Ultranormal case of the Bernstein problem (Q1392316)

Let \(V\) be a quadratic stochastic mapping on the simplex \(\Delta^{n-1}\). The mapping \(V\) is called a Bernstein mapping if \(V^2=V\). The Bernstein problem is to describe all Bernstein mappings explicitly. The case \(n\leq 2\) is trivial. This problem was solved by Bernstein for the case \(n=3\), and he obtained some results for \(n\geq 4\). In earlier papers of the author the Bernstein problem was solved for all \(n\) under the regularity assumption, i.e. \(Vx\) depends only on the values \(f(x)\) (where \(f\) runs over all invariant linear forms). In this work a stochastic quadratic mapping \(V\) and the corresponding Bernstein algebra \(A_V\) is considered, which has the ultranormality property. The algebra \(A_V\) is said to be normal if the quadratic forms \((Vx)_i\) and \((Vx)_j\) \((i\neq j)\) are linearly independent and, for no pair \(k,s\) \((k\neq s)\), all \((Vx)_i\) depend on \(x_k+x_s\) and \(x_j\) only \((j\neq k\), \(j\neq s)\); \(A_V\) is called ultranormal if all coordinate subalgebras are normal. It is proved that every ultranormal stochastic Bernstein algebra is regular. This implies the solution of the Bernstein problem in the ultranormal case.