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The power sum polynomial associated to a multi-subset of the projective plane PG(2,q) is the sum of the (q−1)-th powers of the Rédei factors of the points in the multi-subset. The classification of multi-subsets having the same power sum polynomial passes through the determination of those multi-subsets associated to the zero polynomial, called ghosts. In this paper we provide new classes of ghosts and compute the dimension of the ghost subspace by exploiting the linear code generated by the lines of PG(2,q) and its dual. Moreover, we explicitly enumerate and classify ghosts for planes of order 2, 3, 4.

The core problem of discrete tomography, i.e., the faithful reconstruction of an unknown discrete object from projections along a given set of directions, is ill-posed in general. When further constraints are imposed on the object or on the employed directions, uniqueness of the reconstruction can be obtained. It is the case, for example, of convex lattice sets in Z2, for which a theorem by Gardner and Gritzmann assures the faithful reconstruction when suitable sets of directions are considered. It was conjectured that a similar result holds for the class of hv-convex polyominoes. In this paper we are concerned with this conjecture, providing new 4-tuples of discrete directions that do not lead to a unique reconstruction of hv-convex polyominoes, underlining the relevant structural difference with the class of convex sets. Our result is based on the recursive definition of new hv-convex switching components on discrete sets along four directions.

We present an algorithm that for any given rectangular grid A∈Z2 and set of directions D computes in linear time the values of any function f:A→R outside the convex hull of the union of the switching domains from its line sums in the directions of D. Moreover, the algorithm reconstructs f completely if there are no switching domains. We present a simpler algorithm in case the directions satisfy some monotonicity condition. Finally, for given A we propose how to choose the set D so that only a small number of directions is needed to reconstruct any f from its line sums in the directions of D.

The goal of discrete tomography is to reconstruct an image, seen as a finite set of pixels, by knowing its projections along given directions. Uniqueness of reconstruction cannot be guaranteed in general, because of the existence of the switching components. Therefore, instead of considering the uniqueness problem for the whole image, in this paper we focus on local uniqueness, i.e., we seek what pixels have uniquely determined value. Two different kinds of local uniqueness are presented: one related to the structure of the directions and of the grid supporting the image, having as a sub-case the region of uniqueness (ROU), and the other one depending on the available projections. In the case when projections are taken along two lattice directions, both kinds of uniqueness have been characterized in a graph-theoretical reformulation. This paper is intended to be a starting point in the construction of connections between pixels with uniquely determined value and graphs.