$r$-primitive $k$-normal polynomials over finite fields with last two coefficients prescribed

K. Chatterjee; R.K. Sharma; S.K. Tiwari

REVIEW 3 major objections 4 minor 29 references

$r$-primitive $k$-normal polynomials over finite fields with last two coefficients prescribed

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper proves a sufficient condition that forces finite fields to contain r-primitive k-normal polynomials with any prescribed last two coefficients, and it lists the few possible exceptions for the 3-primitive 1-normal case.

desk verdict Norm obstruction kills Theorem 3.7 as stated; the character-sum machinery is real but the paper overclaims existence for all b. read the letter →

arxiv 2501.04999 v3 pith:PAASPNMB submitted 2025-01-09 math.NT

K. Chatterjee , R.K. Sharma , S.K. Tiwari This is my paper · ORCID

classification math.NT MSC 12E2011T23

keywords r-primitiveelementsk-normalfinitefieldscharactersumsprescribedcoefficientstraceandnormprimesieveprimitivepolynomials

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 asks when a finite field $\mathbb{F}_{q^m}$ contains an element whose minimal polynomial has two prescribed last coefficients, where the element is simultaneously $r$-primitive (multiplicative order $(q^m-1)/r$) and $k$-normal (its conjugates span a space of codimension $k$). The main result is a sufficient condition: if $q^{m/2-k-2}$ exceeds $r$ times a product of square-free divisor counts attached to $q^m-1$ and to $x^m-1$, then such an element exists for every prescribed pair of coefficients. Specializing to $r=3$, $k=1$, and degree $m\ge 7$, the condition, together with a prime sieve, leaves only finitely many exceptional pairs $(q,m)$, all explicitly listed; for $m=1,2,3$ the paper shows no such polynomial exists, and for $m=4$ it gives a necessary condition. The interest is that prescribing the last two coefficients is equivalent to prescribing the norm and the trace of $\xi^{-1}$, so the theorem produces irreducible polynomials with controlled coefficients, a tool relevant to coding and cryptography.

What carries the argument

The engine is a character-sum count $N_{r,k,a,b,c}(Q_e,f)$ for elements $\epsilon$ that are $Q_e$-free, have $\epsilon^r = g\circ\beta$ for an $f$-free $\beta$, satisfy $\operatorname{Tr}(\epsilon^{-r})=ab^{-1}$ and $\operatorname{Norm}(\epsilon)=c$. Writing $r$-primitiveness as $\xi=\epsilon^r$ for primitive $\epsilon$ (Lemma 3.3) and $k$-normality as $\xi=g\circ\beta$ for normal $\beta$ (Lemma 2.1), the paper expresses this count as a sum of four character sums $T_1,T_2,T_3,T_4$; an orthogonality identity (Lemma 2.6) evaluates the $\beta$-sum, and Weil-type mixed exponential sum bounds (Lemma 2.4) estimate the terms, yielding Inequality (8). The prime sieve (Lemma 3.8 and Proposition 3.9) then removes some large prime divisors of $Q$ and some irreducible factors of $x^m-1$, producing the cheaper test Inequality (9) used to cover borderline pairs.

What would settle it

Compute $M$ as the product of the first 9632 primes, count its square-free divisors ($2^{9632}$), and compare with $M^{1/15}$; if $2^{9632}$ is not smaller, then Lemma 4.7 is false and the exception lists for $7\le m\le11$ cannot be considered exhaustive.

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Extended reading notes

Core claim

The central claim is Theorem 3.7: with $q$ a prime power, $r\mid q^m-1$, and $k<m/2$, choose any monic $g\in\mathbb{F}_q[x]$ of degree $k$ dividing $x^m-1$, let $Q$ be the largest divisor of $q^m-1$ coprime to $q-1$, and let $W$ count square-free divisors. If $q^{m/2-k-2} > r\,W(Q)\,W\big((x^m-1)/g\big)$, then for every $a\in\mathbb{F}_q$ and $b\in\mathbb{F}_q^*$ there is an $r$-primitive $k$-normal $\xi\in\mathbb{F}_{q^m}$ with $\operatorname{Tr}_{\mathbb{F}_{q^m}/\mathbb{F}_q}(\xi^{-1})=ab^{-1}$ and $\operatorname{Norm}(\xi)=b$; because $k<m/2$ forces the minimal polynomial to have degree $m$, its last two coefficients are exactly $a$ and $b$. The paper further uses a prime sieve to weaken the inequality, and for $r=3$, $k=1$, $m\ge 7$ it lists all pairs $(q,m)$ that its sufficient conditions do not cover: the lists contain 18 small cases with $q<8$, 20 cases with $q\ge8$ and $8\le m\le11$ (including $(11,10)$), and 40 values of $q$ for $m=7$.

Load-bearing premise

The exhaustive exception lists for degrees 7 through 11 rest on a bound quoted from [1] without proof: every integer with at least 9632 distinct prime factors has fewer than its 1/15-power many square-free divisors; if that bound is false, the lists could be incomplete.

Editorial extensions

If this is right

Whenever Inequality (8) holds for given $q,m,r,k,g$, the field $\mathbb{F}_{q^m}$ contains a degree-$m$ $r$-primitive $k$-normal polynomial realizing every prescribed pair $(a,b)$ with $b\neq0$.
For $r=3$, $k=1$, and $m\ge7$, every pair $(q,m)$ with $3\mid q^m-1$ is covered by Inequality (8) or the sieved Inequality (9), apart from the explicitly listed finite exceptions; in particular, for $m\ge12$ and $q\ge8$ there are no exceptions.
For $m=1,2,3$, no $3$-primitive $1$-normal polynomial with prescribed last two coefficients exists, and for $m=4$ the only possible fields have $q\equiv1\pmod4$ and $3\mid q^4-1$.
The sieve is a genuine improvement: pairs such as $(4,11)$, $(4,14)$, $(4,15)$, $(4,18)$, $(5,16)$, $(5,24)$, and $(7,10)$ satisfy Inequality (9) but not Inequality (8).
The result extends the known primitive-normal case with specified last two coefficients to the wider family of $r$-primitive $k$-normal polynomials.

Reading between the lines

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

The paper does not test the quoted bound numerically, but the minimal candidate to check is the product of the first 9632 primes; comparing its number of square-free divisors, $2^{9632}$, with its $1/15$ power would settle whether the pruning step in Section 4.2 is reliable.
A natural extension is to prescribe more than two coefficients: each extra coefficient should add one rational function to the character sum and weaken the exponent in Inequality (8), following the same proof pattern.
For $m=4$, the paper's necessary condition $q\equiv1\pmod4$ and $3\mid q^4-1$ may also be sufficient; a computer search over small prime powers $q$ could close the open problem.
The restriction $k<m/2$ is doing real work: if $k$ is at least $m/2$, the minimal polynomial can have degree below $m$, so the coefficient-to-trace/norm translation ceases to hold and a different method would be needed.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The main theorem is false as stated. For an r-primitive element, the norm lies in the subgroup of F_q^* of order (q-1)/gcd(r,q-1), not all of F_q^*. For r=3, q=7, that means only norms 1 and -1 are possible (the stress-test's claim that only -1 is attainable is wrong for m≥2, but the subgroup restriction stands). Theorem 3.7 asserts existence for every b ∈ F_q^*, which cannot hold. The proof itself quietly restricts to w-free b and then requires b = c^r with c primitive, which is narrower still—for q=7, b=1 never gets counted. That is an internal contradiction with the authors' own Lemma 3.4, and it breaks the Section 4 application: e.g., q=4, w=3, L=1, so only b=1 is ever a norm, and the paper claims (4,m) ∈ Γ_{a,b}(3,1) for all m except a short list.

What survives? The exponential sum estimates (Lemmas 3.5 and 3.6) and the character-sum decomposition are genuinely new, and the constant accounting in Inequality (8) checks out. The idea to extend Fan-Wang to r-primitive k-normal with last two coefficients is a fair problem. But the main quantitative claim and all the exception lists are overclaims; they need to be re-stated with the norm restricted to the correct subgroup, and the enumeration would then look different.

The unproved Lemma 4.7 (ω(M) ≥ 9632 implies W(M) < M^{1/15}) is a separate concern—it is imported without proof from [1] and it props up the Section 4 reduction. The finite scans are also not reproducible: no code or data, just 'we verify Inequality (10)'. That would be a minor issue in a healthy paper; here it compounds the broken main theorem.

Who is this for? Finite-field people working on primitive normal bases. They would get value from a corrected version, not this one. I would send it to a referee rather than desk reject, because the character-sum core is nontrivial and the mistake is instructive, but I would expect the referee to require a major rewrite: fix the norm subgroup quantifier, prove or properly import Lemma 4.7, and provide reproducible computation.

Referee Report

3 major / 4 minor

Summary. The paper studies r-primitive k-normal elements over finite fields and seeks a sufficient condition for the existence of an r-primitive k-normal polynomial of degree m whose last two coefficients are prescribed. The main tool is a character-sum count, Theorem 3.7, which yields Inequality (8), and a sieve, Proposition 3.9, used to reduce the range in which direct verification is needed. The paper then specializes to r=3, k=1 and claims to determine all possible exceptional pairs (q,m) for m>=7, with separate statements for q<8 and q>=8, plus some results for m<=4. The character-sum derivation is internally detailed and contains no fitted parameters, but the paper's central quantification over all prescribed norms is algebraically incorrect, and the computational classification is not reproducible from the text.

Significance. If corrected, the method would be a useful extension of prior work on primitive normal polynomials with prescribed last two coefficients to r-primitive k-normal polynomials. The derivation of the main character-sum inequality is a genuine strength, and the paper explicitly separates a sufficient condition from the sieve-based reduction. However, the advertised conclusion, namely existence for every prescribed b in F_q^*, is false: the norm of an r-primitive element is constrained to the subgroup of F_q^* of order (q-1)/gcd(r,q-1). This error propagates through Section 4, where several pairs are asserted to lie in Gamma_{a,b}(3,1) for all b. The exception lists also rely on an imported numerical bound and on undocumented finite scans. The contribution as submitted is therefore not acceptable, although a reformulation restricted to attainable norms may be salvageable.

major comments (3)

[Sections 3.1 and 4] The quantification 'for any prescribed a in F_q and b in F_q^*' in Theorem 3.7 is false. Write n=q^m-1, h=n/(q-1), w=gcd(r,q-1), r=wR, q-1=wL with gcd(R,L)=1. For an r-primitive element xi of order n/r, the condition r|n forces R|h; writing h=Rh' gives ord(xi)=Lh' and N(xi)=xi^h. Since gcd(Lh', h)=h', the norm has exact order L=(q-1)/w, not arbitrary order dividing q-1. Thus, for q=7 and r=3, every 3-primitive element has norm -1 in every extension, so b=1 is never attainable; consequently Lemma 4.3's assertion that (7,10) lies in Gamma_{a,b}(3,1) for every b is impossible. The proof counts epsilon with N(epsilon)=c for a primitive c, but the required link c^r=b is never imposed in the statement of Theorem 3.7; this is an algebraic obstruction rather than a missing estimate. The theorem and all Section 4 applications must be reformulated for b of exact order (q-1)/w, i.e. for b in the image of the r-th power map on primitive elements. Relatedly, under the Section 2 definition of 'l-free', Lemma 3.4's 'w-free' assertion is inconsistent: for q=31, m=1, r=3, a 3-primitive element has norm of order 10, yet gcd(3,30/10)=3.
[Section 4.2] The bound 'if omega(M) >= 9632 then W(M) < M^{1/15}' is imported without proof from [1] and is then used as a threshold to eliminate in one stroke every case with omega(q^m-1) >= 9632 and to justify the later reductions for 7 <= m <= 11. The paper does not state the exact lemma from [1] nor show the deduction of the constant 9632. Since a wrong threshold would change the exception lists in Lemmas 4.8 and 4.9, this point is load-bearing for the claimed exhaustive classification and needs either a self-contained proof or a precise statement of the cited result together with the derivation.
[Sections 4.1-4.2] The repeated phrases 'after verifying Inequality (10) and (11)' conceal finite computations that are essential to the exception lists. For example, Lemma 4.9 requires checking all prime powers q up to 780097 for m=7, and Lemmas 4.3 and 4.5 require similar checks for small q and m. The manuscript gives no code, no output tables, and no description of the specific choices of e and f in Proposition 3.9 for each pair. These scans should be made reproducible, for instance by including a Sage or PARI script and its output in an appendix or supplementary file; otherwise the classification cannot be audited.

minor comments (4)

[Section 4.2] The sentence 'This gives us (q,9) in Gamma_{a,b}(3,1)' should read '(q,7) in Gamma_{a,b}(3,1)', since the lemma is about m=7.
[Section 2] The term 'l-free' is defined for elements of F_{q^m}, but the same terminology is later applied to elements of F_q^* in Lemma 3.2 and Lemma 3.4; the analogous definition for F_q^* should be stated explicitly, since the two possible conventions (with respect to order or with respect to index) give different results.
[Throughout] There are several typos and OCR artifacts, including 'abelain' in Section 2, 'integerdivide' in the display of Lemma 2.3, 'POL YNOMIALS' in the title, 'Fore these values' in Lemma 4.5, and 'if and only of it' in Section 2.
[Section 4.2] The symbol r is used both for the order of primitivity and as a real parameter in Lemma 4.1; this overloading is confusing in the rows of Table 1, especially since the table uses real values such as r=8.5. A different symbol for the real exponent would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the main sufficient condition is derived in-paper from character-sum estimates, and the self-cited auxiliary lemmas are parameter-free tools rather than restatements of the conclusion.

full rationale

I found no circular step. The main sufficient condition (Theorem 3.7 and Inequality (8)) is obtained by expanding N_{r,k,a,b,c}(Q_e,f) with standard multiplicative/additive character sums and bounding them via Weil/Castro-Moreno estimates; the inequality is an output of the count, not an input. The enumeration in Section 4 applies the sieve Proposition 3.9 and divisor-function bounds; Lemma 4.7, although quoted from [1] with overlapping authorship, is a parameter-free estimate W(M)<M^{1/15} for omega(M)>=9632 and is not equivalent to the target existence statement, so it is independent support rather than circular. Omitted proofs (Lemma 3.6, Lemma 3.8/Prop 3.9, Lemma 4.7) are tools borrowed from prior work, not assumptions containing the desired conclusion. The only issue I see is a correctness/quantification mismatch, not circularity: Theorem 3.7 states 'for any a in F_q and b in F_q^*', while its proof via Lemma 3.4 can only handle w-free b with w=gcd(r,q-1); hence the theorem is overbroad when w>1. That is a truth defect, not an equivalence of the derivation to its own inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard character sum theory, the domain assumption k<m/2, and two imported results: the sieve from [14,10] and the numerical bound of Lemma 4.7. No free parameters are fitted, and no new entities are introduced.

assumptions (4)

standard math Character sum bounds of Weil and Castro-Moreno (Lemmas 2.3, 2.4) hold for the rational functions theta x^r + t x^{-r} and t x^{-r}.
Lemmas 3.5 and 3.6 prove the non-degeneracy conditions needed for these bounds; this is a standard application.
domain assumption k < m/2 ensures the minimal polynomial of the k-normal element has degree exactly m.
Stated in Section 3.1; if k >= m/2, the element may have smaller degree and the coefficient-prescription equivalence via Lemma 3.1 fails.
domain assumption Lemma 4.7: if omega(M) >= 9632 then W(M) < M^{1/15}.
Quoted as deduced from [1] Lemma 5.1, no proof shown; used to reduce the m=7..11 cases.
domain assumption Sieve Proposition 3.9 and Lemma 3.8, stated without proof and credited to [14,10].
These bounds on the sieved count are external results from the same group; they are used to handle pairs where Inequality (8) fails.

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Pith. "Pith review of $r$-primitive $k$-normal polynomials over finite fields with last two coefficients prescribed." pith.science (2026). https://pith.science/paper/PAASPNMB

@misc{pith2026250104999,
  author       = {Pith},
  title        = {Pith review of: $r$-primitive $k$-normal polynomials over finite fields with last two coefficients prescribed},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PAASPNMB}},
  note         = {Machine review of arXiv:2501.04999}
}

abstract

Let $\xi\in\mathbb{F}_{q^m}$ be an $r$-primitive $k$-normal element over $\mathbb{F}_q$, where $q$ is a prime power and $m$ is a positive integer. The minimal polynomial of $\xi$ is referred to be the $r$-primitive $k$-normal polynomial of $\xi$ over $\mathbb{F}_q$. In this article, we study the existence of an $r$-primitive $k$-normal polynomial over $\mathbb{F}_q$ such that the last two coefficients are prescribed. In this context, first, we prove a sufficient condition which guarantees the existence of such a polynomial. Further, we compute all possible exceptional pairs $(q,m)$ in case of $3$-primitive $1$-normal polynomials for $m\geq 7$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Gupta, R

A. Gupta, R. K. Sharma, and S. D. Cohen, Primitive element pairs with one prescribed trace over a ﬁnite ﬁeld. Finite Fields Appl. 54 (2018) 1–14

work page 2018

[2] [2]

A. K. Sharma, M. Rani, and S. K. Tiwari, Primitive normal p airs with prescribed norm and trace. Finite Fields Appl. 78 101976 (2022)

work page 2022

[3] [3]

A. W eil. On some exponential sums. Proc. Nat. Acad. Sci. U.S.A. 34(5) 204-207, (1948)

work page 1948

[4] [4]

F. N. Castro and C. J. Moreno. Mixed exponential sums over ﬁnite ﬁelds. Proc. Amer. Math. Soc. 128(9) 2529-2537 (2000)

work page 2000

[5] [5]

Shuqin and H

F. Shuqin and H. W enbao, Primitive polynomials over ﬁnit e ﬁelds of characteristic two. Appl. Algebra Engrg. Comm. Comput. 14 (2004) 381–395

work page 2004

[6] [6]

Shuqin and H

F. Shuqin and H. W enbao, p-adic formal series and cohen’s problem. Glasg. Math. J. 46(1) (2004) 47–61

work page 2004

[7] [7]

Kapetanakis and L

G. Kapetanakis and L. Reis, Variations of the primitive n ormal basis theorem. Des. Codes Cryptogr. 87(7) (2019) 1459–1480

work page 2019

[8] [8]

H. W. Lenstra Jr. and R. J. Schoof, Primitive normal bases for ﬁnite ﬁelds. Math. Comp. 48 (1987) 217-231

work page 1987