# Determinantal conditions for homomorphic sensing

## TL;DR

This paper establishes determinantal conditions that guarantee unique recovery of vectors under certain endomorphisms, providing a theoretical foundation for unlabeled sensing in signal processing.

## Contribution

It introduces a dimension bound on a determinantal scheme ensuring injectivity of combined endomorphisms for generic subspaces, generalizing unlabeled sensing results.

## Key findings

- Provides a dimension bound for determinantal schemes ensuring injectivity.
- Offers an abstract proof of the unlabeled sensing theorem.
- Generalizes conditions for homomorphic sensing to broader algebraic settings.

## 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.