On the number of solutions of systems of diagonal equations through diagonal GP-graphs: the general and the Hermitian-form cases

Abstract: For any $m, s \in \mathbb{N}$, we study the number $N_{m\times s,q}(κ, β)$ of solutions $(x_1,\ldots,x_s) \in (\mathbb{F}<em>q)<sup>s$ of the monic system of diagonal equations $$X</em>{1}^{k_i} + \cdots + X_{s}^{k_i}= β<em>i, \qquad (1\le i \le m),$$ with $κ=(k_1,\ldots,k_m) \in \mathbb{N}<sup>m$ and $β=(β_1,\ldots,β_m) \in (\mathbb{F}_q)<sup>m$. 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, i.e. Cayley graphs of the form $$Γ(κ,q) = Cay(\mathbb{F}</em>{q}^{m}, R_κ) \quad \text{where} \quad R_κ = { (x^{{k_1},\ldots,x^{k_m})} : x \in \mathbb{F}<em>{q}^*},$$ with $κ=(k_1,\ldots,k_m)\in \mathbb{N}<sup>{m}$. In particular, we give three different expressions for $N</em>{m\times s,q}(κ, β)$: 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 $N_{m}(s,q) = N_{m\times s,q}(κ<em>\ell, 0)$ of monic homogeneous systems of diagonal equations of the form $$X_1^{{q^{\ell_i}+1}} + \cdots + X_s^{{q^{\ell_i}+1}} = 0 \qquad (1\le i \le m),$$ with $$κ</em>\ell=(\ell_1,\ldots,\ell_m)=(1,3,\ldots,2m-1)$$ and $m\ge 2$, via the known spectrum of Hermitian-form graphs, which can be viewed as diagonal GP-graphs. For any $m,s \in \mathbb{N}$, we give general summation and recursive formulas for $N_m(s,q) \in \mathbb{Z}[q]$. For the small cases $N_{1}(s,q)$, $N_{2}(s,q)$ and $N_{m}(s,q)$, with $1\le s \le 5$, we give explicit expressions.

• The paper develops a graph-theoretic framework using generalized Paley graphs (GP-graphs) to determine exact solution counts of monic systems of diagonal equations over finite fields
• Using GP-graphs, the paper provides three different methods of calculation (walks, adjacency matrices, and spectra), allowing a computational approach.
• The results are specifically applied to both the general and Hermitian-form systems cases, with closed-form, recursive, and hypergeometric expressions derived for Hermitian-form systems

Overview

This paper by Podestá and Videla develops a graph-theoretic framework for computing the exact number of solutions of monic systems of diagonal equations (SDEs) over finite fields. The central object is a new family of Cayley graphs, the diagonal generalized Paley graphs $$\Gamma(\kappa,q) = \mathrm{Cay}(\mathbb{F}_q^m, R_\kappa)$$ with connection set $$R_\kappa = \{(x^{k_1},\ldots,x^{k_m}) : x \in \mathbb{F}_q^*\}$$ for $\kappa = (k_1,\ldots,k_m)$. The authors show that solution counts of systems

$$X_1^{k_i} + \cdots + X_s^{k_i} = \beta_i, \qquad 1 \le i \le m,$$

can be recovered from walks, adjacency matrices, or the spectrum of $\Gamma(\kappa,q)$, and they specialize to Hermitian-form systems where closed-form, recursive, and hypergeometric expressions are obtained.

Diagonal GP-graphs and their spectrum

The graph $\Gamma(\kappa,q)$ is loopless and regular of degree $(q-1)/d_\kappa$, where $d_\kappa = \gcd(k_1,\ldots,k_m,q-1)$. Its eigenvalues admit an explicit description via Weil sums: for each additive character indexed by $\alpha \in (\mathbb{F}_q)^m$,

$$\lambda_{\alpha,\kappa,q} = \frac{1}{d_\kappa}\big(\mathcal{W}_\chi(f_{\alpha,\kappa}) - 1\big),$$

where $$R_\kappa = \{(x^{k_1},\ldots,x^{k_m}) : x \in \mathbb{F}_q^*\}$$0. This is a direct generalization of the classical GP-graph spectrum. As an application, when the graph is undirected (which requires $$R_\kappa = \{(x^{k_1},\ldots,x^{k_m}) : x \in \mathbb{F}_q^*\}$$1 even, or $$R_\kappa = \{(x^{k_1},\ldots,x^{k_m}) : x \in \mathbb{F}_q^*\}$$2 odd with $$R_\kappa = \{(x^{k_1},\ldots,x^{k_m}) : x \in \mathbb{F}_q^*\}$$3 for all $$R_\kappa = \{(x^{k_1},\ldots,x^{k_m}) : x \in \mathbb{F}_q^*\}$$4), the Ihara zeta function is given explicitly through the Ihara–Bass determinant formula, with circuit rank $$R_\kappa = \{(x^{k_1},\ldots,x^{k_m}) : x \in \mathbb{F}_q^*\}$$5 and eigenvalues expressed entirely in terms of Weil sums. The undirectedness hypothesis is a genuine restriction: the zeta-function results do not cover directed instances of these graphs.

Solution counts via walks and adjacency matrices

The key combinatorial bridge is that an $$R_\kappa = \{(x^{k_1},\ldots,x^{k_m}) : x \in \mathbb{F}_q^*\}$$6-walk from $$R_\kappa = \{(x^{k_1},\ldots,x^{k_m}) : x \in \mathbb{F}_q^*\}$$7 to $$R_\kappa = \{(x^{k_1},\ldots,x^{k_m}) : x \in \mathbb{F}_q^*\}$$8 in $$R_\kappa = \{(x^{k_1},\ldots,x^{k_m}) : x \in \mathbb{F}_q^*\}$$9 corresponds to solutions with non-zero coordinates of the system with right-hand side $\kappa = (k_1,\ldots,k_m)$0, with multiplicity exactly $\kappa = (k_1,\ldots,k_m)$1:

$\kappa = (k_1,\ldots,k_m)$2

From this, three equivalent expressions follow. First, a walk-counting formula:

$\kappa = (k_1,\ldots,k_m)$3

Second, an adjacency-matrix formula, $\kappa = (k_1,\ldots,k_m)$4. Third, since the adjacency matrix of a Cayley graph is normal, diagonalization converts this into a spectral expression. A recursive relation also holds: the count for $\kappa = (k_1,\ldots,k_m)$5 variables reduces to counts with non-zero coordinates for smaller numbers of variables $\kappa = (k_1,\ldots,k_m)$6.

A notable structural result connects homogeneous solution counts to the logarithmic expansion of the Ihara zeta function: $\kappa = (k_1,\ldots,k_m)$7 is completely determined by the numbers $\kappa = (k_1,\ldots,k_m)$8, obtained via binomial inversion between total and strictly non-zero solution counts. Thus prime-cycle data of the graph encode the arithmetic of these diagonal systems.

Spectral formulas and Weil bounds

For homogeneous systems, the count depends only on the spectrum:

$\kappa = (k_1,\ldots,k_m)$9

with an equivalent character-sum/Weil-sum form

$$X_1^{k_i} + \cdots + X_s^{k_i} = \beta_i, \qquad 1 \le i \le m,$$0

Under the non-degeneracy assumption on all $$X_1^{k_i} + \cdots + X_s^{k_i} = \beta_i, \qquad 1 \le i \le m,$$1, this yields the uniform Weil-type bound

$$X_1^{k_i} + \cdots + X_s^{k_i} = \beta_i, \qquad 1 \le i \le m,$$2

independent of $$X_1^{k_i} + \cdots + X_s^{k_i} = \beta_i, \qquad 1 \le i \le m,$$3. An instructive worked example — the system $$X_1^{k_i} + \cdots + X_s^{k_i} = \beta_i, \qquad 1 \le i \le m,$$4, $$X_1^{k_i} + \cdots + X_s^{k_i} = \beta_i, \qquad 1 \le i \le m,$$5 over $$X_1^{k_i} + \cdots + X_s^{k_i} = \beta_i, \qquad 1 \le i \le m,$$6 — confirms $$X_1^{k_i} + \cdots + X_s^{k_i} = \beta_i, \qquad 1 \le i \le m,$$7, exactly the Weil-expected value $$X_1^{k_i} + \cdots + X_s^{k_i} = \beta_i, \qquad 1 \le i \le m,$$8, verified simultaneously by the matrix, spectral, and spectral-moment methods. The paper also notes a minor algebraic typo in Theorem 3.3 of Cao–Chou–Gu (2016) concerning the case where both $$X_1^{k_i} + \cdots + X_s^{k_i} = \beta_i, \qquad 1 \le i \le m,$$9 and $\Gamma(\kappa,q)$0 are odd, supplying a corrected piecewise formula consistent with their example.

Hermitian-form systems

The specialization to $\Gamma(\kappa,q)$1 exploits the fact that the diagonal GP-graph $\Gamma(\kappa,q)$2 is isomorphic to the Hermitian-form graph $\Gamma(\kappa,q)$3 with $\Gamma(\kappa,q)$4, whose spectrum was computed by Stanton. Since $\Gamma(\kappa,q)$5, Theorem on the homogeneous case yields the closed formula ($\Gamma(\kappa,q)$6):

$\Gamma(\kappa,q)$7

where the multiplicities $\Gamma(\kappa,q)$8 are explicit positive integers satisfying a one-term recursion. Two consequences deserve emphasis. First, because $\Gamma(\kappa,q)$9 does not depend on $\Gamma(\kappa,q)$0, computing $\Gamma(\kappa,q)$1 once suffices to obtain solution counts for any number of variables over a fixed field. Second, the same spectrum gives the full Ihara zeta function of the Hermitian-form graph as a rational function. A Weil-type corollary gives $\Gamma(\kappa,q)$2.

Small cases are treated explicitly: for $\Gamma(\kappa,q)$3 the closed formula

$\Gamma(\kappa,q)$4

holds in $\Gamma(\kappa,q)$5; for $\Gamma(\kappa,q)$6 a more involved but still closed expression is given; for $\Gamma(\kappa,q)$7 summation formulas and first values are provided. Closed-form treatment for $\Gamma(\kappa,q)$8 and large $\Gamma(\kappa,q)$9 becomes cumbersome, which the authors acknowledge as evidence of intrinsic complexity rather than a deficiency of the method.

Arbitrary $(q-1)/d_\kappa$0 and $(q-1)/d_\kappa$1: recursion, closed forms, and injectivity

Using a functional equation $(q-1)/d_\kappa$2 for the generating polynomial $(q-1)/d_\kappa$3 (with $(q-1)/d_\kappa$4), the paper derives several strong results:

• Uniform small-$(q-1)/d_\kappa$5 formulas: $(q-1)/d_\kappa$6 and $(q-1)/d_\kappa$7 hold for every $(q-1)/d_\kappa$8, with analogous expressions for $(q-1)/d_\kappa$9.
• Two-term recursion valid for all $d_\kappa = \gcd(k_1,\ldots,k_m,q-1)$0: $d_\kappa = \gcd(k_1,\ldots,k_m,q-1)$1.
• Explicit Gaussian-binomial sums: separating parity of $d_\kappa = \gcd(k_1,\ldots,k_m,q-1)$2, $d_\kappa = \gcd(k_1,\ldots,k_m,q-1)$3 equals a finite sum over $d_\kappa = \gcd(k_1,\ldots,k_m,q-1)$4 involving $d_\kappa = \gcd(k_1,\ldots,k_m,q-1)$5-binomial coefficients $d_\kappa = \gcd(k_1,\ldots,k_m,q-1)$6 and factors $d_\kappa = \gcd(k_1,\ldots,k_m,q-1)$7.

Consequently $d_\kappa = \gcd(k_1,\ldots,k_m,q-1)$8 is a monic polynomial in $d_\kappa = \gcd(k_1,\ldots,k_m,q-1)$9 of explicitly known degree $\alpha \in (\mathbb{F}_q)^m$0 for even $\alpha \in (\mathbb{F}_q)^m$1 and $\alpha \in (\mathbb{F}_q)^m$2 for odd $\alpha \in (\mathbb{F}_q)^m$3, with lowest-degree term $\alpha \in (\mathbb{F}_q)^m$4. This degree structure yields an injectivity result: for fixed $\alpha \in (\mathbb{F}_q)^m$5, the map $\alpha \in (\mathbb{F}_q)^m$6 is injective on $\alpha \in (\mathbb{F}_q)^m$7, so no two distinct Hermitian-form systems (with at least two variables) have the same number of solutions. The proof uses the $\alpha \in (\mathbb{F}_q)^m$8-adic valuation to recover $\alpha \in (\mathbb{F}_q)^m$9 and monotonicity in $$\lambda_{\alpha,\kappa,q} = \frac{1}{d_\kappa}\big(\mathcal{W}_\chi(f_{\alpha,\kappa}) - 1\big),$$0 to recover $$\lambda_{\alpha,\kappa,q} = \frac{1}{d_\kappa}\big(\mathcal{W}_\chi(f_{\alpha,\kappa}) - 1\big),$$1.

The divisibility pattern observed in small cases leads the authors to conjecture improved congruences, e.g. $$\lambda_{\alpha,\kappa,q} = \frac{1}{d_\kappa}\big(\mathcal{W}_\chi(f_{\alpha,\kappa}) - 1\big),$$2 and $$\lambda_{\alpha,\kappa,q} = \frac{1}{d_\kappa}\big(\mathcal{W}_\chi(f_{\alpha,\kappa}) - 1\big),$$3; these remain unproven.

Hypergeometric connection

The final section shows that $$\lambda_{\alpha,\kappa,q} = \frac{1}{d_\kappa}\big(\mathcal{W}_\chi(f_{\alpha,\kappa}) - 1\big),$$4 is a terminating basic hypergeometric series, giving

$$\lambda_{\alpha,\kappa,q} = \frac{1}{d_\kappa}\big(\mathcal{W}_\chi(f_{\alpha,\kappa}) - 1\big),$$5

Termination is forced by the upper parameter $$\lambda_{\alpha,\kappa,q} = \frac{1}{d_\kappa}\big(\mathcal{W}_\chi(f_{\alpha,\kappa}) - 1\big),$$6, so the infinite series truncates at degree $$\lambda_{\alpha,\kappa,q} = \frac{1}{d_\kappa}\big(\mathcal{W}_\chi(f_{\alpha,\kappa}) - 1\big),$$7, matching the polynomial exactly. A contiguous relation for $$\lambda_{\alpha,\kappa,q} = \frac{1}{d_\kappa}\big(\mathcal{W}_\chi(f_{\alpha,\kappa}) - 1\big),$$8 recovers the two-term recursion, and the sequence $$\lambda_{\alpha,\kappa,q} = \frac{1}{d_\kappa}\big(\mathcal{W}_\chi(f_{\alpha,\kappa}) - 1\big),$$9 satisfies a second-order linear $$R_\kappa = \{(x^{k_1},\ldots,x^{k_m}) : x \in \mathbb{F}_q^*\}$$00-difference equation, hence is $$R_\kappa = \{(x^{k_1},\ldots,x^{k_m}) : x \in \mathbb{F}_q^*\}$$01-holonomic. The authors connect this to Stanton's affine $$R_\kappa = \{(x^{k_1},\ldots,x^{k_m}) : x \in \mathbb{F}_q^*\}$$02-Krawtchouk polynomials: within the $$R_\kappa = \{(x^{k_1},\ldots,x^{k_m}) : x \in \mathbb{F}_q^*\}$$03-Askey scheme, those $$R_\kappa = \{(x^{k_1},\ldots,x^{k_m}) : x \in \mathbb{F}_q^*\}$$04 spherical functions degenerate into terminating $$R_\kappa = \{(x^{k_1},\ldots,x^{k_m}) : x \in \mathbb{F}_q^*\}$$05 series, so the hypergeometric representation reflects the underlying association-scheme geometry rather than being incidental.

Limitations and open questions

Several qualifications apply. The Ihara zeta function results require undirectedness, excluding odd-characteristic parameters not dividing $$R_\kappa = \{(x^{k_1},\ldots,x^{k_m}) : x \in \mathbb{F}_q^*\}$$06. The Weil-type bounds require the non-degeneracy assumption on all polynomials $$R_\kappa = \{(x^{k_1},\ldots,x^{k_m}) : x \in \mathbb{F}_q^*\}$$07. The general closed-form and recursion results are specific to the Hermitian-family exponents $$R_\kappa = \{(x^{k_1},\ldots,x^{k_m}) : x \in \mathbb{F}_q^*\}$$08; arbitrary exponent vectors $$R_\kappa = \{(x^{k_1},\ldots,x^{k_m}) : x \in \mathbb{F}_q^*\}$$09 are covered only by the spectral/Weil-sum expressions, which still require evaluating Weil sums. The congruence conjectures for $$R_\kappa = \{(x^{k_1},\ldots,x^{k_m}) : x \in \mathbb{F}_q^*\}$$10 and $$R_\kappa = \{(x^{k_1},\ldots,x^{k_m}) : x \in \mathbb{F}_q^*\}$$11 are supported only by computed data up to moderate $$R_\kappa = \{(x^{k_1},\ldots,x^{k_m}) : x \in \mathbb{F}_q^*\}$$12. Finally, the extension to non-monic coefficient matrices $$R_\kappa = \{(x^{k_1},\ldots,x^{k_m}) : x \in \mathbb{F}_q^*\}$$13 is not addressed here, and the companion treatment of the single-equation case $$R_\kappa = \{(x^{k_1},\ldots,x^{k_m}) : x \in \mathbb{F}_q^*\}$$14 via GP-graphs is deferred to forthcoming work.

Conclusion

The paper establishes that exact solution counts for monic diagonal systems over finite fields can be read off from spectral data of diagonal GP-graphs, provides three interchangeable computational formalisms (walks, adjacency matrices, spectra), and delivers complete closed-form, recursive, and hypergeometric descriptions for the Hermitian-form family for arbitrary numbers of equations and variables. The injectivity theorem and the monic-polynomial structure of $$R_\kappa = \{(x^{k_1},\ldots,x^{k_m}) : x \in \mathbb{F}_q^*\}$$15 are concrete structural outputs, while the divisibility conjectures and the directed/non-Hermitian cases remain open problems directly raised by the framework.