Proceedings of the American Mathematical Society

It is shown that λ x n , k ( λ ) \lambda x_{n,k}^{(\lambda )} increases as λ \lambda increases for 0 > λ > 1 , k = 1 , 2 , … , [ n 2 ] 0 > \lambda > 1,k = 1,2, \ldots ,\left [ {\tfrac {n}{2}} \right ] , where x n , k ( λ ) x_{n,k}^{(\lambda )} is the k th k{\text {th}} positive zero of ultraspherical polynomial P n ( λ ) ( x ) P_n^{(\lambda )}(x) .

We give a new characterization of the property of seminormality for 1-dimensional local rings; and with the help of this result we develop extensions of the theorems of Salmon and Bombieri which interpret this property in terms of singularities of algebraic curves.

In this paper we investigate situations in which the space C π ( X ) {C_\pi }(X) of continuous, real-valued functions on X X is a Borel subset of the product space R X {{\mathbf {R}}^X} . We show that for completely regular, nondiscrete spaces, C π ( X ) {C_\pi }(X) cannot be a G δ {G_\delta } , an F σ {F_\sigma } , or a G δ σ {G_{\delta \sigma }} subset of R X {{\mathbf {R}}^X} , but it can be an F σ δ {F_{\sigma \delta }} or G δ σ δ {G_{\delta \sigma \delta }} subset.