Voronenko A.A.

It was proved earlier that product xy is a universal function for the class of linear functions depending on two arguments if k = 6l±1. In the paper we prove that there is no universal polynomials for the class of linear functions depending on two arguments for any even k.

It was proved earlier that product xy is a universal function for the class of linear functions depending on two arguments if k = 6l ± 1. Also it was shown that there is no universal polynomials for the class of linear functions depending on two arguments for any even k. In this paper we prove that polynomial xy + xz + yz is universal for classes of linear functions depending on three arguments for arbitrary odd k and polynomial xy + zw is universal for classes of linear functions depending on four arguments for any k.

It was proved earlier that product xy is a universal function for the class of linear functions depending on two arguments if k = 6l ± 1. In the paper we prove that there is no universal polynomials for the class of linear functions depending on two arguments for any k dividing three. Also we show that there is no universal
polynomials for the class of linear functions depending on three arguments for any even k. Thus the criterion of existence of universal polynomials for the class of linear functions is obtained.