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Counting in Vieta graphs over $\mathbb{F}_p$

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Pith's one-line read Quadratic character sums give exact degree counts for Vieta graphs over finite fields.

desk verdict A broad, mostly careful framework for counting degrees in Vieta graphs over F_p, with solid dimension-3 results and dimension-4 tables that rest on a few unproved counts. read the letter →

arxiv 2608.03097 v1 pith:RMLUVL63 submitted 2026-08-04 math.NT math.CO

Bogdan Nica This is my paper · ORCID
classification math.NTmath.CO MSC 11D2505C2511L10
keywords VietagraphMarkoffgeneralizedcubicquadraticcharactersumsJacobsthaldegreedistributionMarkoff-Hurwitzquarticfinitefields
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper builds a counting machine for Vieta graphs: finite graphs whose vertices are solutions over the prime field to a symmetric polynomial equation that is quadratic in each variable, with edges given by Vieta flips between the two roots. It shows that the two basic census questions—how many vertices, and how many vertices of each degree—can be answered by a small set of auxiliary counts, and that these counts reduce to quadratic character sums in one fewer variable. The method yields explicit formulas for the generalized Markoff cubic in four one-parameter families, and for two quartic families in four variables. A sympathetic reader would care because the Markoff graph modulo a prime is a widely studied object, and this gives a systematic, explicit approach to a broad class of such graphs.

What carries the argument

The machinery is the family of counts recording vertices fixed by at least k of the n Vieta flips, together with an inclusion-exclusion lemma that converts the sequence of these counts into the full degree distribution. The Discriminant Lemma is the workhorse: it expresses the two lowest counts and an associated character sum as lower-dimensional counts involving the discriminant of the defining polynomial viewed as a quadratic in one variable, dropping the dimension by one. Evaluation of the resulting sums uses standard quadratic character sums, a quartic-to-cubic reduction lemma, and Jacobsthal sums, whose values are known through representations of p as sums of two or three squares. In dimension 3, the key algebraic object is a 'main cubic' whose quadratic character sum is the only non-elementary ingredient in the count of degree-deficient vertices; the families are chosen so that this cubic is non-separable, or is a shift of a monomial cubic, or so that the relevant Jacobsthal sum falls in a small explicitly computable list.

What would settle it

Directly enumerate all solutions over a small prime field, say p = 7, for one of the quartics (x+y+z+t+1)^2 = c x y z t with c = 8, -16, or 32, and compare the numbers of vertices of degree 0, 1, 2, 3 with the paper's tables; a mismatch would show the unproved counts in Lemma 15.1 to be wrong, and the degree tables with them.

Watch

Extended reading notes

Core claim

The paper's central claim is that for a symmetric polynomial of degree 2 in each variable, the degree distribution of the Vieta graph over the field with p elements is determined by the counts of vertices fixed by at least k Vieta flips, and that these counts can be computed explicitly via quadratic character sums for several natural families. In dimension 3, the generalized Markoff cubic admits closed-form counts in the Cayley family, the K-family, the Fricke family at four distinguished level parameters, and the J-family in cases where the relevant Jacobsthal sum is known. In dimension 4, explicit degree distributions are given for Cayley-type quartics at specific parameter values and for Markoff-Hurwitz quartics. Along the way the paper proves that among all generalized Markoff cubics the Vieta graph is regular only for two small exceptional instances, and that nodes of the underlying cubic surface occur only in the Cayley and K families.

Load-bearing premise

For the four-variable Cayley-type quartics, the degree-distribution theorems rest on three counts that the paper states without proof; if any of those counts is wrong, the corresponding degree tables fail.

Editorial extensions

If this is right

  • For the generalized Markoff cubic, the Vieta graph is regular only for one cubic over the field of 5 elements and one over the field of 7 elements; for every other parameter pair and every prime p the graph has deficient vertices.
  • The Cayley family is exceptional in the census: its graphs have about twice as many deficient vertices as the generic family, and the Cayley cubic is the only generalized Markoff cubic whose Vieta graph has no degree-2 vertices for any p.
  • Nodal cubics occur only in the Cayley family (generally three nodes) and the K-family (always one node), with the Cayley cubic the unique case with four nodes; a particular intersection cubic has a single node.
  • In dimension 4, the Markoff-Hurwitz quartic has essentially two Vieta graphs depending only on the quadratic signature of its parameter, and when p is congruent to 3 modulo 4 it has no vertices of degree 1 or 2.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the edge count depends only on the first two counts, every explicit formula for those counts in the paper immediately yields the total number of edges in the corresponding Vieta graph, a quantity the paper does not tabulate.
  • The character-sum method suggests a recipe for finding further countable families: impose an algebraic condition on the main cubic, such as being, up to a shift, of the form x^3 + b x. The author notes this condition is more complicated and lacks a complete classification; finding one would add new explicit families.
  • The unproved higher counts for the Cayley-type quartics could be checked independently by brute force for small primes; this would form a cheap test of the four-variable degree tables.
  • The observed coincidence that a distinguished set of real parameters lies in the Cayley family, with the same quadratic signature appearing in the vertex count, points to a possible arithmetic echo of real-dynamics phenomena, but the paper offers no explanation.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper introduces Vieta graphs over F_p associated to symmetric polynomials f that are quadratic in each variable, with vertices the solutions and edges given by Vieta flips. The main methodological contribution is a reduction of the degree distribution to counts N_k(f) of vertices fixed by at least k flips, together with a 'Discriminant Lemma' that lowers the dimension of the relevant character-sum computations. In dimension 3, the paper studies the generalized Markoff cubic M(C,D): x^2+y^2+z^2=xyz-C(x+y+z)+D, computes N_0 through N_3, and gives explicit degree tables for the Cayley family D=4-2C-C^2/4, the K family C=K^2+2K, D=-(2K^3+3K^2), and selected Fricke and J family members, using Jacobsthal sums. In dimension 4, it treats Cayley-type quartics (x+y+z+t+F)^2=E xyzt and Markoff-Hurwitz quartics x^2+y^2+z^2+t^2=2Exyzt, producing degree distribution tables in Theorems 15.2, 15.3 and 16.2. The paper is written as a systematic framework with many explicit finite-field formulas; the dimension-3 part is largely self-contained, while the dimension-4 part relies on counts that are stated without full proof.

Significance. Conditional on the completeness of the proofs, the paper delivers a substantial toolbox for a natural class of finite graphs of algebraic origin. The three-variable analysis is detailed and appears correct: the character-sum evaluations for the Cayley and K families are fully derived, the Fricke and J families are handled through explicit Jacobsthal-sum identities, and the tables are internally consistent. The paper also does a service by connecting Carlitz's classical point counts to Vieta-flip degree distributions. I verified algebraically that the N_k values stated in Lemma 15.1 reproduce Theorem 15.2 when substituted into Table 9, so the announced tables are consistent with the asserted counts. The main value of the paper, if the missing computations are supplied or verified, would be as a reference framework and a set of explicit formulas for further work on Markoff-type graphs in higher dimensions.

major comments (2)
  1. [§15, Lemma 15.1 and Theorems 15.2–15.3] The proof of Lemma 15.1 explicitly states that the counts for N2(f), N3(f), and N4(f) are 'elementary, and somewhat tedious' and are 'all omitted and left to the reader.' These counts are load-bearing: Theorems 15.2 and 15.3 and Table 10 are obtained from them by the linear combinations of Lemma 3.1, and the degree-0 column is exactly N4(f). An error in any one of these unproved counts would change the announced degree distributions for the Cayley-type quartics. I checked that substituting the stated N_k values into Table 9 reproduces Theorem 15.2, but this only verifies the algebra, not the counts themselves. Please either provide complete derivations for N2, N3, and N4, or add an independent small-prime verification (e.g., direct enumeration of the Vieta graph for p=5,7,11) in a table or appendix.
  2. [§16, Counting Lemma 16.1] The N1 computation for the Markoff-Hurwitz quartics is the most delicate step in Part 4 and is not fully carried out. It depends on the evaluation S2 = σ(2)(4A2(p)^2 - 2p) in (16.14), obtained by a chain of nontrivial manipulations including a double-sign change and a change of variables, and on 'simple manipulations, left to the reader' in the N0 part. A sign error in (16.14) or in the reductions (16.11)–(16.15) would propagate directly into the degree-3 column of Theorem 16.2. Please expand these computations, or at minimum provide a verification by direct enumeration for small primes in both congruence classes p≡1 mod 4 and p≡3 mod 4.
minor comments (5)
  1. [§14, after equation (14.2)] The sentence 'The J family meets the K family when J=±2 (correspondingly, K=−1) or J=−1 (correspondingly, K=2)' has the K-values interchanged: J=±2 gives C=8, D=−28, which is the K=2 cubic M(8,−28), while J=−1 gives C=D=−1, which is the K=−1 cubic M(−1,−1).
  2. [§9, final paragraph] The assertion that κ and λ are non-separable if and only if C and D are parameterized as in (9.1) is stated without proof; since it is not needed for the counting lemmas, it should either be proved in a short appendix or be explicitly labeled as an observation.
  3. [§16, end of the N1 proof] The sentence 'This completes the proof of Theorem 16.1' should refer to Lemma 16.1, since that is the statement being proved.
  4. [§12 and §14, use of A3(p)] In Table 7 and Theorem 14.3, the integer A3(p) is used without restating its normalization A≡−1 mod 3 from Lemma 12.2; a parenthetical reminder would avoid ambiguity.
  5. [§15, Table 10] The Iverson bracket 'JF 2E= 16K' is typographically ambiguous; it should be written as JF^2E=16K to avoid reading the exponent as 2E.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the N_k counts are computed independently, and the degree tables follow by inclusion–exclusion rather than by definition.

full rationale

The paper's central derivation is self-contained in the relevant sense. The degree distributions are obtained from the N_k counts via Lemma 3.1, a straightforward inclusion–exclusion identity for graphs generated by involutions; the N_k counts themselves are defined as fixed-point counts of Vieta flips, not as degree-distribution quantities, so the relation is a genuine mathematical derivation rather than a tautology. The N_0, N_1, N_2, N_3, and N_4 counts in dimensions 3 and 4 are computed through the Discriminant Lemma, direct elimination, and quadratic character sums; where the paper relies on external results, those results are independent benchmarks: Carlitz's point counts are classical and cited as such, and the Jacobsthal sum evaluations used for the Fricke and J families are standard facts that the paper records from the author's monograph [26] but does not derive from the conclusions being asserted. The unproved N_2–N_4 counts in Lemma 15.1 are a genuine proof gap, but they are asserted inputs, not quantities fitted from or defined by the degree tables, so they constitute a completeness/correctness risk rather than circularity. Similarly, the restriction to the Cayley, K, Fricke, and J families is an explicit design choice that makes the auxiliary cubics split or reduces the character sums to known Jacobsthal sums; this is not a case of smuggling the answer into the ansatz. No fitted parameter is relabeled as a prediction, no load-bearing uniqueness theorem is imported from the author's own work, and no known result is merely renamed. Accordingly, the paper's explicit counting results do not reduce by construction to their inputs.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The counting framework depends on standard character-sum evaluations and on Carlitz's external counts; no free parameters are fitted to data, and no new physical entities are posited. The main auxiliary choices are the parametric families, made to render the Jacobsthal sums tractable.

assumptions (5)
  • standard math Standard complete character sum evaluations: (4.3), (4.5), Lemma 4.1.
    Used throughout to count roots of quadratics; these are classical results, e.g., [26, Thm.1.10, Thm.2.4].
  • standard math Hasse-Weil bound for cubic character sums: |Σ σ(λ(x))| ≤ 2√p.
    Used in Corollary 7.3 to restrict p. Standard consequence of Weil's bound.
  • domain assumption Equations are symmetric, degree 2 in each variable, and regular: ∂_i^2 f never vanishes on V(f).
    Regularity ensures the Vieta flip (2.3) is fully defined. The paper restricts to equations of the form (2.5), where ∂_i^2 f = 2.
  • standard math Carlitz's counting results [8, 9, 10] are correct.
    The paper cites Carlitz for N0 counts of several families; these are external published results.
  • ad hoc to paper The parameter families (Cayley, K, Fricke, J) are chosen so that the relevant character sums reduce to known Jacobsthal sums.
    These choices are made to make the counts explicitly computable; they are not forced by the general problem. This shapes the scope of the results.

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Pith. "Pith review of Counting in Vieta graphs over $\mathbb{F}_p$." pith.science (2026). https://pith.science/paper/RMLUVL63

@misc{pith2026260803097,
  author       = {Pith},
  title        = {Pith review of: Counting in Vieta graphs over $\mathbbF_p$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RMLUVL63}},
  note         = {Machine review of arXiv:2608.03097}
}
abstract

We introduce and study a finite simple graph of algebraic origin: the Vieta graph on the solution set over $\mathbb{F}_p$ to a symmetric, multivariate equation which is quadratic in each variable. This construction is a broad generalization of the Markoff graph over $\mathbb{F}_p$, extensively studied in the recent literature. We give a systematic approach, partly based on quadratic character sums, to the following basic counting questions: how many vertices does a Vieta graph have, and what is the degree distribution? We focus on explicit counts, addressing the low-dimensional cases in three and four variables.

Figures

Figures reproduced from arXiv: 2608.03097 by the authors.

Figure 1
Figure 1. The Markoff graph over F5. There are 41 vertices, all of degree 3 except for the isolated vertex (0, 0, 0). Another Markoff-type graph over Fp was considered by de Courcy-Ireland, Lit￾man, and Mizuno [14]. It is defined by the equation x 2 + y 2 + z 2 (1.4) + K1yz + K2xz + K3xy = (3 + K1 + K2 + K3)xyz where K1+K2+K3 ̸= −3. For K1 = K2 = K3 = 0, we recover the Markov equation. As a diophantine equation, (1.4) has bee… view at source ↗
Figure 2
Figure 2. The Vieta graph of the cubic M(0, 2) over F5, a con￾nected 3-regular graph on 16 vertices. This means that the cubic d(x 3 − x 2 ) + 16(x − 1)2 evaluates to a non-square in F ∗ p whenever x ̸= 0, 1. When p = 5 we find d = 3, and when p = 7 we find d = 1. Recalling that d = D − D• and D• = 4, as C = 0, this means that D = 2 when p = 5, respectively D = 5 when p = 7. □ 8. The N2 count The counts for N2 and N3, obtaine… view at source ↗
Figure 3
Figure 3. The K family meets the Fricke family (C = 0) and the Cayley family (D = 4−2C −C 2/4) in the real (C, D) plane. Note: the axes are not equally scaled. It was convenient to introduce the K family in relation to the modified Markov equation (1.5), which was recently studied in the literature. But that is not the main reason why the K family deserves attention for the discussion at hand. Two structural facts make the K … view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: The Vieta graph of the Cayley cubic M(0, 4) over F5. There are N3 = 4 isolated vertices. Proof. We show that a node can only occur in the Cayley family and in the K family; and that when it does occur, a node has one of the following two types: (i) (−2, −2, 2−C/2), (−2…
Figure 5
Figure 5. Figure 5: The Vieta graph of the cubic M(8, −28) over F5. The only isolated vertex is (−2, −2, −2). 11. Degree distributions: the Cayley family and the K family We can finally derive the explicit degree distribution for some distinguished fam￾ilies of 3-dimensional Vieta graphs.…
Figure 6
Figure 6. Figure 6: The Vieta graph of the cubic M(−1, −1) over F5. The deficient vertices are all of degree 2 (highlighted in light grey), except for the one isolated vertex [PITH_FULL_IMAGE:figures/full_fig_p027_6.png]
Figure 7
Figure 7. Figure 7: The Vieta graph of the Fricke cubic M(0, 8) over F5. This is a connected graph on 36 vertices; there are 12 deficient vertices, all of degree 2 (highlighted in light grey). Counting Lemma 14.1. Consider the J family of cubics, where J ̸= ±2, −1. Then the Nk counts are …
Figure 8
Figure 8. Figure 8: The J family meets the Cayley family and the K family in the real (C, D) plane. Note: the axes are not equally scaled. Proof. We are in the non-Cayley case D ̸= D•, since J ̸= ±2. In this proof, we only work with the main cubic λ(x); we recall that N0(κ) = N0(λ), by Le…
Figure 9
Figure 9. Figure 9: The Vieta graph of the Cayley cubic M(−8/3, −20/3) over F5. The leaf vertices are highlighted in white. total N0 deficient 4N1 − 6N2 + 4N3 − N4 degree 3 4(N1 − 3N2 + 3N3 − N4) degree 2 6(N2 − 2N3 + N4) degree 1 4(N3 − N4) degree 0 N4 [PITH_FULL_IMAGE:figures/full_fig_…

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