Polynomials In Fq Research Articles

Irreducible polynomials from a cubic transformation

Let R(x)=g(x)/h(x) be a rational expression of degree three over the finite field Fq. We count the irreducible polynomials in Fq[x], of a given degree, that have the form h(x)deg⁡f⋅f(R(x)) for some f(x)∈Fq[x]. As an example of application of our results, we recover the number of irreducible transformation shift registers of order three, which were computed by Jiang and Yang in 2017.

Jordan–Landau theorem for matrices over finite fields

Given a positive integer r and a prime power q, we estimate the probability that the characteristic polynomial fA(t) of a random matrix A in GLn(Fq) is square-free with r (monic) irreducible factors when n is large. We also estimate the analogous probability that fA(t) has r irreducible factors counting with multiplicity. In either case, the main term (log⁡n)r−1((r−1)!n)−1 and the error term O((log⁡n)r−2n−1), whose implied constant only depends on r but not on q nor n, coincide with the probability that a random permutation on n letters is a product of r disjoint cycles. The main ingredient of our proof is a recursion argument due to S. D. Cohen, which was previously used to estimate the probability that a random degree n monic polynomial in Fq[t] is square-free with r irreducible factors and the analogous probability that the polynomial has r irreducible factors counting with multiplicity. We obtain our result by carefully modifying Cohen's recursion argument in the matrix setting, using Reiner's theorem that counts the number of n×n matrices with a fixed characteristic polynomial over Fq.

Iterated constructions of completely normal polynomials

The Rσ,t-transform introduced by Bassa and Menares can be used to construct families of irreducible polynomials in Fq[x]. This iterative construction is a generalization of Cohen's R-transform. For this transform, Chapman proved that under some conditions, the polynomials in the resulting family are completely normal. In this paper we establish conditions ensuring that the polynomials obtained by using the Rσ,t-transform are completely normal polynomials and we give a simple proof of Chapman's result.

On the compactification of the Drinfeld modular curve of level [formula omitted

Let p be a rational prime and q a power of p. Let n be a non-constant monic polynomial in Fq[t] which has a prime factor of degree prime to q−1. In this paper, we define a Drinfeld modular curve Y1Δ(n) over A[1/n] and study the structure around cusps of its compactification X1Δ(n), in a parallel way to Katz-Mazur's work on classical modular curves. Using them, we also define a Hodge bundle over X1Δ(n) such that Drinfeld modular forms of level Γ1(n), weight k and some type are identified with global sections of its k-th tensor power.

Polynomial analogue of the Smarandache function

In the integer case, the Smarandache function of a positive integer n is defined to be the smallest positive integer k such that n divides the factorial k!. In this paper, we first define a natural order for polynomials in Fq[t] over a finite field Fq and then define the Smarandache function of a non-zero polynomial f∈Fq[t], denoted by S(f), to be the smallest polynomial g such that f divides the Carlitz factorial of g. In particular, we establish an analogue of a problem of Erdős, which implies that for almost all polynomials f, S(f)=td, where d is the maximal degree of the irreducible factors of f.

Computing isomorphisms and embeddings of finite fields

Let q be a prime power and let F q be a field with q elements. Let f and g be irreducible polynomials in F q [ X ], with deg f dividing deg g. Define k = F q [ X ]/ f and K = F q [ X ]/ g , then there is an embedding ϕ : k [EQUATION] K , unique up to F q -automorphisms of k. Our goal is to describe algorithms to efficiently represent and evaluate one such embedding.

The Furstenberg–Sárközy theorem for intersective polynomials in function fields

Let Fq[t] be the polynomial ring over the finite field Fq of q elements. For a natural number N≥2, let GN be the set of all polynomials in Fq[t] of degree less than N. Let h(x)∈Fq[t][x] be an intersective polynomial, that is, which satisfies (A−A)∩(h(Fq[t])∖{0})≠∅ for any A⊆Fq[t] with limsupN→∞|A∩GN|qN>0. Let B⊆GN. Suppose that (B−B)∩(h(Fq[t])∖{0})=∅ and 2≤degh<p, the characteristic of Fq. It is proved that |B|≤CqN(log⁡NN)1k−1, where C≥1 is a constant depending only on h(x) and q.

Weighted distribution of points on cyclic covers of the projective line over finite fields

For any prescribed finite field Fq and integers m≥2, we study the distribution of the number of Fq-points on (possibly singular) affine curves Cf:ym=f(x), where f is randomly chosen from a fixed collection F(Fq) of polynomials in Fq[x]. These equations are affine models of cyclic m-covers of the projective line. Previously, different authors obtained asymptotic results about distributions of Fq-points on curves associated to certain collections of polynomials f. We obtain analogous results for infinitely many new examples of collections F(Fq) and study how the asymptotic distribution depends on the choice of the collection of polynomials F(Fq).

Applications of the Hasse–Weil bound to permutation polynomials

Riemann's hypothesis on function fields over a finite field implies the Hasse–Weil bound for the number of zeros of an absolutely irreducible bi-variate polynomial over a finite field. The Hasse–Weil bound has extensive applications in the arithmetic of finite fields. In this paper, we use the Hasse–Weil bound to prove two results on permutation polynomials over Fq where q is sufficiently large. To facilitate these applications, the absolute irreducibility of certain polynomials in Fq[X,Y] is established.

A group action on multivariate polynomials over finite fields

Let Fq be the finite field with q elements, where q is a power of a prime p. Recently, a particular action of the group GL2(Fq) on irreducible polynomials in Fq[x] has been introduced and many questions concerning the invariant polynomials have been discussed. In this paper, we give a natural extension of this action on the polynomial ring Fq[x1,…,xn] and study the algebraic properties of the invariant elements.

Smooth Polynomial Solutions to a Ternary Additive Equation

AbstractLet Fq[T] be the ring of polynomials over the finite field of q elements and Y a large integer. We say a polynomial in Fq[T] is Y-smooth if all of its irreducible factors are of degree at most Y. We show that a ternary additive equation a + b = c over Y-smooth polynomials has many solutions. As an application, if S is the set of first s primes in Fq[T] and s is large, we prove that the S-unit equation u + v = 1 has at least exp solutions.

Normal bases and irreducible polynomials

A normal basis of Fqn over Fq is a basis of the form {α,αq,…,αqn−1}. An irreducible polynomial in Fq[x] is called an N-polynomial if its roots are linearly independent over Fq. Let p be the characteristic of Fq. Pelis et al. showed that every monic irreducible polynomial with degree n and nonzero trace is an N-polynomial provided that n is either a power of p or a prime different from p and q is a primitive root modulo n. Chang et al. proved that the converse is also true. By comparing the number of N-polynomials with that of irreducible polynomials with nonzero traces, we present an alternative treatment to this problem and show that all the results mentioned above can be easily deduced from our main theorem.

On the difference between permutation polynomials

The well-known Chowla and Zassenhaus conjecture, proved by Cohen in 1990, states that if p>(d2−3d+4)2, then there is no complete mapping polynomial f in Fp[x] of degree d≥2. For arbitrary finite fields Fq, a similar non-existence result was obtained recently by Işık, Topuzoğlu and Winterhof in terms of the Carlitz rank of f.Cohen, Mullen and Shiue generalized the Chowla–Zassenhaus–Cohen Theorem significantly in 1995, by considering differences of permutation polynomials. More precisely, they showed that if f and f+g are both permutation polynomials of degree d≥2 over Fp, with p>(d2−3d+4)2, then the degree k of g satisfies k≥3d/5, unless g is constant. In this article, assuming f and f+g are permutation polynomials in Fq[x], we give lower bounds for the Carlitz rank of f in terms of q and k. Our results generalize the above mentioned result of Işık et al. We also show for a special class of permutation polynomials f of Carlitz rank n≥1 that if f+xk is a permutation over Fq, with gcd⁡(k+1,q−1)=1, then k≥(q−n)/(n+3).

On the irreducibility of Fibonacci and Lucas polynomials over finite fields

In this paper, we give the necessary and sufficient condition that Fibonacci and Lucas polynomials are irreducible in Fq[T]. As applications of our results, under the GRH, we prove that there are infinitely many irreducible Fibonacci and Lucas polynomials in Fq[T] for some q.

A note on the spectrum of linearized Wenger graphs

Let Fq be a finite field of order q=pe, where p is a positive prime. For m≥1, let P and L be two copies of Fqm+1. To each m-tuple g=(g2,…,gm+1) of polynomials in Fq[x,y], we consider the bipartite graph Wq(g). The vertex set V of Wq(g) is P∪L. The edge set E of Wq(g) consists of (p,l)∈P×L satisfying p2+l2=g2(p1,l1),p3+l3=g3(p1,l1),…,pm+1+lm+1=gm+1(p1,l1),where p=(p1,p2,…,pm+1)∈P and l=(l1,l2,…,lm+1)∈L. Wq(g) is called linearized Wenger graph when g=(xy,xpy,…,xpm−1y). In this paper, we determine the eigenvalues of linearized Wenger graph and their multiplicities in the case of m<e, which is an open problem put forward by Cao et al. (2015).

ON THREE-VARIABLE EXPANDERS OVER FINITE FIELDS

Let 𝔽q be the finite field with q elements and P(x, y) be a polynomial in 𝔽q[x, y]. Using additive character sum estimates, we study expander property of the function x1 + P(x2, x3). We give an alternative proof using spectra of sum–product graphs in the case of P(x, y) = xy, and also extend the problem in the setting of finite cyclic rings.

Quasi-cyclic codes of index 2 and skew polynomial rings over finite fields

Let θ be the Frobenius automorphism of the finite field Fql over its subfield Fq, Fql[Y;θ] the skew polynomial ring and Fql[Y;θ]/〈Yl−1〉 the quotient ring of Fql[Y;θ] modulo its ideal 〈Yl−1〉. We construct a specific Fq-algebra isomorphism from Fql[Y;θ]/〈Yl−1〉 onto the matrix ring Ml(Fq), and investigate factorizations of polynomials in Fq[X] over Fq and Fql when l is a prime integer. Then we present an algorithm to calculate monic factors of Xm−1 in Fq2[Y;θ]/〈Y2−1〉[X], and construct a class of quasi-cyclic codes of length 2m and index 2 over Fq from these monic factors by use of an Fq-algebra isomorphism from Fq2[Y;θ]/〈Y2−1〉[X] onto M2(Fq)[X].

Square-free values of polynomials over the rational function field

We study representation of square-free polynomials in the polynomial ring Fq[t] over a finite field Fq by polynomials in Fq[t][x]. This is a function field version of the well-studied problem of representing square-free integers by integer polynomials, where it is conjectured that a separable polynomial f∈Z[x] takes infinitely many square-free values, barring some simple exceptional cases, in fact that the integers a for which f(a) is square-free have a positive density. We show that if f(x)∈Fq[t][x] is separable, with square-free content, of bounded degree and height, and n is fixed, then as q→∞, for almost all monic polynomials a(t) of degree n, the polynomial f(a) is square-free.

On the characterization of minimal value set polynomials

We give an explicit characterization of all minimal value set polynomials in Fq[x] whose set of values is a subfield Fq′ of Fq. We show that the set of such polynomials, together with the constants of Fq′, is an Fq′-vector space of dimension 2[Fq:Fq′]. Our approach not only provides the exact number of such polynomials, but also yields a construction of new examples of minimal value set polynomials for some other fixed value sets. In the latter case, we also derive a non-trivial lower bound for the number of such polynomials.

On an exponential predicate in polynomials over finite fields

We show that the theory of the set of polynomials in F q [ t ] \mathbb {F}_q[t] , where F q \mathbb {F}_q is a finite field, in a language including addition and a predicate for the relation “ x x is a power of t t ” is model-complete and therefore decidable.