Papers (Published/Accepted for publication)

45. Normal points on Artin-Schreier curves over finite fields (with G. Kapetanakis), Comptes Rendus Mathematique, v. 363, p. 541-554, 2025.
In Paper 32, we extended Vinogradov's concept of freeness to multiplicative cyclic groups by introducing a new parameter. This allowed us to study the existence of K-rational points on curves of the form Y^n = F(X) whose coordinates are generators of the multiplicative cyclic group of the finite field K. In this paper, we develop an additive counterpart. We observe that if K/k is a finite extension of finite fields, then K has a natural cyclic k[X]-module structure induced by the generator of the Galois group Gal(K/k), leading to a corresponding notion of additive freeness. We obtain results similar to those in Paper 32, extending them to certain curves. Specifically, under mild conditions on a polynomial F in k[X], where k is a prime field of order p, we show that the Artin-Schreier curve defined by the equation y^p - y = F(X) contains a rational point whose coordinates are normal elements of K (i.e., generators of the cyclic k[X]-module K).

44. Existence of normal elements with prescribed norms (with A. Fernandes and D. Panario), Acta Arithmetica, 2025 (to appear).

This paper is primarily motivated by Cohen's 1990 work, which completely solved the problem of the existence of primitive elements with prescribed trace over finite fields. We observe that finite fields possess two "twin" cyclic structures, and in this context, the natural counterpart to Cohen's problem is the existence of normal elements with prescribed norm. In this paper, we fully solve this problem and provide asymptotic results for the case where the norm over more than one intermediate extension is prescribed. 

43. Iterating additive polynomials over finite fields, Proceedings of the Edinburgh Mathematical Society, v. 68 (3), pp. 795 - 810, 2025.

For a finite field F and an additive polynomial L in F[x], let s(n) denote the degree of the splitting field of the n-th iterate of L. In 1999, Odoni proved that s(n) grows linearly with respect to n. In this paper, we recover Odoni's theorem in a more general form, extending it to affine polynomials. Specifically, we show that in many cases, a nice closed formula for s(n) can be derived. As applications, we provide results on the factorization of iterated additive polynomials and periodic points under additive polynomial maps over finite fields. 

42. On polynomials over finite fields that are free of binomials (with F. Brochero and S. Ribas), Designs, Codes and Cryptography, v. 93, pp. 1795–1807, 2025.

In Paper 31, we provide a new bound on multiplicative character sums over affine subspaces of finite fields. As a main application, we discuss the existence of primitive k-normal elements—generators of finite fields whose Galois conjugates generate a vector space of codimension k. Our results reveal that for sufficiently large q, the existence of such an element in the n-degree extension of Fq is equivalent to the existence of a divisor of xⁿ - 1 over Fq of degree k that does not divide any binomial xᵗ - a with t < n. These polynomials are referred to as "free of binomials." In this work, we explore the existence of such polynomials, providing general results that are significantly improved under the condition that every prime factor of n divides q - 1. Additionally, we establish a connection between polynomials free of binomials and certain classical linear codes. 

41. Stable binomials over finite fields (with A. Fernandes and D. Panario), Finite Fields and Their Applications, v. 101, 2025.

In this paper, we study the binomials xᵗ - a over finite fields whose iterations are all irreducible (such polynomials are called stable). In particular, we explore the relationship between the stability of these binomials and the values of the quadratic character evaluated at elements of the forward orbit of z = 0 under the map z ↦ zᵗ - a. This recovers a previous result for the case t = 2, and building upon the work of Shparlinski et al., we provide nontrivial bounds on the size of the orbit in the case where the binomial is stable. Our method primarily relies on Capelli's Lemma and bounds on character sums. 

40. Prescribing traces of primitive elements in finite fields,  WAIFI 2024 (International Workshop on the Arithmetic of Finite Fields), Lecture Notes in Computer Science, Springer, 2025. 

In this paper, we provide an asymptotic general result on the existence of primitive elements whose traces over any family of intermediate extensions are prescribed. In particular, we asymptotically fill the gap left in the results of Paper 30. 

39. Nilpotent linearized polynomials over finite fields, revisited (with D. Panario), Finite Fields And Their Applications, v. 97, 2024.

This paper continues the work from Paper 5, where we introduced Nilpotent Linearized Polynomials (NLPs). Here, we provide a complete characterization of NLPs that are binomials and explore constructions of permutations of finite fields using NLPs. In particular, we present several classes of additive polynomials that permute infinitely many extensions of their field of definition. Additionally, we further investigate NLPs over binary fields and introduce a new family of NLPs related to trace orthogonal bases over finite fields. 

38. Constructing permutation polynomials from permutation polynomials of subfields (with Q. Wang), Finite Fields And Their Applications, v. 96, 2024.

This paper presents a general method for producing permutations of finite fields from permutations of subfields. The method is highly effective and requires solving only very simple linear equations defined by trace functions. Moreover, a more general framework is developed, unifying many recent works in the subject. 

37. Digraphs of power maps over finite nilpotent groups (with A. Fernandes), Discrete Mathematics, v. 347(7), 2024.

This paper examines issues related to the digraphs induced by power maps g → gᵗ over finite groups. In particular, we show that finite nilpotent groups are precisely the groups for which these digraphs exhibit a special symmetry that has appeared frequently in the literature (often related to the maps discussed in Paper 12). Additionally, we introduce an interesting equivalence relation on the class of finite nilpotent groups arising from these digraphs and discuss the extent to which this relation differs from group isomorphism. 

36. The average density of K-normal elements over finite fields, Designs, Codes and Cryptography, 2023.

This paper continues the work in Paper 24, where we discuss a mean value theorem for the density of special elements in finite fields (viewed as an arithmetic function). Using basic tools from the theory of arithmetic functions, we prove that, for a fixed prime power q, the density fₖ(n) of k-normal elements in the finite field extension GF(qⁿ) / GF(q) is an arithmetic function with a positive mean value. 

35. On the functional graph of the power map over finite groups (with C. Qureshi), Discrete Mathematics v. 346(7), 2023.

We provide an explicit description of the functional graph of power maps g → gᵗ over certain classes of finite groups, including abelian groups and flower groups, which are introduced in this work. This study is primarily motivated by the case where the underlying group is the projective general linear group of order 2 over a finite field (which we prove to be a flower group).

34. Paley-like graphs over finite fields from vector spaces, Finite Fields And Their Applications v. 88, 2023.

Motivated by the well-known Paley graphs and their generalizations, this work studies the clique number of a Paley-like graph arising from vector spaces over finite fields, where we modify the operations from the traditional setting of Paley graphs. We provide bounds for the clique number of these graphs and obtain more precise results in some special cases. 

33. Counting distinct functional graphs from linear finite dynamical systems, Linear Algebra and its Applications v. 656, 409-420, 2023.

This work provides bounds for the number N(n) of non-isomorphic functional graphs arising from linear maps over the vector space Fⁿ, where F is a finite field. Specifically, as n grows, we prove that log log N(n) ~ log n, thereby closing the "gap" in a result by Bach and Bridy (2013), who showed that the quotient log log N(n) / log n lies within the interval [1/2, 1] for large n. 

32. The existence of Fq-primitive points on curves using freeness (with S. D. Cohen and G. Kapetanakis), Comptes Rendus Mathematique, v. 360, p. 641-652, 2022.

Freeness is a key concept in the context of primitive elements in finite fields, or more generally, the multiplicative structure of elements in finite fields. In this work, we generalize and extend this notion to a broader setting, including abstract finite cyclic groups. As an application, we use the traditional character sum method to examine the existence of special rational points on certain curves over finite fields. 

31. Character sums over affine spaces and applications, Finite Fields And Their Applications, v. 83, 2022. 

This paper presents a new nontrivial bound for multiplicative character sums over affine spaces, primarily derived from a remarkable result by N. Katz. We also provide several applications related to special primitive elements in finite fields and, in particular, offer significant improvements over previous works. 

30. Generators of finite fields with prescribed traces (with S. Ribas),  Journal of the Australian Mathematical Society  v. 112(3), p. 355-366, 2022.

In this paper, we investigate the existence of primitive elements in finite fields whose trace over a set of intermediate extensions is prescribed. This work is primarily motivated by S.D. Cohen's 2003 study, which considers the case of a single intermediate extension. Using the traditional character sum method, we derive both asymptotic and concrete existence results. 

29. The additive index of polynomials over finite fields (with Q. Wang), Finite Fields and Their Applications v. 79, 2022. 

The index of a polynomial over a finite field is a positive integer that provides insight into the interaction between the polynomial (viewed as a map over the finite field) and the cyclic structure of the multiplicative group of the field. In this paper, we replace the multiplicative group with an additive group (i.e., a vector subspace) and introduce an analogue of the index, which is now a polynomial rather than an integer. We then explore several classical problems in the theory of polynomials over finite fields—such as value sets, permutations, and character sum bounds—through the lens of the additive index. 

28. On primitive elements of finite fields avoiding affine hyperplanes (with A. Fernandes), Finite Fields And Their Applications v. 76, 2021.

This paper explores the existence of primitive elements in finite fields that do not belong to a given set of affine hyperplanes in general position. The problem is primarily motivated by previous works on special elements in finite fields with "missing digits." 

27. Arithmetic constraints of polynomial maps through discrete logarithms, Journal of Number Theory 229, p. 432-443, 2021.

This paper explores the distribution of the elements P₁(z), ..., Pₖ(z) through cyclotomic cosets, where P₁, ..., Pₖ are given polynomials over a finite field and "z" runs over the elements of a finite field. Using Weil's bound, we prove that, unless the polynomials Pᵢ exhibit some multiplicative dependence, the "random variables" Pᵢ(z) are independent. We also present some minor applications, recovering previously known results. 

26. Minimal value set polynomials over fields of size p^3 (with H. Borges), Proceedings of the American Mathematical Society 149, p. 3639-3649, 2021.

This paper provides a complete description of the minimal value set polynomials (MVSPs) over fields of size p³. Previously, this question was only resolved for fields of size p and p². As an application, we use these results to characterize a special family of curves over finite fields. 

25. Mobius-Frobenius maps on irreducible polynomials (with F. Brochero and D. Oliveira), Bulletin of the Australian Mathematical Society, v.105(1), p. 66-77, 2021.

This paper is primarily motivated by earlier works of mine, where an action of the group PGL(2, q) on irreducible polynomials over a finite field was studied. In this work, we extend the action to the semidirect product PTL(2, q) = PGL(2, q) × Gal(F_q^n / F_q) on the set of irreducible polynomials over F_q^n. We present results on the characterization and number of fixed points under this action and highlight the contrast between this setting and the simpler case of the action of the group PGL(2, q). 

24. Mean value theorems for a class of density-like arithmetic functions,  International Journal of Number Theory, v.17(4), p. 1013-1027, 2021. 

In this paper, we apply basic techniques from Analytic Number Theory to prove that a family of arithmetic functions has a well-defined mean value. Under a mild additional condition, we show that this mean value is positive. This result is primarily motivated by the behavior of two arithmetic functions that measure the density of primitive and normal elements in finite field extensions. As an application, we derive statistical results on the density of normal elements in finite fields, providing improvements to previously known results in this area. 

23. On linearly chinese field extensions (with C. Greither), Communications in Algebra v. 49(5), p. 1884-1894, 2021.

Given a field extension L/K of finite degree, we investigate the existence of elements in L whose trace over intermediate extensions is prescribed. In the affirmative case, we further explore the dimension of the set of such elements, which forms a K-affine subspace. Our main approach is to translate the problem into a system of modular equations over an appropriate group algebra. We provide both positive and negative results and fully solve the problem when the extension L/K is abelian. 

22. On polynomials x^n − 1 over binary fields whose irreducible factors are binomials and trinomials (with D. Oliveira), Finite Fields And Their Applications, v. 73, 2021.

In 2015, Brochero, Giraldo Vergara, and Batista de Oliveira proved that, under certain conditions on the positive integer n, the irreducible factors of the binomial Xⁿ - 1 over a finite field are all binomials and trinomials. Motivated by this result, we explore the converse in this work. Specifically, we characterize the integers n for which the binomial Xⁿ - 1 splits into irreducible binomials and trinomials over the binary fields F₂ and F₄. Additionally, we provide some general results and propose open problems for future research. 

21.  On the factorization of iterated polynomials,  REVISTA MATEMÁTICA IBEROAMERICANA v. 36(7) , p. 1957–1978, 2020. 

In this work, we study the growth of several arithmetic functions related to the factorization of iterated polynomials of the form f(gⁿ(X)), including the number of distinct roots, the extremal degrees of the irreducible factors (both minimum and maximum), and the number of distinct irreducible factors. We focus on the cases where g(X) is either a monomial or a linearized polynomial, providing interesting examples that demonstrate the sharpness of some of our results. In particular, we improve on results previously obtained by Gomez-Perez, Ostafe, and Shparlinski (2014). 

20. Counting solutions of special linear equations over finite fields, Finite Fields And Their Applications v. 68, 2020.

In this paper, we investigate the number of solutions to the equation L₁(X₁) + ... + Lₙ(Xₙ) = b over a finite field, where the Lᵢ's are linearized polynomials and the Xᵢ's are restricted to take values in certain intermediate extensions. We provide a characterization of the values of b for which this equation has a solution and, in the affirmative case, we give the exact number of solutions. The key idea is to combine basic linear algebra results with the arithmetic structure of the polynomial ring Fq[X]. 

19. Permutations from an arithmetic setting (with S. Ribas), Discrete Mathematics, v. 343 (8), 2020.

In this work, we introduce a class of permutations of the ring Z_n whose cycle decomposition is determined by a divisor of n. Specifically, we provide an explicit description of the cycle decomposition of such permutations, as well as their compositional inverses. As an application, we give the cycle decomposition of several well-known classes of permutation polynomials over finite fields. 

18. The dynamics of permutations on irreducible polynomials (with Q. Wang), Finite Fields and Their Applications v.64, 2020.

In this paper, we introduce a natural action of the group of permutations on the finite field Fqn on the set of irreducible polynomials over a finite field. We present several results regarding the fixed points of this action and propose a method for generating k-degree irreducible polynomials from a single k-degree irreducible polynomial. 

17. On the existence and number of invariant polynomials, Finite Fields and Their Applications v. 61, 2020.

This work is the final part of a three-paper series exploring a natural action of the group PGL(2, q) on the set of irreducible polynomials over finite fields. In contrast to the known asymptotic formulas for the number of irreducible polynomials of fixed degree that are invariant under a generic element of PGL(2, q), this paper provides exact formulas (some of which were previously known for particular cases). Additionally, we prove that no irreducible polynomial of degree at least 3 can be invariant under the action of a noncyclic subgroup of PGL(2, q)—a result that was only known before in the case where the subgroup is the entire PGL(2, q)). 

16. Construction of irreducible polynomials through rational transformations (with D. Panario and Q. Wang), Journal of Pure and Applied Algebra v. 224 (5), 2020.

This work is the second in a three-paper series examining a natural action of the group PGL(2, q) on the set of irreducible polynomials over finite fields. Building on results from the first paper, we know that certain irreducible polynomials arise from special rational transformations, some of which have been previously studied in the literature. In this paper, we investigate the irreducibility of iterates of a special family introduced in Paper 15. Our approach leverages the concept of the spin of polynomials over finite fields, which serves as a natural generalization of the well-known Capelli Lemma.

15. Mobius-like maps on irreducible polynomials and rational transformations,  Journal of Pure and Applied Algebra, v. 224(1), p. 169-180, 2020.

This work is the first in a three-paper series exploring a natural action of the group PGL(2, q) on the set of irreducible polynomials over finite fields. Previous studies have shown that polynomials invariant under specific elements of PGL(2, q) arise naturally from certain rational transformations. Notably, when considering the inversion x → 1/x, we obtain the well-known self-reciprocal polynomials. In this paper, we extend this result to a generic element of PGL(2, q). 

14. Factorization of composed polynomials and applications (with F. Brochero and L. Silva), Discrete Mathematics v. 342(12), 2019 .  

In this paper, we present a general procedure for factoring composed polynomials of the form f(Xⁿ) over finite fields, building upon the work of Brochero, Giraldo Vergara, and Batista de Oliveira (2015). Specifically, we extend several earlier results on this topic. 

13. On the dimension of permutation vector spaces, Bulletin of the Australian Mathematical Society, v. 100(2), p. 256-267, 2019.

Given a cyclic Galois extension L/K of degree N and an element u = (u₁, ..., uN) in L (viewed as a K-vector space), we examine the action of the group of affine maps modulo N on the coordinates of u. We derive a formula for the dimension of the K-vector space generated by the orbit of a single element u under this action. This formula involves the greatest common divisor (GCD) of certain univariate polynomials over K and is particularly related to cyclotomic polynomials over general fields. As an application, we present additional combinatorial results when K is either the field of rationals or a finite field.  

12. Dynamics of the a-map over residually finite Dedekind Domains (with C. Qureshi), Journal of Number Theory, v. 204, p. 134-154, 2019.

In this paper, we present a unified approach to describing the functional graphs of a broad class of finite dynamical systems that have been studied in recent years. Our framework recovers numerous known results, including those related to power maps on cyclic groups and linear maps over finite fields. Additionally, we establish new results concerning the structure of the trees that appear in these functional graphs, which may be of independent interest. 

11.  Existence results on k-normal elements over finite fields, REVISTA MATEMÁTICA IBEROAMERICANA, v. 35 (3), p. 805-822, 2019.  

This paper completes a three-part series on k-normal elements in finite fields (continuing from Papers 7 and 9). We present general results on the existence of primitive elements that are also k-normal. The core approach is to construct k-normal elements explicitly from normal elements, thereby avoiding the need to define a characteristic function for the set of k-normal elements in the character sum method. Our results include both asymptotic estimates and explicit constructions, and in several cases, we show that these results are asymptotically sharp.

10. Factorization of a class of composed polynomials, Designs, Codes and Cryptography v. 87 (7), p. 1657-1671, 2019.  

In 1955, Butler described the degree distribution in the factorization of composed polynomials of the form f(Xⁿ) over finite fields. His results relied on classical arithmetic functions such as Euler’s totient function and the multiplicative order modulo an integer. In this paper, we present an analogue of Butler’s theorem for composed polynomials of the form f(L(X)), where L(X) is a special linearized polynomial. Our main result closely mirrors Butler’s, with the classical arithmetic functions replaced by their polynomial analogues. We also discuss applications, including alternative proofs of known results, new iterative constructions of irreducible polynomials, and an explicit factorization of the composed polynomial f(X^q − X). 

9. Variations of the Primitive Normal Basis Theorem (with G. Kapetanakis), Designs, Codes and Cryptography v. 87 (7), p. 1459-1480, 2019. 

This paper is a natural continuation of Paper 7. We present several variations of the Primitive Normal Basis Theorem and resolve a conjecture proposed by Mullen and Anderson. 

8.  The functional graph of linear maps over finite fields and applications (with D. Panario), Designs, Codes and Cryptography v. 87 (2-3), p. 437–453 , 2019.

In this paper, we study the functional graphs associated with a special class of linearized polynomials over finite fields. We give particular attention to cases where these polynomials induce permutations of the field, recovering known results and presenting new findings motivated by practical applications. 

7.   Existence of primitive 1-normal elements in finite fields (with D.Thomson), Finite Fields And Their Applications v. 51, p. 238-269, 2018. 

In 2013, Huczynska, Mullen, Panario, and Thomson introduced the concept of k-normal elements over finite fields. Motivated by the celebrated Primitive Normal Basis Theorem, they posed the problem of the existence of primitive 1-normal elements in finite field extensions—a question also featured in a broader conjecture by Mullen and Anderson. Using the character sum method, we resolve this problem in a significantly stronger form.. 

6. A group action on multivariate polynomials over finite fields, Finite Fields And Their Applications v. 51, p. 218-237, 2018. 

In this work, we study the subring of multivariate polynomials over a finite field that are invariant under affine transformations of the form xi → ai·xi + bi. We explore both algebraic and combinatorial aspects of this "fixed point ring," including its structural properties and the minimal number of generators required. In particular, we establish a connection between the generator problem and a classical zero-sum problem in finite cyclic groups. 

5. Nilpotent linearized polynomials over finite fields and applications, Finite Fields And Their Applications v. 50, p. 279-292, 2018.

In this work, we introduce the concept of nilpotent linearized polynomials over finite fields and provide a partial characterization from different perspectives, including their polynomial and matrix representations. We present a method for constructing permutation polynomials from these nilpotent polynomials, with a particular focus on generating involutions over binary fields. Additionally, we discuss aspects of the cycle decomposition of the resulting permutations. 

4. Factoring polynomials of the form f(x^n)\in Fq[x] (with F. Brochero), Finite Fields And Their Applications v. 49, p. 166-179, 2018.

In this paper we provide explicit and algorithmic results on the factorization of composed polynomials f(x^n) over finite fields, where the integer n satisfying special conditions.  

3.  The action of GL2(Fq) on irreducible polynomials over Fq, revisited, Journal of Pure and Applied Algebra v. 222 (5), p. 1087-1094, 2018.

In this work, we explore an action of the group GL(2, q) on the set of monic irreducible polynomials over finite fields, previously introduced in the literature. Our main focus is to characterize and count the fixed points of this action—that is, the polynomials that remain invariant under the action of an element or, more generally, a cyclic subgroup of GL(2, q). We present new results, recover and unify existing ones, and also provide corrections to some inaccuracies found in earlier work. 

2. On the multiplicative order of the roots of bX^{q^r+1}-aX^{q^r}+dX-c (with F. Brochero, T. Garefalakis and E. Tzanakis), Finite Fields And Their Applications v. 47, p. 33-45, 2017.

In this paper we provide a lower bound the multiplciative order of certain elements arising from extensions Fq[X]/(F(X)), where F(X) is a "generic" irreducible factor of a polynomial of the form bX^{q^r+1}-aX^{q^r}+dX-c.

1. Elements of high order in Artin-Schreier extensions of finite fields (with F. Brochero), Finite Fields And Their Applications  v. 41, p. 24-33, 2016. 

In this paper we provide a lower bound for the multiplicative order of certain elements in Artin-Schreier extensions Fq[X]/(X^p-X-a).