Symmetric tensors with applications to Hilbert schemes

Let A[X]U be a fraction ring of the polynomial ring A[X] in the variable X over a commutative ring A. We show that the Hilbert functor {Hilb}^n{A[X]_U} is represented by an affine scheme ${\text{Symm}}_A^n(A[X{]}_U)$ give as the ring of symmetric tensors of ${\otimes }_A^{n}A[X{]}_U$. The universal family is given as ${\text{Symm}}_A^{n-1}(A[X{]}_U){\times }_A\text{Spec}(A[X{]}_U)$.