Sum Complexes and Uncertainty Numbers

TL;DR

This paper investigates the homology of certain simplicial complexes constructed from subsets of finite fields and relates these topological properties to the uncertainty numbers of those subsets.

Contribution

It determines the homology groups of complexes built from subsets of finite fields and links these to the uncertainty numbers, providing a new topological perspective.

Findings

Homology groups of X_{A,k} are explicitly determined.

If |A| ≤ k, then H_{k-1}(X_{A,k};F_p)=0.

Homological characterization of uncertainty numbers in finite fields.

Abstract

Let p be a prime and let A be a subset of F_p. For k<p let X_{A,k} be the (k-1)-dimensional complex on the vertex set F_p with a full (k-2)-skeleton whose (k-1)-faces are k-subsets S of F_p such that the sum of the elements of S belongs to A. The homology groups of X_{A,k} with field coefficients are determined. In particular it is shown that if |A| \leq k then H_{k-1}(X_{A,k};F_p)=0. This implies a homological characterization of uncertainty numbers of subsets of F_p.