  Unlabeled sensing is the problem of solving a linear system of equations,
where the right-hand-side vector is known only up to a permutation. In this
work, we study fields of rational functions related to symmetric polynomials
and their images under a linear projection of the variables; as a consequence,
we establish that the solution to an n-dimensional unlabeled sensing problem
with generic data can be obtained as the unique solution to a system of n + 1
polynomial equations of degrees 1, 2, . . . , n + 1 in n unknowns. Besides the
new theoretical insights, this development offers the potential for scaling up
algebraic unlabeled sensing algorithms.
