Papers (Published/Accepted for publication)

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

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

This paper discusses some issues on the digraphs induced by power maps g --> g^t over finite groups. In particular, we show that finite nilpotent groups are precisely the groups for which such digraphs yield a special symmetry that it has appeared several times in the literature (most of them closely related to the maps discussed in Paper 12). Moreover, we introduce an interesting equivalence relation on the class of finite nilpotent groups that arises from these digraphs and discuss how far this relation is from the isomorphism of groups.

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

This paper is a continuation of Paper 24, where we discuss a mean value theorem for the density of special elements in finite fields (regarded as an arithmetic function). By employing some basic tools from theory of Arithmetic Functions, we prove that, for a fixed prime power q, the density f_k(n) of k-normal elements in the finite field extension GF(q^n)/GF(q) is an arithmetic function of 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 --> g^t over some classes of finite groups, including abelian groups and flower groups, which are introduced in this work. This was mainly motivated by the case where the underlying group is the projective general linear group of order 2 over a finite field.

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, in this work we study the clique number of a Paley-like graph arising from vector spaces over finite fields: here we switch the operations from the traditional setting in Paley graphs. We obtain 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 improvements on bounds for the number of non isomorphic functional graphs arising from linear maps over finite fields. 

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 an important 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 extend this notion to a more general setting, including abstract finite cyclic groups. As an application, we employ the traditional character sum method to study 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 provides a new nontrivial bound for multiplicative character sums over affine spaces, mainly derived from a beautiful result of N. Katz. We also provide some applications concerning special primitive elements in finite fields and, in particular, we obtain nice improvements on 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 explore the existence of primitive elements in finite fields whose trace over a set of intermediate extensions is prescribed. This is mainly motivated by a work of S.D. Cohen (2003) where the case of one intermediate extension is considered. By employing the traditional character sum method, we obtain 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 some information on the interaction of the polynomial (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 by an additive group (i.e., a vector subspace) and obtain an analogue of the index, which is a polynomial instead of an integer. We then study many classical problems in the theory of polynomials over finite fields (value sets, permutations and character sum bounds) in terms 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 mainly motivated by past 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 P_1(z), ..., P_k(z) thorugh cyclotomic cosets, where P_1, ..., P_k are (given) polynomials over a finite field and "z" runs over the elements of a finite field. With the help of Weil's bound we prove that, unless the polynomials P_i exhibit some multiplicative dependence, the "random variables" P_i(z) are independent. We also provide some minor applications, recovering past 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 (MVSP's) over fields of size p^3. Before, we only had the solution to this question for fields of size p and p^2. As an application, we 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 mainly motivated by earlier works of mine, where an action of the group PGL(2, q) over irreducible polynomials over a finite field is considered. Here we take a natural action of the semidirect product PTL(2, q)=PGL(2, q) x Gal(Fq^n/F_q) on the set of irreducible polynomials over Fq^n. We obtain results on the characterization and number of fixed points and exhibit the constrast between this setting and the simple setting 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 employ some basic Analytic Number Theory techniques and prove that a family of arithmetic functions has mean value and, with a mild extra condition, such value is positive. This is mainly motivated by the behaviour of two arithmetic functions that measure the density of primitive and normal elements in finite field extensions. As an application, we obtain statistical results on the density of normal elements in finite fields, enhancing the known results in this direction. 

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 explore the existence of elements in L whose trace over intermediate extensions are prescribed and, in affirmative case, we explore the dimension of the set of elements (which is a K-affine subspace). Our main tool is to translate the problem to a sistem of modular equations over an appropriate group algebra. We obtain many positive and negative answers and completely 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 X^n-1 over a finite field are all binomials and trinomials. Motivated by the latter, in this work we explore the converse of this result. In particular, we characterize the integers n for which the binomial x^n-1 splits into irreducible binomials and trinomials over the binary fields F_2 and F_4. We also provide some general minor 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 some arithmetic functions related to the factorization of iterated polynomials f(g^n(X)) such as number of distinct roots, extremal degree of the irreducible factors (min and max) and number of distinct irreducible factors. The cases where g(X) is a monomial or a linearized polynomial yield interesting examples, showing that some of our results are relatively sharp. In particular, we improve some results of 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 explore the number of solutions to equations L_1(X_1)+...+L_n(X_n)=b over a finite field, where the L_i's are linearized polynomials and the X_i's can only take values in certain intermediate extensions. We provide a characterization of b's such that this equation has a solution and, in affirmative case, we provide 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 constrained by a divisor of n. In particular we provide an explicit description of the cycle decomposition of such permutations and also their compositional inverses. As an application we provide the cycle decomposition of some 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 of a finite field Fq^n on the set of irreducible polynomials over a finite field. We provide many results concerning the fixed points of this action.

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

This work is the final piece on a 3-paper series concerning a natural action of the group PGL(2, q) on the set of irreducible polynomials over finite fields. In constrast to a known asymptotic formulae for the number of irreducible polynomials of fixed degree that are invariant by a generic element of PGL(2, q), in this paper we provide exact formulae (some particular cases were already known). Moreover we also prove that no irreducible polynomial of degree at least 3 can be invariant by the action of a noncyclic subgroup of PGL(2, q) (before that, this was only known for the case where the subgroup is the whole 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 piece on a 3-paper series concerning a natural action of the group PGL(2, q) on the set of irreducible polynomials over finite fields. From the first paper, we know that there are irreducible polynomials arising from some special rational transformations and, in fact, some of them were previsouly considered in the literature. In this paper we explore the irreducibility of iterations of a special family that was introduced in Paper 15. Our approach relies on the concept of the spin of polynomials over finite fields, that can be seen as a nice 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 piece on a 3-paper series concerning a natural action of the group PGL(2, q) on the set of irreducible polynomials over finite fields. Some past works have proved that the polynomials that are invariant by  specific element of  PGL(2, q) arise naturally from some rational transformations. Most 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 provide a general procedure to factorize composed polynomials f(X^n) over finite fields, mainly based on a work of Brochero, Giraldo Vergara, and Batista de Oliveira (2015). In particular, we extend some earlier works on this issue.

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=(u_1, ..., u_N) in L (regarded as an K-vector space), we consider the action of the group of affine maps modulo N on the coordiantes of u. We obtain a formula for the dimension of the K-vector space generated by the orbit of a single element u under the previous action. This formula depends on the GCD of certain univariate polynomials over the field K and, in particular, it is connected to cyclotomic polynomials over general fields. As an application, we obtain further 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 provide a unified approach to describe the functional graph of a series of finite dynamical systems that have been considered in the past few years. In particular, we recover many past results, including the power map on cyclic groups and linear maps over finite fields. We also provide further results on the class of trees that appear in these functional graphs, which might be of further interest. 

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

This paper is the final piece of the 3-papers series on k-normal elements in finite fields (Papers 7 and 9). Here we provide some general results on the existence of primitive elements in finite fields that are also k-normal. The main idea is to explicitly construct k-normal elements from normal elements, avoiding to build the characteristic function for the set of k-normal elements in the character sum method. We obtain both asymptotic and concrete results and, in some cases, we show that our 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 provided the degree distribution of the factorization of composed polynomials f(X^n) over finite fields. The description depends on some arithmetic functions such as Euler Totient function and the multiplicative order modulo an integer. In this paper we provide an interesting analogue of this result, considering composed polynomial f(L(X)), where L(X) is a special linearized polynomial. In particular our main theorem resembles Butler's main theorem, and the arithmetic functions are replaced by their polynomial analogues. We provide some applications including alternative proofs for known results, new iterated constructions of  irreducible polynomials and the explicit factorization of composed polynomials 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 provide some variations of the Primitive Normal Basis Theorem, solving a conjecture proposed by Mullen and Anderson. Moreover, we introduce the concept of r-primitive elements in finite fields: this notation is widely used now and, most notably, it has motivated further works from several researchers. 

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 explore the description of functional graphs associated to a special class of linearized polynomials over finite fields. We provide more detailed results when these polynomials yield permutations of the finite field, recovering past results and providing some new results motivated by practical issues.

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 Huckzynska, Mullen, Panario and Thomson introduced the concept of k-normal elements over finite fields. Motivated by the celebrated Primitive Normal Basis Theorem, they propose the problem on the existence of primitive 1-normal elements in finite fields extensions. This problem is also part of a conjecture proposed by Mullen and Anderson. By employing the character sum method, we were able to solve the problem in a fairly 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 maps x_i --> a_ix_i+b_i. We explore algebraic and combinatorial questions, including the structure of the "fixed point ring" and the minimal number of generators for this ring. In particular, we relate the latter to a classic 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 partially describe them from distinct points of view (polynomial representation and matrix representation). We provide a method to produce permutation polynomials from such polynomials, including the construction of involutions over binary fields. We further discuss some issues on the cycle decomposition of the permutations that we obtain. 

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 satisfies some 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 that was previsouly introduced. The main problem is to characterize and count the fixed points (i.e., polynomials that are invariant by an element or, more generally, by a subgroup of GL(2, q)). We obtain new results, recover some past results and provide corrections to a past 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 arising from Artin-Schreier extensions Fq[X]/(X^p-X-a).