Determinantal conditions for homomorphic sensing

Abstract

With $k$ an infinite field and ${\tau }_1,{\tau }_2$ endomorphisms of $k^m$, we provide a dimension bound on an open locus of a determinantal scheme, under which, for a general subspace $V\subseteq k^m$ of dimension $n\le m/2$, for $v_1,v_2\in V$ we have ${\tau }_1(v_1)={\tau }_2(v_2)$ only if $v_1=v_2$. Specializing to permutations composed by coordinate projections, we obtain an abstract proof of the unlabeled sensing theorem.

enhanced technical clarity at some places, 15 pages