A note on locality of algebras

In [15], the second author stated two conjectures characterizing algebraic commutative local algebras with unit over a field F. In this paper we settle Conjecture 1 from [15] in positive, and Conjecture 2 in positive when charF=0 and in negative otherwise. We also discuss locality of arbitrary commutative algebras with unit in this context. As an application, we propose deterministic and randomized polynomial-time algorithms that test locality of finite-dimensional algebras over algebraic number fields.