arXiv subjects

Hongfeng Wu

Publications and source records attributed to Hongfeng Wu.

At least 19 recordsLinked to original sources

arXiv2026

Cyclotomic polynomials with trinomial irreducible factors over finite fields

We classify the pairs $(q,n)$, where $q$ is a prime power, $n>1$, and $\gcd(n,q)=1$, for which every irreducible factor of $Φ_n(x)$ over $\F_q$ has exactly three nonzero terms. The positive support exponents of these factors have a common greatest common divisor, which gives a unique reduction by a power substitution. The reduced degree is two in odd characteristic and belongs to $\{2,3,4\}$ in characteristic two; the cubic and quartic cases arise by scalar substitution from the layers of orders $7$ and $15$. This yields necessary and sufficient arithmetic conditions on $(q,n)$ and explicit factorizations. Among the classified families, precisely those arising from quartic bases have a divisor layer containing an irreducible factor with more than three terms.

math.NT↗

arXiv2026

Equal Hamming Weights at Fixed Root Order and Applications to Constacyclic MDS Codes

For a fixed positive integer $n>1$, we study the prime powers $q$ coprime to $n$ for which all monic irreducible factors of $Φ_n$ over $\F_q$ have the same coefficient Hamming weight. Cyclotomic orbit coefficients give a finite, effective classification of these fields. Outside finitely many computable characteristics, every factor has the support of its characteristic-zero orbit polynomial; the uniform condition $p>2^{φ(n)}$ suffices. In each remaining characteristic, extension exponents reduce to divisors of $\ord_n(p)$, and an integer variance identity expressed by polynomial gcd degrees decides equal weight. The same coefficient data determine the full weight distribution. As an application, we prove that an MDS constacyclic code with an irreducible generator forces every irreducible polynomial of the same root order to have full support, including factors associated with different constacyclic constants. This yields a uniform gcd and norm obstruction. We develop its short-multiplier extension, give a prime-order cubic family, and establish an explicit quadratic-residue exclusion without a prime-square alphabet or a near-square length hypothesis.

cs.IT↗

arXiv2026

Finite Fields: Root Sets, Frobenius, and Algebraic Structure

This monograph develops finite field theory through root sets, Frobenius orbits, trace duality, and cyclic modules, with applications to polynomial constructions and algebraic coding theory. Root-set descent and base extension are expressed through finite reduced algebras, while linearized polynomials connect normal bases, invariant subspaces, and linear recurrences. Coefficient pairings and derivative formulas recover information lost by trace pairings in quotient algebras with repeated factors. Cyclotomic cosets are studied through additive stabilizers, power substitution, and compatible factor lifting, including extensions to Galois rings. A central theme connects low-degree representations of subfields with conjugate root blocks, polynomial composition, and recovery from relative trace data. Return maps on root fibers determine factor degrees and organize constructions of irreducible polynomials, affine symmetries, and permutation polynomials. Applications include cyclic and constacyclic codes, BCH and Goppa distance bounds, rank-metric codes, trace codes, and Fourier support. Character orthogonality and Gauss and Jacobi sums yield counting formulas, a proof of the primitive normal basis theorem, and recurrences for extension-field character sums. Gaussian periods connect these methods with short linear relations in multiplicative subgroups. Proofs of Weil bounds for polynomial character sums combine auxiliary polynomials with generating polynomials and estimates for their reciprocal roots. The emphasis throughout is on explicit arguments connecting field structure, orbit structure, and linear dependence.

math.NT↗

arXiv2026

Support periods of polynomials and support-primitive decomposition of constacyclic codes over Galois rings

Let $A=\GR(p^e,r)$ have residue field $\F_q$, and let $λ\in A^\times$. We relate coefficient compression of monic divisors of $X^N-λ$ to support-primitive decomposition of constacyclic codes. At a simple-root length, the support period of every nonconstant monic divisor is preserved at each coefficient precision and equals the order of the multiplicative stabilizer of its root set. For a cyclic Frobenius-orbit polynomial, the multiple equal-difference structure determines its support period, core degree, and least relative degree of an unramified extension over which all its basic-irreducible factors are binomials. Coordinate projections give every constacyclic code over a finite commutative ring a unique maximal interleaving into copies of one support-primitive code. At simple-root lengths over $A$, the intrinsic period of a code is the greatest common divisor of the periods of its torsion codes. The decomposition yields distance and duality formulas and the arithmetic Singleton bound for free codes. At every length, a nonzero code over $A$ has the same minimum Hamming distance as its highest torsion code, which gives a refined bound. At repeated-root lengths, identical torsion filtrations can occur with different intrinsic periods, while a class of nonzero codes obtained by substitution from simple-root generators retains the same core and minimum distance. Trace matrices recover the coefficients of basic-irreducible cores using unit pivots, and local Hensel corrections lift compressed factors directly. For $X^n-1$ with $p\nmid n$, lifting and output have quasi-linear bit cost in the compressed representation when the arithmetic data and residue factorizations are given.

cs.IT↗

arXiv2026

Covering finite fields by translates of multiplicative subgroups

Let $G_d$ be the subgroup of index $d>1$ in $\mathbb{F}_q^\times$, and put $G_d^0=G_d\cup\{0\}$. For a covering $A+G_d^0=\mathbb{F}_q$ with $\#A=s$, we prove $q\le d^{8s/d}$ by shifting the roots of a Bonferroni polynomial. Although a uniform bound in $s$ alone must allow exponential growth, our estimate confines such growth to bounded indices and gives polynomial bounds when $s/d$ is bounded. A cubic specialization gives bounds of degree at most eight for $s\le2d$. For shift sets of logarithmic size, higher moment estimates give a uniform covering-probability formula, including the critical window. Together with the index restriction, this determines the covering indices of almost every shift set of size $\lceil k\log q\rceil$ for each fixed integer $k\ge2$. A subfield construction attains the order of the uniform restriction on the index. We determine its covering number exactly and prove minimality under subgroup inclusion. We also characterize equality in the elementary counting bound by subfield cosets. Fixing a subfield as the shift set in a quadratic extension leads to a sharper projection bound, attained by an infinite family with unique representations outside the shift subfield.

math.NT↗

arXiv2026

Affine Symmetries of Polynomial Roots with Multiplicities and Applications to Constacyclic Codes

Let $f(X)$ be a polynomial over a finite field, and let its defining function record the multiplicities of its roots. In this paper, we introduce the affine stabilizer group and the affine stabilizer group scheme of the defining function of $f(X)$. These objects characterize the affine symmetries of the roots of $f(X)$ together with their multiplicities. We establish structural descriptions of these affine stabilizers and relate them to the coefficient structure of $f(X)$. In particular, the affine stabilizer group determines quotient structures of the defining function, which yield canonical descriptions of the corresponding semi-invariant polynomial spaces. The resulting coefficient constraints provide explicit restrictions on the possible supports of polynomials with prescribed affine symmetries. The affine stabilizer group scheme further captures infinitesimal symmetries arising from multiple roots and gives refined coefficient constraints beyond those obtained from ordinary affine transformations. We apply this framework to generator polynomials of constacyclic codes over finite fields. The affine stabilizer structure of a generator polynomial induces subcodes defined by semi-invariant conditions and provides constraints on their dimensions and Hamming distances. In the special case of multiplicative symmetries, the construction specializes to the support-primitive decomposition of constacyclic codes.

cs.IT↗

arXiv2026

Finite fields whose members are the sum of a potent and a 5-potent

We show that there are only finitely many finite fields whose members are the sum of an $n$-potent element and a $5$-potent element. Combining this with the algorithmic results provided by S.D. Cohen {\it et al.}, we confirm in the affirmative the conjecture posed in \cite{Cohen} concerning all finite fields satisfying this condition. Furthermore, we obtain several elementary results related to the more general problem of decomposing every element of a finite field into the sum of an $m$-potent and an $n$-potent (as formulated in \cite{Cohen}), proving that the number of finite fields satisfying this condition is finite.

math.NT↗

arXiv2026

Two-Line Elimination and the Weierstrass Group Law in Arbitrary Characteristic

Let $K$ be a field and let $E/K$ be a nonsingular projective curve in general Weierstrass form. We prove associativity of its chord--tangent operation by combining polynomial factorization, formal differentiation, and quadratic interpolation. The main identity eliminates the two nonvertical lines $Y=\ell(X)$ and $Y=m(X)$ occurring in the successive additions $S=P\oplus Q$ and $T=S\oplus R$. There are $c\in K$ and a monic quadratic polynomial $h\in K[X]$ such that \[ \ell+m+\Apol=c(X-x_S), \qquad \Gpol+\ell m=(X-x_S)h, \] and \[ (Y-\ell)(Y-m) =(X-x_S)(h-cY) +Y^2+\Apol(X)Y-\Gpol(X). \] When $c\ne0$, the graph of $q=h/c$ contains $P,Q,R$, and $-T$. The two bracketings determine the same quadratic by ordinary or Hermite interpolation and therefore yield the same final point. The proof applies in every characteristic, since tangency is expressed through the formal partial derivatives of the Weierstrass equation. For the short Weierstrass form, a separate concise proof identifies $q$ with the interpolating parabola in Zwegers's construction.

math.NT↗

arXiv2026

Direct Group Laws on Classical Genus-One Models

This exposition constructs group laws by residual intersection on short Weierstrass cubics, the curves $(u^2+u)(v^2+v)=d$, Edwards curves, Hessian curves, Jacobi quartics curves, Jacobi intersections curves, Huff curves, and a symmetric QRT biquadratic. Lines give the operation on plane cubics, while planes give the operation on smooth intersections of two quadrics in $\PP^3$. Associativity is proved by comparing two products of three line or plane equations. After the common zeros are cancelled, equality of the remaining zeros follows from the elementary fact that a smooth projective curve of genus one admits no nonconstant rational function with one simple pole. The formulas for $(u^2+u)(v^2+v)=d$ are valid in every characteristic and are written explicitly in characteristics $2$ and $3$; the other models are treated in characteristic different from $2$ and $3$.

math.AG↗

arXiv2026

Singular Cholesky Fibers over Finite Fields

For a finite field $\F_q$, consider the triangular Cholesky map $Γ_{n,q}(U)=U^TU$ from upper triangular matrices to symmetric matrices. Generalized Cholesky theory describes the regular locus on which all leading principal minors are nonzero, but it does not determine the multiplicities or root ranks in a fiber over a singular target. We develop a fixed-target fiber theory for this singular boundary and answer three questions posed by Cooper and Whitlatch. Our principal results concern the zero fiber. Over $\F_2$ we construct an explicit, invertible, rank-preserving recursive bijection between square-zero upper triangular matrices and upper triangular matrices satisfying $U^TU=0$. Over every finite field we determine the entire rank distribution of this zero fiber: in even characteristic the square-zero recurrence, and hence the rank-refined equinumerosity, persists over every $\F_{2^e}$, whereas in odd characteristic its failure is measured by an explicit quadratic-character correction. We place these results in a uniform framework by proving an exact first-pivot recursion for the fiber cardinality over an arbitrary symmetric target. Its regular specialization gives the constant fiber sizes on leading-principal-minor cones, while its zero-pivot branch explains the new singular behavior. For binary diagonal targets we further compress the rank-refined count to $O(n^2)$ integer-arithmetic transitions and show that it depends on the order, not merely the number, of the diagonal entries. Executable implementations and exhaustive low-order checks are collected in an appendix.

math.RA↗

arXiv2026

Peirce Stability of Null Polynomials and a Sharp Radical Bound for Finite Rings

Let $R$ be a finite associative ring with identity, and let $\Nul(R)$ denote the set of polynomials in a central indeterminate that vanish identically on $R$ under right evaluation. We establish two results concerning the failure of $\Nul(R)$ to be a right ideal. First, if $e$ is an idempotent, $f=1-e$, and either $eRf=0$ or $fRe=0$, then right multiplication by $e$ preserves $\Nul(R)$. Thus both off-diagonal Peirce components must be nonzero whenever $e$ witnesses a failure of right stability. Second, if $J$ is a finite nilpotent ideal whose additive group is a $2$-group and $e,f$ are complementary idempotents such that \[ eJf\neq0,\qquad fJe\neq0,\qquad J^3\neq0, \] then $|J|\geq32$. Werner's theorem that $\Jac(R)^3=0$ implies the two-sidedness of $\Nul(R)$ therefore yields $|R|\geq128$ whenever $\Nul(R)$ is not two-sided. We also construct a tiled matrix ring of characteristic $4$ and order $128$ whose null ideal is not two-sided.

math.RA↗

arXiv2026

Consecutive non-square non-primitive tuples in finite fields

Let $q$ be an odd prime power and put \[ θ_q=\frac{φ(q-1)}{q-1}. \] An element of $\Fq$ is called non-square non-primitive, or \emph{NSNP}, if it is both a non-square and a non-primitive element. Let $\ell$ be an odd prime divisor of $q-1$. We obtain a character-sum estimate for the number of translates of an arbitrary finite set lying in the set of non-square $\ell$th powers. Combining this estimate with a finite computation, we prove that $θ_q<4/15$ guarantees three consecutive NSNP elements. On the boundary $θ_q=4/15$, the only exceptions are $q\in\{31,61,121\}$. As a further application, we show that for $\operatorname{char}\Fq>3$, the inequality $θ_q<8/35$ guarantees four consecutive NSNP elements, with $q=211$ as the unique exception on the boundary $θ_q=8/35$. Thus both constants are best possible.

math.NT↗

arXiv2026

The Optimal Coefficients-Based Criterion for Primitive Quadratic Polynomials over Finite Fields

Let $\mathbb F_q$ be a finite field and consider quadratic polynomials $f(X)=X^2+bX+c\in\mathbb F_q[X]$ with primitive constant term $c$. We construct an optimal coefficients-based determining polynomial for primitive quadratic polynomials over every finite field. More precisely, for every primitive $c\in\mathbb F_q$, its specialization is the unique monic square-free polynomial whose roots are exactly the coefficients $b$ for which $X^2+bX+c$ is primitive. The construction is based on Lucas polynomials and Lucas atoms over finite fields. In odd characteristic, the optimal determining polynomial is the $(q+1)$-st Lucas atom. In characteristic $2$, this Lucas atom has multiplicity $2$ in the variable $B$, and the optimal polynomial is obtained by removing this multiplicity through the inverse Frobenius over $A=\mathbb F_q[C]/(Φ_{q-1}(C))$. We prove that the determining polynomial is unique for each fixed primitive constant term and, globally, unique as an element of $A[B]$. We also give equivalent criteria involving only recursively computable Lucas polynomials and, as an application, a first-zero coefficient description of the binomial order and order of an irreducible quadratic polynomial.

math.NT↗

arXiv2026

Explicit Representatives and Sizes of Cyclotomic Cosets II: Cyclotomic Systems and Applications to Constacyclic Codes

Let $q=p^e$ be a prime power, let $\ell$ be a prime with $\gcd(\ell,qn)=1$, and let $n$ be a positive integer. While $q$-cyclotomic cosets are usually considered modulo a fixed integer, we consider their behavior along the sequence $n,\ell n,\ell^2n,\ldots$ of moduli. To describe the resulting compatibility among cyclotomic cosets, we introduce the $\ell$-adic $q$-cyclotomic system with base module $n$, defined as the projective limit of the spaces of $q$-cyclotomic cosets modulo $\ell^i n$. We associate to each $q$-cyclotomic coset modulo $n$ a cyclotomic $\ell$-adic integer and use its $\ell$-adic expansion to describe all compatible liftings through the successive levels. This leads to a classification of the corresponding sequences according to their splitting and stable behaviors, together with explicit formulas for the representatives and sizes of their components. We treat separately the cases of odd $\ell$ and $\ell=2$. These results are then used to determine representatives and sizes of $q$-cyclotomic cosets for arbitrary admissible parameters. As an application, we obtain explicit irreducible factorizations of binomials over finite fields and use them to describe the corresponding constacyclic codes.

math.NT↗

arXiv2026

A New Characteristic-Uniform Model for Elliptic Curves -- Theory, Arithmetic, and Applications

We develop a characteristic-uniform arithmetic theory for \[ \mathcal C_d:\quad (u^2+u)(v^2+v)=d. \] For $d(1-16d)\ne0$, its smooth $(2,2)$-completion has four rational boundary points forming $\mathbb Z/4\mathbb Z$, an intrinsic $D_8$-action, the inverse $-(u,v)=(u,-v-1)$, and the native Kummer map $κ_d(u,v)=(u+1)$. Working natively, we derive complete full-point and differential laws, Kummer ladders and recovery, halving, tripling, $2P+Q$, division polynomials, isogenies, CM endomorphisms, and pairings, with dedicated formulas in characteristics two and three. Classical models provide proof and optimization dictionaries while all endpoints remain native. For Cd25519, the Kummer line is exactly the X25519 line, and a native Segre recoding realizes the optimized complete $a=-1$ Edwards full-point dependency graph. We further study \[ \begin{gathered} \mathcal C_{a,b,d}:(u^2+u+a)(v^2+v+b)=d,\quad \mathcal T_{a,d}:(u^2+u+a)(v^2+v)=d,\\ \mathcal R_{τ,σ,κ}:(x^2-τ)(y^2-σ)=κxy,\quad \mathcal Q_{α,β,γ}:x^2y^2+α(x^2+y^2)+βxy+γ=0. \end{gathered} \] For these product, one-sided twisted, reciprocal, and QRT families, we determine their genus-one geometry, finite-field forms, arithmetic, and isogenies. On each smooth QRT fibre, the Vieta--McMillan map is a fixed elliptic translation. Marking $D$ gives the state $P\mapsto(κ(P),κ(P+D))$, with maps realizing $n\mapsto mn+r$. This yields logarithmic ladders and an elliptic Lucas calculus with nonlinear addition, fast-index doubling, state-division polynomials, and bridges to elliptic divisibility sequences, sigma functions, and elliptic nets. In characteristic two, every ordinary pointed elliptic curve over a perfect field admits the binary state model. We further develop the $\mathcal C_d$ platform for isogeny-based cryptography.

math.NT↗

arXiv2026

First Subconstituents of Orthogonal Graphs in Even Characteristic:Automorphisms, Cliques, Cores, and Binary Triple Transitivity

In even characteristic, the first subconstituent of an orthogonal graph at a singular point is shown to be the complement of an affine polar graph. This identification yields the full automorphism group as an affine semisimilarity group and provides explicit extensions to the ambient graph. Maximum independent sets are affine cosets of maximal totally singular subspaces, giving the independence and chromatic numbers. In plus type, Delsarte cliques correspond bijectively to ovoids through the base point, leading to a sharp rank threshold. Over $\mathbb{F}_2$, we determine the exact maximum clique number in every dimension, prove that all maximum cliques form a single orbit, and obtain the core of every first subconstituent. A uniform Witt-extension argument gives a short proof of triple transitivity for the binary local graphs.

math.CO↗

arXiv2026

Primitive Quadratic Polynomials in Additive Coset Families

Let $q$ be an odd prime power. For $μ\in\mathbb F_q^\times$ and an additive coset $\barα\in\mathbb F_{q^2}/\mathbb F_q$, consider the family \[ \{x^2+μx-α:α\in\barα\}, \] where each polynomial is regarded over its coefficient field $\mathbb F_q(α)$. We prove that, for \[ q\notin{7,11,13,19,29,31,41,43}, \] every such family contains a primitive polynomial. For the zero coset, the result follows from Cohen's prescribed-trace theorem. For a nonzero coset, after a natural normalization the roots are parameterized by two smooth projective conics arising from the two $q^2$-Frobenius eigenspaces in $\mathbb F_{q^4}$. Their affine $\mathbb F_q$-point counts, $q-1$ and $q+1$, correspond respectively to the reducible and irreducible members of the family. On the irreducible root conic, $q^2$-Frobenius induces a fixed-point-free involution on rational points. Passing to the quotient conic allows the relative norm of the root function to descend to a rational function. Tensor induction then yields order-sensitive character-sum bounds with constants $6,8$ on the root conic and the sharper constants $2,4$ after norm descent. Combining these estimates with a double-core prime sieve and an exact residue-cover verification completes the finite range. As consequences, Gow and McGuire's Conjecture~1 holds for every odd prime power $q\ne13$, while their Conjectures~2 and~3 hold for every odd prime power $q>43$; the threshold $43$ is sharp.

math.NT↗

arXiv2026

The Arithmetic Singleton Bound on the Hamming Distances of Simple-rooted Constacyclic Codes over Finite Fields

In this work, We introduce a new upper bound on the Hamming distance of simple-root constacyclic codes over finite fields, which we call the arithmetic Singleton bound. The main technical tool is the notion of a multiple equal-difference (MED) representation. Via the MED representations of the defining set of the generator polynomial of a simple-root constacyclic code, we obtain a family of upper bounds on its Hamming distance, among which the weakest one coincides with the Singleton bound, while the strongest one is defined to be the arithmetic Singleton bound for this code. Consequently, the arithmetic Singleton bound is always at least as strong as the classical Singleton bound, and is in fact strictly stronger in numerous nontrivial cases. The arithmetic Singleton bound partially measures the restriction on the Hamming distance of a simple-root constacyclic code imposed by its arithmetic structure. In particular, for an irreducible constacyclic code, the MED representations of the defining set of its generator polynomial are completely determined, via which the arithmetic Singleton bound is computed concretely. Finally for any simple-root cyclic code the arithmetic Singleton bound and the BCH bound are compared.

cs.IT↗