Let $\varphi _1,\ldots ,\varphi _r\in {\mathbb Z}[z_1,\ldots z_k]$ be integral linear combinations of elementary symmetric polynomials with $\text {deg}(\varphi _j)=k_j\ (1\le j\le r)$, where $1\le k_1<k_2<\cdots <k_r=k$. Subject to the condition $k_1+\cdots +k_r\ge \tfrac {1}{2}k(k-~1)+2$, we show that there is a paucity of nondiagonal solutions to the Diophantine system $\varphi _j({\mathbf x})=\varphi _j({\mathbf y})\ (1\le j\le r)$.

We consider the algebra of holomorphic functions on L∞ that are symmetric, i.e. that are invariant under composition of the variable with any measure-preserving bijection of [0, 1]. Its spectrum is identified with the collection of scalar sequences such that is bounded and turns to be separable. All this follows from our main result that the subalgebra of symmetric polynomials on L∞ has a natural algebraic basis.

In the spectrum of the algebra of symmetric analytic functions of bounded type on ℓp, 1 ≤ p < +∞, and along the same lines as the general non-symmetric case, we define and study a convolution operation and give a formula for the ‘radius’ function. It is also proved that the algebra of analytic functions of bounded type on ℓ1 is isometrically isomorphic to an algebra of symmetric analytic functions on a polydisc of ℓ1. We also consider the existence of algebraic projections between algebras of symmetric polynomials and the corresponding subspace of subsymmetric polynomials.

A characterization of the space of symmetric Laurent polynomials of type (BC)n, which vanish on a certain set of submanifolds, is given by using the Koornwinder–Macdonald polynomials. A similar characterization was given previously for symmetric polynomials of type An by using the Macdonald polynomials. We use a new method which exploits the duality relation. The method simplifies a part of the proof in the An case.