for $n\times n$ matrix algebras over fields of characteristic 0 is confirmed for $n\leq 3$, with the paper focusing on methods to search for such polynomials, particularly proving no central polynomials in two variables exist for $4\times 4$ matrices at degrees $\leq 12$.

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arXiv:2601.07750v2 Announce Type: replace Abstract: Formanek made the conjecture that the minimal degree of the central polynomials for the $n\times n$ matrix algebra over a field of characteristic 0 is $(n^2+3n-2)/2$ and this is true for $n\leq 3$. For $n=4$ there are examples of central polynomials of degree $13=(4^2+3\cdot 4-2)/2$ and we do not know whether there are central polynomials of lower degree. In this paper we discuss methods for searching for central polynomials of low degree and prove that the algebra of $4\times 4$ matrices does not have central polynomials in two variables of degree $\leq 12$. As a byproduct of our computations we obtain that this algebra does not have also polynomial identities in two variables of degree $\leq 12$.