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We study finite-dimensional representations of hyper loop algebras over non-algebraically closed fields. The main results concern the classification of the irreducible representations, the construction of the Weyl modules, base change, tensor products of irreducible and Weyl modules, and the block decomposition of the underlying abelian category. Several results are interestingly related to the study of irreducible representations of polynomial algebras and Galois theory.

The Brauer monoid is studied by the notion of 0-cohomology. We investigate the impact of invertible elements of modifications on the structure of the Brauer monoid, especially for finite fields.

Let $K$ be a field, $a, b\in K$ and $ab\neq 0$. Let us consider the polynomials $g_{1}(x)=x^n+ax+b, g_{2}(x)=x^n+ax^2+bx$, where $n$ is a fixed positive integer. In this paper we show that for each $k\geq 2$ the hypersurface given by the equation \begin{equation*} S_{k}^{i}: u^2=\prod_{j=1}^{k}g_{i}(x_{j}),\quad i=1, 2. \end{equation*} contains a rational curve. Using the above and Woestijne's recent results \cite{Woe} we show how one can construct a rational point different from the point at infinity on the curves $C_{i}:y^2=g_{i}(x), (i=1, 2)$ defined over a finite field, in polynomial time.

The leader-following consensus problem of multi-agent systems over finite fields ${\mathbb F}_p$ is considered in this paper. Dynamics of each agent is governed by a linear equation over ${\mathbb F}_p$, where a distributed control protocol is utilized by the followers.Sufficient and/or necessary conditions on system matrices and graph weights in ${\mathbb F}_p$ are provided for the followers to track the leader.

The study of solutions to polynomial equations over finite fields has a long history in mathematics and is an interesting area of contemporary research. In recent years the subject has found important applications in the modelling of problems from applied mathematical fields such as signal analysis, system theory, coding theory and cryptology. In this connection it is of interest to know criteria for the existence of squares and other powers in arbitrary finite fields. Making good use of polynomial division in polynomial rings over finite fields, we have examined a classical criterion of Euler for squares in odd prime fields, giving it a formulation which is apt for generalization to arbitrary finite fields and powers. Our proof uses algebra rather than classical number theory, which makes it convenient when presenting basic methods of applied algebra in the classroom.

Let ${\mathbf P}^2$ denote the projective plane over a finite field ${\mathbb F}_q$. A pair of nonsingular conics $({\mathcal A}, {\mathcal B})$ in the plane is said to satisfy the Poncelet triangle condition if, considered as conics in ${\mathbf P}^2({\overline{\mathbb F}}_q)$, they intersect transverally and there exists a triangle inscribed in ${\mathcal A}$ and circumscribed around ${\mathcal B}$. It is shown in this article that a randomly chosen pair of conics satisfies the triangle condition with asymptotic probability $1/q$. We also make a conjecture based upon computer experimentation which predicts this probability for tetragons, pentagons and so on up to enneagons.

There exists an absolute constant $C$ with the following property. Let $A \subseteq \mathbb{F}_p$ be a set in the prime order finite field with $p$ elements. Suppose that $|A| > C p^{5/8}$. The set \[ (A \pm A)(A \pm A) = \{(a_1 \pm a_2)(a_3 \pm a_4) : a_1,a_2,a_3,a_4 \in A\} \] contains at least $p/2$ elements.

In this note, we study linear determinantal representations of smooth plane cubics over finite fields. We give an explicit formula of linear determinantal representations corresponding to rational points. Using Schoof's formula, we count the number of projective equivalence classes of smooth plane cubics over a finite field admitting prescribed number of equivalence classes of linear determinantal representations. As an application, we determine isomorphism classes of smooth plane cubics over a finite field with 0, 1 or 2 equivalence classes of linear determinantal representations.

We obtain explicit factorizations of reduced period polynomials of degree $2^m$, $m\ge 4$, for finite fields of characteristic $p\equiv 3$ or $5\pmod{8}$. This extends the results of G. Myerson, who considered the cases $m=1$ and $m=2$, and S. Gurak, who studied the case $m=3$.

In this work we extend our study on a link between automaticity and certain algebraic power series over finite fields. Our starting point is a family of sequences in a finite field of characteristic $2$, recently introduced by the first author in connection with algebraic continued fractions. By including it in a large family of recurrent sequences in an arbitrary finite field, we prove its automaticity. Then we give a criterion on automatic sequences, generalizing a previous result and this allows us to present new families of automatic sequences in an arbitrary finite field.

In this note we show that any basic abelian variety with additional structures over an arbitrary algebraically closed field of characteristic $p>0$ is isogenous to another one defined over a finite field. We also show that the category of abelian varieties over finite fields up to isogeny can be embedded into the category of basic abelian varieties with suitable endomorphism structures. Using this connection, we derive a new mass formula for a finite orbit of polarized abelian surfaces over a finite field.

Let $R$ be a commutative and unital semi-local ring in which 2 is invertible. In this note, we show that anisotropic quadratic spaces over $R$ remain anisotropic after base change to any odd-degree finite \'{e}tale extension of $R$. This generalization of the classical Artin-Springer theorem (concerning the situation where $R$ is a field) was previously established in the case where all residue fields of $R$ are infinite by I. Panin and U. Rehmann. The more general result presented here permits to extend a fundamental isotropy criterion of I. Panin and K. Pimenov for quadratic spaces over regular semi-local domains containing a field of characteristic $\neq 2$ to the case where the ring has at least one residue field which is finite.

Permutation polynomials over finite fields constitute an active research area and have applications in many areas of science and engineering. In this paper, two conjectures on permutation polynomials proposed recently by Wu and Li [19] are settled. Moreover, a new class of permutation trinomials of the form $x+\gamma \textup{Tr}_{q^n/q}(x^k)$ is also presented, which generalizes two examples of [10].

We study arithmetical and geometrical properties of {\it maximal curves}, that is, curves defined over the finite field $\mathbb F_{q^2}$ whose number of $\mathbb F_{q^2}$-rational points reachs the Hasse-Weil upper bound. Under a hypothesis on non-gaps at rational points we prove that maximal curves are $\mathbb F_{q^2}$-isomorphic to $y^q+y=x^m$ for some $m\in \mathbb Z^+$.

We study arithmetical and geometrical properties of {\it maximal curves}, that is, curves defined over the finite field $\mathbb F_{q^2}$ whose number of $\mathbb F_{q^2}$-rational points reachs the Hasse-Weil upper bound. Under a hypothesis on non-gaps at rational points we prove that maximal curves are $\mathbb F_{q^2}$-isomorphic to $y^q+y=x^m$ for some $m\in \mathbb Z^+$.

The Lie algebra of planar vector fields with coefficients from the field of rational functions over an algebraically closed field of characteristic zero is considered. We find all finite-dimensional Lie algebras that can be realized as subalgebras of this algebra.

Schubert calculus provides algebraic tools to solve enumerative problems. There have been several applied problems in systems theory, linear algebra and physics which were studied by means of Schubert calculus. The method is most powerful when the base field is algebraically closed. In this article we first review some of the successes Schubert calculus had in the past. Then we show how the problem of decoding of subspace codes used in random network coding can be formulated as a problem in Schubert calculus. Since for this application the base field has to be assumed to be a finite field new techniques will have to be developed in the future.

Let $\mathbb{F}_{q}$ be the finite field of characteristic $p$ containing $q = p^{r}$ elements and $f(x)=ax^{n} + x^{m}$ a binomial with coefficients in this field. If some conditions on the gcd of $n-m$ an $q-1$ are satisfied then this polynomial does not permute the elements of the field. We prove in particular that if $f(x) = ax^{n} + x^{m}$ permutes $\mathbb{F}_{p}$, where $n>m>0$ and $a \in {\mathbb{F}_{p}}^{*}$, then $p -1 \leq (d -1)d$, where $d = {{gcd}}(n-m,p-1)$, and that this bound of $p$ in term of $d$ only, is sharp. We show as well how to obtain in certain cases a permutation binomial over a subfield of $\mathbb{F}_{q}$ from a permutation binomial over $\mathbb{F}_{q}$.

We give a new way to study recursive towers of curves over a finite field, defined from a bottom curve $\Cun$ and a correspondence $\Cdeux$ on $\Cun$.In particular, we study their asymptotic behavior. A close examination of singularities leads to a necessary condition for a tower to be asymptotically good. Then, spectral theory on a directed graph and considerations on the class of $\Cdeux$ in $\NS (\Cun \times \Cun)$ lead to the fact that, under some mild assumptions, a recursive tower which does not reach Drinfeld-Vladut bound cannot be optimal in Tsfasmann-Vladut sense. Results are applied to the Bezerra-Garcia-Stichtenoth tower along the paper for illustration.

We give a new way to study recursive towers of curves over a finite field, defined from a bottom curve $\Cun$ and a correspondence $\Cdeux$ on $\Cun$.In particular, we study their asymptotic behavior. A close examination of singularities leads to a necessary condition for a tower to be asymptotically good. Then, spectral theory on a directed graph and considerations on the class of $\Cdeux$ in $\NS (\Cun \times \Cun)$ lead to the fact that, under some mild assumptions, a recursive tower which does not reach Drinfeld-Vladut bound cannot be optimal in Tsfasmann-Vladut sense. Results are applied to the Bezerra-Garcia-Stichtenoth tower along the paper for illustration.

In this paper we study random walks on the hypergroup of circles in a finite field of prime order p = 4l + 3. We investigating the behavior of random walks on this hypergroup, the equilibrium distribution and the mixing times. We use two different approaches - comparision of Dirichlet forms (geometric bound of eigenvalues), and coupling methods, to show that the mixing time of random walks on hypergroup of circles is only linear.

For a prime number $p$, we show that if two certain canonical finite quotients of a finitely generated Bloch-Kato pro-$p$ group $G$ coincide, then $G$ has a very simple structure, i.e., $G$ is a $p$-adic analytic pro-$p$ group. This result has a remarkable Galois-theoretic consequence: if the two corresponding canonical finite extensions $F^{(3)}/F$ and $F^{\{3\}}/F$ of a field $F$ -- with $F$ containing a primitive $p$-th root of unity -- coincide, then $F$ is $p$-rigid. The proof relies only on group-theoretic tools, and on certain properties of Bloch-Kato pro-$p$ groups.

We provide an index bound for character sums of polynomials over finite fields. This improves the Weil bound for high degree polynomials with small indices, as well as polynomials with large indices that are generated by cyclotomic mappings of small indices. As an application, we also give some general bounds for numbers of solutions of some Artin-Schreier equations and mininum weights of some cyclic codes.

In this paper we study special Fibonacci quaternions and special generalized Fibonacci-Lucas quaternions in quaternion algebras over finite fields.

Given a finite endomorphism $\varphi$ of a variety $X$ defined over the field of fractions $K$ of a Dedekind domain, we study the extension $K(\varphi^{-\infty}(\alpha)) : = \bigcup_{n \geq 1} K(\varphi^{-n}(\alpha))$ generated by the preimages of $\alpha$ under all iterates of $\varphi$. In particular when $\varphi$ is post-critically finite, i.e., there exists a non-empty, Zariski-open $W \subseteq X$ such that $\varphi^{-1}(W) \subseteq W$ and $\varphi : W \to X$ is \'etale, we prove that $K(\varphi^{-\infty}(\alpha))$ is ramified over only finitely many primes of $K$. This provides a large supply of infinite extensions with restricted ramification, and generalizes results of Aitken-Hajir-Maire in the case $X = \mathbb{A}^1$ and Cullinan-Hajir, Jones-Manes in the case $X = \mathbb{P}^1$. Moreover, we conjecture that this finite ramification condition characterizes post-critically finite morphisms, and we give an entirely new result showing this for $X = \mathbb{P}^1$. The proof relies on Faltings' theorem and a local argument.

We assign functorially a $\mathbb{Z}$-lattice with semisimple Frobenius action to each abelian variety over $\mathbb{F}_p$. This establishes an equivalence of categories that describes abelian varieties over $\mathbb{F}_p$ avoiding $\sqrt{p}$ as an eigenvalue of Frobenius in terms of simple commutative algebra. The result extends the isomorphism classification of Waterhouse and Deligne's equivalence for ordinary abelian varieties.

We give a construction of singular curves with many rational points over finite fields. This construction enables us to prove some results on the maximum number of rational points on an absolutely irreducible projective algebraic curve defined over Fq of geometric genus g and arithmetic genus $\pi$.

We propose and rigorously analyze two randomized algorithms to factor univariate polynomials over finite fields using rank $2$ Drinfeld modules. The first algorithm estimates the degree of an irreducible factor of a polynomial from Euler-Poincare characteristics of random Drinfeld modules. Knowledge of a factor degree allows one to rapidly extract all factors of that degree. As a consequence, the problem of factoring polynomials over finite fields in time nearly linear in the degree is reduced to finding Euler-Poincare characteristics of random Drinfeld modules with high probability. Notably, the worst case complexity of polynomial factorization over finite fields is reduced to the average case complexity of a problem concerning Drinfeld modules. The second algorithm is a random Drinfeld module analogue of Berlekamp's algorithm. During the course of its analysis, we prove a new bound on degree distributions in factorization patterns of polynomials over finite fields in certain short intervals.

Long-range spatial correlations in the velocity and energy fields of a granular fluid are discussed in the framework of a 1d lattice model. The dynamics of the velocity field occurs through nearest-neighbour inelastic collisions that conserve momentum but dissipate energy. A set of equations for the fluctuating hydrodynamics of the velocity and energy mesoscopic fields give a first approximation for (i) the velocity structure factor and (ii) the finite-size correction to the Haff law, both in the homogeneous cooling regime. At a more refined level, we have derived the equations for the two-site velocity correlations and the total energy fluctuations. First, we seek a perturbative solution thereof, in powers of the inverse of system size. On the one hand, when scaled with the granular temperature, the velocity correlations tend to a stationary value in the long time limit. On the other hand, the scaled standard deviation of the total energy diverges, that is, the system shows multiscaling. Second, we find an exact solution for the velocity correlations in terms of the spectrum of eigenvalues of a certain matrix. The results of numerical simulations of the microscopic model confirm our theoretical results, including the above described multiscaling phenomenon.

Golomb and Welch conjectured in 1970 that there only exist perfect Lee codes for radius $t=1$ or dimension $n=1, 2$. It is admitted that the existence and the construction of quasi-perfect Lee codes have to be studied since they are the best alternative to the perfect codes. In this paper we firstly highlight the relationships between subset sums, Cayley graphs, and Lee linear codes and present some results. Next, we present a new constructive method for constructing quasi-perfect Lee codes. Our approach uses subsets derived from some quadratic curves over finite fields (in odd characteristic) to derive two classes of $2$-quasi-perfect Lee codes are given over the space $\mathbb{Z}_p^n$ for $n=\frac{p^k+1}{2}$ $(\text{with} ~p\equiv 1, -5 \mod 12 \text{and} k \text{is any integer}, \text{or} p\equiv -1, 5 \mod 12 \text{and} k \text{is an even integer})$ and $n=\frac{p^k-1}{2}$ $(\text{with}p\equiv -1, 5 \mod 12, k \text{is an odd integer} \text{and} p^k>12)$, where $p$ is an odd prime. Our codes encompass the quasi-perfect Lee codes constructed recently by Camarero and Mart\'inez. Furthermore, we solve a conjecture proposed by Camarero and Mart\'inez (in "quasi-perfect Lee codes of radius $2$ and arbitrarily large dimension", IEEE Trans. Inf. Theory, vol. 62, no. 3, 2016) by proving that the related Cayley graphs are Ramanujan or almost Ramanujan. The Lee codes presented in this paper have applications to constrained and partial-response channels, in flash memories and decision diagrams.

We state an improved version of the conjecture of Langlands and Rapoport, and we prove the conjecture for a large class of Shimura varieties. In particular, we obtain the first proof of the (original) conjecture for Shimura varieties of PEL-type.

We investigate stable operations in supersingular elliptic cohomology using isogenies of supersingular elliptic curves over finite fields. Our main results provide a framework in which we give a conceptually simple proof of an elliptic cohomology version of the Morava change of rings theorem and also gives models for explicit stable operations in terms of isogenies and morphisms in certain enlarged isogeny categories. We relate our work to that of G. Robert on the Hecke algebra structure of the ring of supersingular modular forms.

We investigate stable operations in supersingular elliptic cohomology using isogenies of supersingular elliptic curves over finite fields. Our main results provide a framework in which we give a conceptually simple proof of an elliptic cohomology version of the Morava change of rings theorem and also gives models for explicit stable operations in terms of isogenies and morphisms in certain enlarged isogeny categories. We relate our work to that of G. Robert on the Hecke algebra structure of the ring of supersingular modular forms.

Currently, the best upper bounds on the number of rational points on an absolutely irreducible, smooth, projective algebraic curve of genus g defined over a finite field F_q come either from Serre's refinement of the Weil bound if the genus is small compared to q, or from Oesterle's optimization of the explicit formulae method if the genus is large. This paper presents three methods for improving these bounds. The arguments used are the indecomposability of the theta divisor of a curve, Galois descent, and Honda-Tate theory. Examples of improvements on the bounds include lowering them for a wide range of small genus when q=2^3, 2^5, 2^{13}, 3^3, 3^5, 5^3, 5^7, and when q=2^{2s}, s>1. For large genera, isolated improvements are obtained for q=3,8,9.

The classical Lagrange inversion formula is extended to analytic and non--analytic inversion problems on non--Archimedean fields. We give some applications to the field of formal Laurent series in $n$ variables, where the non--analytic inversion formula gives explicit formal solutions of general semilinear differential and $q$--difference equations. We will be interested in linearization problems for germs of diffeomorphisms (Siegel center problem) and vector fields. In addition to analytic results, we give sufficient condition for the linearization to belong to some Classes of ultradifferentiable germs, closed under composition and derivation, including Gevrey Classes. We prove that Bruno's condition is sufficient for the linearization to belong to the same Class of the germ, whereas new conditions weaker than Bruno's one are introduced if one allows the linearization to be less regular than the germ. This generalizes to dimension $n> 1$ some results of [CarlettiMarmi]. Our formulation of the Lagrange inversion formula by mean of trees, allows us to point out the strong similarities existing between the two linearization problems, formulated (essentially) with the same functional equation. For analytic vector fields of $\C^2$ we prove a quantitative estimate of a previous qualitative result of [MatteiMoussu] and we compare it with a result of [YoccozPerezMarco].

Let ${\mathbb F}_q$ be a finite field of characteristic two and ${\mathbb F}_q(X_1,...,X_n)$ a rational function field. We use matrix methods to obtain explicit transcendental bases of the invariant subfields of orthogonal groups and pseudo-symplectic groups on ${\mathbb F}_q(X_1,...,X_n)$ over ${\mathbb F}_q$.

We obtain an asymptotic formula on the Odlyzko-Stanley enumeration problem. Let $N_m^*(k,b)$ be the number of $k$-subsets $S\subseteq F_p^*$ such that $\sum_{x\in S}x^m=b$. If $m 0$ such that | N_m^*(k,b)-p^{-1}{p-1 \choose k}|\leq {p^{1-\epsilon}+mk-m \choose k}. In addition, let $\gamma'(m,p)$ denote the distinct Waring's number $(\mod p)$, the smallest positive integer $k$ such that every integer is a sum of m-th powers of $k$-distinct elements $(\mod p)$. The above bound implies that there is a constant $\epsilon(\delta)>0$ such for any prime $p$ and any $m

In this paper we systematically study various properties of the distance graph in ${\Bbb F}_q^d$, the $d$-dimensional vector space over the finite field ${\Bbb F}_q$ with $q$ elements. In the process we compute the diameter of distance graphs and show that sufficiently large subsets of $d$-dimensional vector spaces over finite fields contain every possible finite configurations.

We prove the generalised Tate conjecture for H^3 of products of elliptic curves over finite fields, by slightly modifying an argument of M. Spiess concerning the Tate conjecture. We prove it fully if the elliptic curves run among at most 3 isogeny classes. We also show how things become more intricate from H^4 onwards, for more that 3 isogeny classes.

We prove that if $G$ is an Abelian group and $A_1,\ldots,A_k \subseteq G$ satisfy $m A_i=G$ (the $m$-fold sumset), then $A_1+\ldots+A_k=G$ provided that $k \ge c_m \log n$. This generalizes a result of Alon, Linial, and Meshulam [Additive bases of vector spaces over prime fields. J. Combin. Theory Ser. A, 57(2):203--210, 1991] regarding the so called additive bases.

We estimate the number of possible types degree patterns of $k$-lacunary polynomials of degree $t < p$ which split completely modulo $p$. The result is based on a combination of a bound on the number of zeros of lacunary polynomials with some graph theory arguments.

A basis of the ideal of the complement of a linear subspace in a projective space over a finite field is given. As an application, the second largest number of points of plane curves of degree $d$ over the finite field of $q$ elements is also given for $d\geq q+1$.

The article contents suggestions on how to perform the Fast Fourier Transform over Large Finite Fields. The technique is to use the fact that the multiplicative groups of specific prime fields are surprisingly composite.

We describe the "generic" part of the character ring of general linear groups over a finite field in terms of quiver representations.

We construct an intermediate cohmology between motivic cohomology and Weil-etale cohomology. Using this, the Bass conjecture on finite generation of motivic cohomology, and the Beilinson-Tate on the finite generation of Weil-etale cohomology are related.

We construct MDS Euclidean and Hermitian self-dual codes over large finite fields of odd and even characteristics. Our codes arise from cyclic and negacyclic duadic codes.

Given a polynomial f and a finite field F one can construct a directed graph where the vertices are the values in the finite field, and emanating from each vertex is an edge joining the vertex to its image under f. When f is a Chebyshev polynomial of prime degree, the graphs display an unusual degree of symmetry. In this paper we provide a complete description of these graphs, and also provide some examples of how these graphs can be used to determine the decomposition of primes in certain field extensions.

For n=1,2,3,... let N_n(q) denote the number of monic irreducible polynomials over the finite field F_q. We mainly show that the sequence N_n(q)^{1/n} (n>e^{3+7/(q-1)^2}) is strictly increasing and the sequence N_{n+1}(q)^{1/(n+1)}/N_n(q)^{1/n} (n>=5.835*10^{14}) is strictly decreasing. We also prove that if q>8 then N_{n+1}(q)/N_n(q) (n=1,2,3,...) is strictly increasing.

We give effective bounds for the class number of any algebraic function field of genus $g$ defined over a finite field. These bounds depend on the possibly partial information on the number of places on each degree $\leq g$. Such bounds are especially useful for estimating the class number of function fields in towers of function fields over finite fields. We give examples in the case of asymptotically good towers. In particular we estimate the class number of function fields which are steps of towers having one or several positive Tsfasman-Vladut invariants. Note that the study is not done asymptotically, but for each individual step of the towers for which we determine precise parameters.