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Power Sum Varieties over Finite Fields

  • Nardo Giménez
  • Juan Francisco Gottig
  • Miguel Palacios
  • Mariana Pérez
  • Melina Privitelli
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Abstract

We study a family of affine algebraic varieties over a finite field 𝔽q defined by systems of power sum polynomials. Our main goal is to investigate their geometric and arithmetic properties, with particular emphasis on the structure and dimension of their singular loci. Using tools from the theory of symmetric polynomials and generalized Vandermonde determinants, we establish explicit conditions under which these varieties are complete intersections and prove that, in a broad range of parameters, their singular loci have codimension at least two. As a consequence, we derive normality and irreducibility results for both the affine varieties and their projective closures. Building on this geometric analysis, we obtain estimates for the number of 𝔽q-rational points and deduce effective existence results for solutions of systems of symmetric power sum equations over finite fields. Finally, we apply these results to the study of Waring's problem and to the analysis of the cost of computing a 𝔽q-rational point of such systems.

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Cite

BibTeXRIS
BibTeX
@misc{gimenez2026power,
  title = {{Power Sum Varieties over Finite Fields}},
  author = {Nardo Giménez and Juan Francisco Gottig and Miguel Palacios and Mariana Pérez and Melina Privitelli},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.32056},
  eprint = {2610.32056},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.32056},
}
RIS
TY  - UNPB
TI  - Power Sum Varieties over Finite Fields
AU  - Nardo Giménez
AU  - Juan Francisco Gottig
AU  - Miguel Palacios
AU  - Mariana Pérez
AU  - Melina Privitelli
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2610.32056
UR  - https://arxiv.org/abs/2610.32056
AB  - We study a family of affine algebraic varieties over a finite field 𝔽_q defined by systems of power sum polynomials. Our main goal is to investigate their geometric and arithmetic properties, with particular emphasis on the structure and dimension of their singular loci. Using tools from the theory of symmetric polynomials and generalized Vandermonde determinants, we establish explicit conditions under which these varieties are complete intersections and prove that, in a broad range of parameters, their singular loci have codimension at least two. As a consequence, we derive normality and irreducibility results for both the affine varieties and their projective closures. Building on this geometric analysis, we obtain estimates for the number of 𝔽_q-rational points and deduce effective existence results for solutions of systems of symmetric power sum equations over finite fields. Finally, we apply these results to the study of Waring's problem and to the analysis of the cost of computing a 𝔽_q-rational point of such systems.
ER  -