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REVIEW 3 major objections 5 minor 14 references

Primitive pairs of rational functions with prescribed traces over finite fields

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read For $m\ge 9$, every pair $(q,m)$ outside a short explicit list admits a nonzero $\varepsilon$ whose images under two rational functions are primitive and whose traces are prescribed.

desk verdict The paper's sufficient condition is solid, but Theorem 1.1's exception list is only a list of sieve failures, not a classification of non-existence. read the letter →

arxiv 2411.15568 v1 pith:QDGEJHMH submitted 2024-11-23 math.NT

Shikhamoni Nath , Dhiren Kumar Basnet This is my paper · ORCID
classification math.NT MSC 12E2011T23
keywords finitefieldsprimitiveelementspairsrationalfunctionsprescribedtracescharactersumsKloostermansievemethods
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

The paper establishes a sufficient condition for a finite field $\mathbb{F}_{q^m}$ to contain a nonzero element $\varepsilon$ whose images under two rational functions $f,g$ are both primitive, while the traces of $\varepsilon$ and $\varepsilon^{-1}$ equal prescribed values $a,b\in\mathbb{F}_q$. The condition is expressed in terms of the degree sums of $f$ and $g$ and the number $W(q^m-1)$ of squarefree divisors of $q^m-1$. Using it, the authors prove that such an $\varepsilon$ exists in all but finitely many fields $\mathbb{F}_{q^m}$ over $\mathbb{F}_q$. When both degree sums are $3$, they sharpen this to a classification: for every prime power $q$ and every $m\ge 9$, the guarantee holds outside an explicit list of exceptional pairs.

What carries the argument

The counting function $N_{f,g,a,b}(e_1,e_2)$ counts nonzero elements $\varepsilon$ for which $f(\varepsilon)$ is $e_1$-free, $g(\varepsilon)$ is $e_2$-free, and both trace conditions hold, where an element is $e$-free when it is not a $d$-th power for any non-trivial divisor $d$ of $e$. Expansion through the characteristic functions (2.1) and (2.2) turns this count into sums of mixed character sums $\chi_{f,g}(d_1,d_2,u,v)$. Lemmas 2.1, 2.2 and 2.3 bound those sums, producing the sufficient condition (3.1) and its simpler version (3.2). The sieving inequality of Theorem 3.3 then upgrades counts from a divisor $d$ to the full $q^m-1$, and Lemma 4.3 bounds $w(n)$ to make the numerical check for $m_1=m_2=3$ finite.

What would settle it

In $\mathbb{F}_{2^{17}}$, take $f(x)=x^3+x$ and $g(x)=x^3+x^2$, and exhaustively check the 131071 nonzero $\varepsilon$ for each of the four choices $a,b\in\mathbb{F}_2$. If some choice yields no $\varepsilon$ with $f(\varepsilon)$ and $g(\varepsilon)$ both primitive and the two trace equations satisfied, Theorem 1.1 is false for the non-exceptional pair $(2,17)$; if all four choices succeed, the theorem survives this test.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: with degree sums $m_1=m_2=3$, for every prime power $q$ and $m\ge 9$ except the pairs listed in the theorem, and for every pair $f,g$ in the class $\mathcal{R}$ (rational functions not of the form $c\,x^j h(x)^d$ with $d>1$ dividing $q^m-1$), there is a nonzero $\varepsilon\in\mathbb{F}_{q^m}$ such that $(f(\varepsilon),g(\varepsilon))$ is a primitive pair and $\operatorname{Tr}_{\mathbb{F}_{q^m}/\mathbb{F}_q}(\varepsilon)=a$, $\operatorname{Tr}_{\mathbb{F}_{q^m}/\mathbb{F}_q}(\varepsilon^{-1})=b$ for any prescribed $a,b\in\mathbb{F}_q$. The paper also proves Proposition 4.2: for $m\ge 5$, all but finitely many fields $\mathbb{F}_{q^m}$ contain such an element. The element $\varepsilon$ itself is not required to be primitive; only its two rational-function values are required to generate $\mathbb{F}_{q^m}^*$.

Load-bearing premise

The character-sum estimates quoted in Section 3 must hold with exactly the constants used there, because the sufficient condition and the resulting list of exceptional pairs are built on those constants.

Editorial extensions

If this is right

  • For every prime power $q$ and $m\ge 9$ outside the listed exceptional pairs, the existence of $\varepsilon$ is guaranteed for all admissible $f,g$ of degree sums $3$ and all prescribed traces $a,b$, with no search required.
  • For $m\ge 109$ the numerical criterion (4.5) holds for every prime power $q$, so the theorem is unconditional in that range.
  • For the listed exceptional pairs the method leaves existence undecided; each case is small enough to be settled by direct computation.
  • The general sufficient condition (3.2) applies to arbitrary degree sums $m_1,m_2$, so larger degree sums merely delay the threshold $q^{m/2-2}$ rather than requiring new machinery.

Reading between the lines

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

  • Beyond the paper, the same characteristic-function expansion with an extra multiplicative condition suggests a sufficient condition for $\varepsilon$ itself to be primitive in addition to having primitive images.
  • A natural extension to $k$ rational functions should replace $W(q^m-1)^2$ by $W(q^m-1)^k$ in the sieve, preserving an all-but-finitely-many conclusion.
  • The exceptional list records pairs where the sieve does not prove existence, not pairs where existence is disproved; direct computation over those small fields would show which cases genuinely fail.
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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

3 major / 5 minor

Summary. The paper studies the existence of a nonzero element \epsilon in F_{q^m} such that, for rational functions f,g in a certain class R, the pair (f(\epsilon),g(\epsilon)) is primitive and the traces of \epsilon and \epsilon^{-1} take prescribed values a,b in F_q. The authors derive a sufficient condition using multiplicative and additive character sums, apply a prime sieve, and then specialize to the case where the degree sums of f and g are both 3. The main theorem, Theorem 1.1, asserts that for every prime power q and every m \ge 9, all pairs (q,m) belong to the set A^m_p except for an explicitly listed finite set of cases. The proof combines analytic estimates with a finite computation over m \le 108.

Significance. If the main theorem were established, the paper would give a useful extension of primitive-pair results to rational functions with trace constraints, and the authors make a serious effort to combine standard tools: Cohen--Huczynska sieving, Weil-type character sum bounds, and estimates for the divisor function W(M). The sufficient condition itself, in the form of Theorem 3.3, is a reasonable technical contribution. However, the central classification is not proved: the list of ``possible exceptional pairs'' is only a list of cases where the sieve fails, and one of the character-sum bounds used to derive the sufficient condition is false. These problems affect the main theorem directly, so the paper cannot be accepted in its present form.

major comments (3)
  1. [Section 4, end, and Theorem 1.1] The proof of Theorem 1.1 does not establish the exception list. The text says that for the listed pairs no suitable d satisfying (3.3) could be found, and those pairs are ``considered as possible exceptional pairs.'' Failure of a sufficient condition is not a proof of nonexistence: to conclude (q,m) \notin A^m_p one needs an independent upper bound showing that N_{f,g,a,b}(q^m-1,q^m-1)=0 for the relevant choices, or a field-theoretic obstruction. No such argument is given. The abstract's wording ``possible exceptional pairs'' is accurate, but Theorem 1.1 changes this to a definite assertion, and the logical gap is load-bearing for the main theorem.
  2. [Section 3, Case IV] The claimed bound on |\chi_{f,g}(d_1,d_2,u,v)| in Case IV is false as stated. Take a cubic h(x) with three distinct nonzero roots, choose c \in F_{q^m}^* with \chi_d(c)=1 for some prime d dividing q^m-1, and set f(x)=h(x), g(x)=c/h(x). Both rational functions have degree sum 3 and satisfy the defining condition of R. Then for d_1=d_2=d and u=v=0, \chi_{f,g}(d,d,0,0)=\sum_{\epsilon\in F_{q^m}\setminus S} \chi_d(h(\epsilon))\chi_d(c/h(\epsilon)) = \chi_d(c)(q^m-|S|)=q^m-|S|, which contradicts the bound (m_1+m_2-1)q^{m/2} printed in that subsection. Since these Case IV estimates feed directly into the lower bound (3.1) and hence into the sieve inequality (3.3), the general sufficient condition is not established.
  3. [Section 4, finite verification] The finite verification from m'=109 down to m=9 is not reproducible from the text. The proof refers to a ``trial and error'' determination of m', to lower bounds q_{m,i}, and to a search for d satisfying (3.3), but it does not give the algorithm, the intermediate tables, or the code used. The listed exception set is the substantive content of Theorem 1.1, and without the computational data being available, the classification cannot be checked independently.
minor comments (5)
  1. [Section 4, list of possible exceptional pairs] The pair (4,12) appears twice in the list; Theorem 1.1 lists q=4 for m=12 only once.
  2. [Abstract and Theorem 1.1] The abstract says ``possible exceptional pairs'' while Theorem 1.1 says ``except the following cases''; these should be aligned, since the proof only identifies possible exceptions.
  3. [Section 4, Eq. (4.4) and following display] The displayed formulas appear to have lost exponent formatting: they should read q^{m/2-2} \ge 2^{2(2+w(q^m-1))} and (x+1)^{1/2-2/m} \ge 2^{4+2.77\log x/\log\log x}. As printed, the right-hand sides appear as ordinary products, which is not consistent with the preceding bounds.
  4. [Section 4, Proposition 4.2] In Lemma 4.1, D depends on M=q^m-1 through the primes p_i \le 2\nu dividing M, but the proof of Proposition 4.2 treats D as a constant when taking logarithms. This can be repaired by bounding D uniformly over all primes \le 2\nu, but as written the monotonicity claim is not fully justified.
  5. [Section 1, definition of A^m_p] The phrase ``for some prescribed values a,b'' is ambiguous. The sufficient condition in Section 3 is independent of a,b and of the particular rational functions, suggesting that Theorem 1.1 is intended as a universal statement; the definition should say so explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the sufficient condition and sieve bounds are derived from external lemmas; the unproved exception list is a proof gap, not a circular reduction.

full rationale

The paper's derivation chain is self-contained relative to its cited external results. The character-sum estimates in Section 3 (Lemmas 2.1, 2.2, 2.3) come from published literature (Lidl-Niederreiter, Castro-Moreno, Laishram-Sarma-Sharma), and the sufficient condition (3.1)/(3.2) is obtained by summing those estimates, not by assuming the desired conclusion. Lemma 3.1 and Lemma 3.2, leading to the sieving inequality (3.3), are also attributed to external sources (Cohen-Huczynska, Gupta-Sharma-Cohen), and the numerical verification in Section 4 checks these independently stated inequalities. No parameter is fitted to a subset of the data and then renamed a prediction, and no load-bearing claim is justified solely by a self-citation; indeed the reference list contains no work by the present authors. The genuine weakness is logical rather than circular: the text ends with 'The pairs for which we could not find such d are considered as possible exceptional pairs and thus we complete the proof of Theorem 1.1.' Failure of a sufficient condition does not prove nonexistence, so the listed 'possible exceptional pairs' are not shown to be actual exceptions in Theorem 1.1. This is an unsupported inference (and the duplicate (4,12) suggests a computational slip), but it is not an equivalence-by-construction between input and output, nor does any fitted or self-cited quantity force the theorem. Therefore the circularity score is 0.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The proof rests on four external bounds and a non-degeneracy class; no new entities are introduced. Two auxiliary thresholds are chosen by hand in the finiteness argument.

free parameters (2)
  • nu = greater than 4
    Chosen in Lemma 4.1's estimate W(M)≤D M^{1/ν}; ν>4 makes the denominator in inequality (4.3) positive and closes the finiteness proof.
  • m_prime = 109
    Threshold chosen by trial and error so that inequality (4.5) holds for all x≥x'; it is a proof artifact, not fitted to data.
assumptions (3)
  • domain assumption The Weil-type character sum bounds in Lemmas 2.1, 2.2, and 2.3 hold as stated for rational functions over F_{q^m} and are applicable to the sums χ_{f,g}(d1,d2,u,v) with error terms controlled by degree sums m1,m2.
    The paper relies on these external bounds throughout Section 3 to derive the sufficient condition (3.1).
  • domain assumption The sieving inequalities Lemma 3.1 and Lemma 3.2 (from refs [13] and [9]) apply to the counting function N_{f,g,a,b} with the given constants, including the factor 2(m1+m2+2)W(d)^2.
    Used to prove the prime sieve inequality Theorem 3.3, which is the basis for the finite exceptional pair classification.
  • domain assumption The non-degeneracy restriction f,g ∈ R, meaning neither has the form c x^j h(x)^d for any d>1 dividing q^m-1, is sufficient for all the character sum bounds used.
    The paper defines R in the introduction and uses this restriction to make the bounds of Lemmas 2.1 and 2.2 valid; the theorem is stated only for this class.

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Pith. "Pith review of Primitive pairs of rational functions with prescribed traces over finite fields." pith.science (2026). https://pith.science/paper/QDGEJHMH

@misc{pith2026241115568,
  author       = {Pith},
  title        = {Pith review of: Primitive pairs of rational functions with prescribed traces over finite fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QDGEJHMH}},
  note         = {Machine review of arXiv:2411.15568}
}
abstract

Let $q$ be a positive integral power of some prime $p$ and $\mathbb{F}_{q^m}$ be a finite field with $q^m$ elements for some $m \in \mathbb{N}$. Here we establish a sufficient condition for the existence of a non-zero element $\epsilon \in \mathbb{F}_{q^m}$, such that $(f(\epsilon), g(\epsilon))$ is a primitive pair in $\mathbb{F}_{q^m}$ with two prescribed traces, $\Tr_{{\mathbb{F}_{q^m}}/{\mathbb{F}_q}}(\epsilon)=a$ and $\Tr_{{\mathbb{F}_{q^m}}/{\mathbb{F}_q}}(\epsilon^{-1})=b$, where $f(x), g(x) \in \mathbb{F}_{q^m}(x)$ are rational functions with some restrictions and $a, b \in \mathbb{F}_q$. Also, we show that there exists an element $\epsilon \in \mathbb{F}_{q^m}$ satisfying our desired properties in all but finitely many fields $\mathbb{F}_{q^m}$ over $\mathbb{F}_q$. We also calculate possible exceptional pairs explicitly for $m\geq 9$, when degree sums of both the rational functions are taken to be 3.

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Works this paper leans on

14 extracted references · 14 canonical work pages

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