Unique Constrution of Formanek Central Polynomial

Abstract:

      Formanek constructed the first central polynomial. i .e. G₁ G₂ … G_n where G₁= G(x, y₁,…, y_n), G₂= G(x, y₂, y₃,…, y_n, y₁) etc. Are called Forma nek's polynomials. Rosset in his nota [3] raised the question that whether all sgmmetic polynomials in G_i also give central polynomials. He showed that the basic sgmmetric polynomials in G_i are not all central. In this paper we shall show, for n≥3 each polynomial in G_j is not central, except f(G₁ … G_n, where f(x)is a polynomial at x. Hence Formanek's central polynomial is unique in some sense.