Solution of the Bernstein Problem in the Non-regular Case

Abstract

This paper describes explicitly all non-regular non-degenerate simplicial stochastic Bernstein algebras. Consequently, the Bernstein problem (S. N. Bernstein, Science Ukraine1 (1992), 14–19) in the non-degenerate case is settled, since the regular and exceptional cases have already been examined by Y. Lyubich in the 1970s. Notice that from this result it is possible to explicitly describe every non-regular simplicial algebra (A,Δ) since the simplicial subalgebra (〈supp(A2)〉, [supp(A2)]) is non-degenerate. Also we prove the relevant Lyubich's conjecture (1992, Yu I. Lyubich, Biomathematics22, 232) in an affirmative way: all normal simplicial stochastic Bernstein algebras are regular.