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.

翻译：无标注感知问题旨在求解一个线性方程组，其右端向量仅已知一个置换关系。在本研究中，我们探讨了与对称多项式相关的有理函数域及其在变量线性投影下的像；由此我们证明，对于具有一般性数据的n维无标注感知问题，其解可作为n个未知量中次数分别为1, 2, ..., n+1的n+1个多项式方程组的唯一解而获得。除了新的理论见解外，这一进展为扩展代数无标注感知算法的规模提供了可能。