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.

Pagani, S. M. C., Pianta, S., Power sum polynomials in a discrete tomography perspective, Paper, in Discrete Geometry and Mathematical Morphology, (Uppsala, Svezia (online), 24-27 May 2021), Springer International Publishing, Cham 2021:12708 325-337. 10.1007/978-3-030-76657-3_23 [http://hdl.handle.net/10807/179376]

Power sum polynomials in a discrete tomography perspective

Pagani, Silvia Maria Carla;PIanta, Silvia
2021

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.
Inglese
Discrete Geometry and Mathematical Morphology
DGMM 2021
Uppsala, Svezia (online)
Paper
24-mag-2021
27-mag-2021
978-3-030-76656-6
Springer International Publishing
Pagani, S. M. C., Pianta, S., Power sum polynomials in a discrete tomography perspective, Paper, in Discrete Geometry and Mathematical Morphology, (Uppsala, Svezia (online), 24-27 May 2021), Springer International Publishing, Cham 2021:12708 325-337. 10.1007/978-3-030-76657-3_23 [http://hdl.handle.net/10807/179376]
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/10807/179376
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