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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. [ABSTRACT FROM AUTHOR]

In discrete tomographic image reconstruction, projections are taken along a finite set S of valid directions for a working grid A. In general, uniqueness cannot be achieved in the whole grid A. Usually, some information on the object to be reconstructed is introduced, that, sometimes, allows possible ambiguities to be removed. From a different perspective, one aims in finding subregions of A where uniqueness can be guaranteed, and obtained in linear time, only from the knowledge of S. When S consists of two lattice directions, the shape of any such region of uniqueness, say ROU, have been completely characterized in previous works by means of a double Euclidean division algorithm called DEDA. Results have been later extended to special triples of directions, under a suitable assumption on their entries. In this paper we remove the previous assumption, so providing a complete characterization of the shape of the ROU for such kind of triples. We also show that the employed strategy can be even applied to more general sets of three directions, where the corresponding ROU can be characterized as well. Independently of the combinatorial interest of the problem, the result can be exploited to define in advance, namely before using any kind of radiation, suitable sets of directions that allow regions of interest to be included in the corresponding ROU. Results have been proved in all details, and several experiments are considered, in order to support the theoretical steps and to clarify possible applications. [ABSTRACT FROM AUTHOR]