On the number of solutions of systems of diagonal equations through diagonal GP-graphs: the general and the Hermitian-form cases
Abstract
For any m, s ∈ ℕ, we study the number Nm× s,q(κ, β) of solutions (x1,…,xs) ∈ (𝔽q)s of the monic system of diagonal equations X1ki + ⋯ + Xski= βi, (1≤ i ≤ m), with κ=(k1,…,km) ∈ ℕm and β=(β1,…,βm) ∈ (𝔽q)m. We show that this number can be obtained in terms of some data of \textit{diagonal} GP-graphs Γ(κ,q). This is a new family of graphs that we introduce here, Γ(κ,q), with κ= (k1,…,km) ∈ ℕm, is the directed graph with vertex set the finite field 𝔽q and there is an arc from u to v if and only if v−u ∈ Rκ = { (xk1,…,xkm) : x ∈ 𝔽q*}. In particular, we give three different expressions for Nm× s,q(κ, β): one in terms of walks, another in terms of adjacency matrices of Γ(κ,q) and the last one in terms of the spectrum of Γ(κ,q). Finally, we explicitly derive combinatorial formulas for the number of solutions Nm(s,q) = Nm× s,q(κℓ, 0) of monic homogeneous systems of diagonal equations of the form X1qℓi+1 + ⋯ + Xsqℓi+1 = 0 (1≤ i ≤ m), with κℓ=(ℓ1,…,ℓm)=(1,3,…,2m−1) and m≥ 2, via the known spectrum of Hermitian-form graphs, which can be viewed as diagonal GP-graphs. For any m,s ∈ ℕ, we give general summation and recursive formulas for Nm(s,q) ∈ ℤ[q]. For the small cases N1(s,q), N2(s,q) and Nm(s,q), with 1≤ s ≤ 5, we give explicit expressions.
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BibTeX
@misc{podesta2026number,
title = {{On the number of solutions of systems of diagonal equations through diagonal GP-graphs: the general and the Hermitian-form cases}},
author = {Ricardo A. Podestá and Denis E. Videla},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.19507},
eprint = {2608.19507},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.19507},
}
RIS
TY - UNPB
TI - On the number of solutions of systems of diagonal equations through diagonal GP-graphs: the general and the Hermitian-form cases
AU - Ricardo A. Podestá
AU - Denis E. Videla
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2608.19507
UR - https://arxiv.org/abs/2608.19507
AB - For any m, s ∈ ℕ, we study the number N_m× s,q(κ, β) of solutions (x_1,…,x_s) ∈ (𝔽_q)^s of the monic system of diagonal equations X_1^k_i + ⋯ + X_s^k_i= β_i, (1≤ i ≤ m), with κ=(k_1,…,k_m) ∈ ℕ^m and β=(β_1,…,β_m) ∈ (𝔽_q)^m. We show that this number can be obtained in terms of some data of \textit{diagonal} GP-graphs Γ(κ,q). This is a new family of graphs that we introduce here, Γ(κ,q), with κ= (k_1,…,k_m) ∈ ℕ^m, is the directed graph with vertex set the finite field 𝔽_q and there is an arc from u to v if and only if v−u ∈ R_κ = { (x^k_1,…,x^k_m) : x ∈ 𝔽_q^*}. In particular, we give three different expressions for N_m× s,q(κ, β): one in terms of walks, another in terms of adjacency matrices of Γ(κ,q) and the last one in terms of the spectrum of Γ(κ,q). Finally, we explicitly derive combinatorial formulas for the number of solutions N_m(s,q) = N_m× s,q(κ_ℓ, 0) of monic homogeneous systems of diagonal equations of the form X_1^q^ℓ_i+1 + ⋯ + X_s^q^ℓ_i+1 = 0 (1≤ i ≤ m), with κ_ℓ=(ℓ_1,…,ℓ_m)=(1,3,…,2m−1) and m≥ 2, via the known spectrum of Hermitian-form graphs, which can be viewed as diagonal GP-graphs. For any m,s ∈ ℕ, we give general summation and recursive formulas for N_m(s,q) ∈ ℤ[q]. For the small cases N_1(s,q), N_2(s,q) and N_m(s,q), with 1≤ s ≤ 5, we give explicit expressions.
ER -