Power sum polynomials in a discrete tomography perspective

TL;DR

This paper explores the concept of power sum polynomials in finite projective spaces, linking them to discrete tomography, and introduces the notion of ghosts, providing new insights into solution sets and their properties.

Contribution

It redefines ghosts within finite geometry, analyzes their properties in PG(2,q), and computes their quantity for prime q, advancing understanding of power sum polynomials in discrete tomography.

Findings

Ghosts can be characterized as subsets with null power sum polynomial.

Adding ghosts via multiset sum preserves the power sum polynomial.

Number of ghosts in PG(2,q) computed for prime q.

Abstract

For a point of the projective space $\PG(n,q)$, its R\'edei factor is the linear polynomial in $n+1$ variables, whose coefficients are the point coordinates. The power sum polynomial of a subset $S$ of $\PG(n,q)$ is the sum of the $(q-1)$-th powers of the R\'edei factors of the points of $S$. The fact that many subsets may share the same power sum polynomial offers a natural connection to discrete tomography. In this paper we deal with the two-dimensional case and show that the notion of ghost, whose employment enables to find all solutions of the tomographic problem, can be rephrased in the finite geometry context, where subsets with null power sum polynomial are called ghosts as well. In the latter case, one can add ghosts still preserving the power sum polynomial by means of the multiset sum (modulo the field characteristic). We prove some general results on ghosts in $\PG(2,q)$ and compute their number in case $q$ is a prime.