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Regarding non-unique factorization of integer-valued polynomials over a discrete valuation domain (R , M) with finite residue field, it is known that there exist absolutely irreducible elements, that is, irreducible elements all of whose powers factor uniquely, and non-absolutely irreducible elements. We completely and constructively characterize the absolutely irreducible elements among split integer-valued polynomials. They correspond bijectively to finite sets, which we call balanced , characterized by a combinatorial property regarding the distribution of their elements among residue classes of powers of M. For each such balanced set as the set of roots of a split polynomial, there exists a unique vector of multiplicities and a unique constant so that the corresponding product of monic linear factors times the constant is an absolutely irreducible integer-valued polynomial. This also yields sufficient criteria for integer-valued polynomials over Dedekind domains to be absolutely irreducible. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity. An element r ∈ R is said to be absolutely irreducible in R if for all natural numbers n > 1, rn has essentially only one factorization namely r n = r ⋯ r. If r ∈ R is irreducible in R but for some n > 1, rn has other factorizations distinct from r n = r ⋯ r , then r is called non-absolutely irreducible. In this paper, we construct non-absolutely irreducible elements in the ring Int (Z) = { f ∈ Q [ x ] | f (Z) ⊆ Z } of integer-valued polynomials. We also give generalizations of these constructions. [ABSTRACT FROM AUTHOR]

Almost Perfect Nonlinear (APN) functions are very useful in cryptography, when they are used as S-Boxes due to their good resistance to differential cryptanalysis. Also, some of the curves and surfaces defined by the corresponding nonlinear functions have many rational points and have applications to Algebraic-Geometric (AG) codes (Janwa and Wilson in Applied algebra, algebraic algorithms and error-correcting codes (San Juan, PR, 1993), vol 673, pp 180–194, Springer, Berlin, 1993, in IEEE Trans Inform Theory, accepted). An APN function f : F 2 n → F 2 n is called an exceptional APN if it is APN on infinitely many extensions of F 2 n . Aubry et al. (Contemp Math 518:23–31, 2010) conjectured that the only exceptional APN functions are the Gold and the Kasami–Welch monomial functions. They established that a polynomial function of odd degree is not an exceptional APN provided the degree is not a Gold number (2 k + 1) or a Kasami–Welch number (2 2 k - 2 k + 1) . When the degree of the polynomial function is a Gold number, several partial results are known. Here, we prove the relative primeness conjecture of the Gold degree polynomials. This result helps us substantially to make advances toward the resolution of the exceptional APN conjecture in the Gold degree case. We prove that Gold degree polynomials of the form x 2 k + 1 + h (x) , where deg (h) is an odd integer, cannot be exceptional APN (with a few natural exceptions). We also prove that the absolutely irreducible components of the Kasami–Welch degree curves intersect transversally at a particular point. Consequently, we prove that Kasami–Welch degree functions of type x 2 2 k - 2 k + 1 + h (x) produce absolutely irreducible surfaces in the case deg (h) ≡ 3 (mod 4) ; and also deg (h) ≡ 2 m - 1 + 1 (mod 2 m) under certain conditions. As a result, we show that the exceptional APN conjecture is true for these classes as well. [ABSTRACT FROM AUTHOR]

An irreducible element of a commutative ring is absolutely irreducible if no power of it has more than one (essentially different) factorization into irreducibles. In the case of the ring Int (D) = { f ∈ K [ x ] | f (D) ⊆ D } , of integer-valued polynomials on a principal ideal domain D with quotient field K, we give an easy to verify graph-theoretic sufficient condition for an element to be absolutely irreducible and show a partial converse: the condition is necessary and sufficient for polynomials with square-free denominator. [ABSTRACT FROM AUTHOR]

Planar functions are mappings from a finite field F q to itself with an extremal differential property. Such functions give rise to finite projective planes and other combinatorial objects. There is a subtle difference between the definitions of these functions depending on the parity of q and we consider the case that q is even. We classify polynomials of degree at most q 1 / 4 that induce planar functions on F q , by showing that such polynomials are precisely those in which the degree of every monomial is a power of two. As a corollary we obtain a complete classification of exceptional planar polynomials, namely polynomials over F q that induce planar functions on infinitely many extensions of F q. The proof strategy is to study the number of F q -rational points of an algebraic curve attached to a putative planar function. Our methods also give a simple proof of a new partial result for the classification of almost perfect nonlinear functions. [ABSTRACT FROM AUTHOR]

In Section 8, we saw the role of absolute irreducibility in determining the number of zeroes of a bivariate polynomial over a finite field. Let us consider the following algorithmic question: given a bivariate polynomial f(x,y) how do we determine whether or not it is absolutely irreducible? In this section, we will see how partial derivatives help in converting this apparently highly nonlinear problem into a simple task in linear algebra. This then gives an efficient algorithm for determining absolute irreducibility. [ABSTRACT FROM AUTHOR]

For any subset $Z \subseteq {\mathbb {Q}}$ , consider the set $S_Z$ of subfields $L\subseteq {\overline {\mathbb {Q}}}$ which contain a co-infinite subset $C \subseteq L$ that is universally definable in L such that $C \cap {\mathbb {Q}}=Z$. Placing a natural topology on the set ${\operatorname {Sub}({\overline {\mathbb {Q}}})}$ of subfields of ${\overline {\mathbb {Q}}}$ , we show that if Z is not thin in ${\mathbb {Q}}$ , then $S_Z$ is meager in ${\operatorname {Sub}({\overline {\mathbb {Q}}})}$. Here, thin and meager both mean "small", in terms of arithmetic geometry and topology, respectively. For example, this implies that only a meager set of fields L have the property that the ring of algebraic integers $\mathcal {O}_L$ is universally definable in L. The main tools are Hilbert's Irreducibility Theorem and a new normal form theorem for existential definitions. The normal form theorem, which may be of independent interest, says roughly that every $\exists $ -definable subset of an algebraic extension of ${\mathbb Q}$ is a finite union of single points and projections of hypersurfaces defined by absolutely irreducible polynomials. [ABSTRACT FROM AUTHOR]

We consider the polynomials of the form P(x) = xk - γTr(x) over 픽 qn for n ≥ 2. We show that P(x) is not a permutation of 픽qn in the case gcd(k, qn - 1) > 1. Our proof uses an absolutely irreducible curve over 픽qn and the number of rational points on it. [ABSTRACT FROM AUTHOR]

An alternative proof of the necessary conditions on A , B ∈ q 2 * for f (X) = X + A X 1 + q 2 (q − 1) / 4 + B X 1 + 3 q 2 (q − 1) / 4 to be a permutation polynomial in q 2 , q even, is given. This proof involves standard arguments from algebraic geometry over finite fields and fast symbolic computations. [ABSTRACT FROM AUTHOR]

How often is a quintic polynomial solvable by radicals? We establish that the number of such polynomials, monic and irreducible with integer coefficients in [-H,H], is O(H^{3.91}). More generally, we show that if n \geqslant 3 and n \notin \{ 7, 8, 10 \} then there are O(H^{n-1.017}) monic, irreducible polynomials of degree n with integer coefficients in [-H,H] and Galois group not containing A_n. Save for the alternating group and degrees 7,8,10, this establishes a 1936 conjecture of van der Waerden. [ABSTRACT FROM AUTHOR]

Planar functions are special functions from a finite field to itself that give rise to finite projective planes and other combinatorial objects. We consider polynomials over a finite field $$K$$ that induce planar functions on infinitely many extensions of $$K$$ ; we call such polynomials exceptional planar. Exceptional planar monomials have been recently classified. In this paper we establish a partial classification of exceptional planar polynomials. This includes results for the classical planar functions on finite fields of odd characteristic and for the recently proposed planar functions on finite fields of characteristic two. [ABSTRACT FROM AUTHOR]

Ovoids of the parabolic quadric Q(6, q) of PG (6 , q) have been largely studied in the last 40 years. They can only occur if q is an odd prime power and there are two known families of ovoids of Q(6, q), the Thas-Kantor ovoids and the Ree-Tits ovoids, both for q a power of 3. It is well known that to any ovoid of Q(6, q) two polynomials f 1 (X , Y , Z) , f 2 (X , Y , Z) can be associated. In this paper we classify ovoids of Q(6, q) with max { deg (f 1) , deg (f 2) } < (1 6.3 q) 3 13 - 1 . [ABSTRACT FROM AUTHOR]

We prove a general result concerning the paucity of integer points on a certain family of 4-dimensional affine hypersurfaces. As a consequence, we deduce that integer-valued polynomials have small asymmetric additive energy. [ABSTRACT FROM AUTHOR]

We study the class of univariate polynomials β k (X) , introduced by Carlitz, with coefficients in the algebraic function field F q (t) over the finite field F q with q elements. It is implicit in the work of Carlitz that these polynomials form an F q [ t ] -module basis of the ring Int (F q [ t ]) = { f ∈ F q (t) [ X ] | f (F q [ t ]) ⊆ F q [ t ] } of integer-valued polynomials on the polynomial ring F q [ t ]. This stands in close analogy to the famous fact that a Z -module basis of the ring Int (Z) is given by the binomial polynomials ( X k ). We prove, for k = q s , where s is a non-negative integer, that β k is irreducible in Int (F q [ t ]) and that it is even absolutely irreducible, that is, all of its powers β k m with m > 0 factor uniquely as products of irreducible elements of this ring. As we show, this result is optimal in the sense that β k is not even irreducible if k is not a power of q. [ABSTRACT FROM AUTHOR]

In this paper, we study the permutation properties of the class of trinomials of the form f(x) = x4q+1 + λ1xq+4 + λ2x2q+3 ∈ Fq²[x] where λ1, λ2 ∈ Fq and they are not simultaneously zero. We find all necessary and sufficient conditions on λ1 and λ2 such that f(x) permutes Fq², where q is odd and q = 22k+1, k ∈ N. [ABSTRACT FROM AUTHOR]

Permutation rational functions over finite fields have attracted high interest in recent years. However, only a few of them have been exhibited. This article studies a class of permutation rational functions constructed using trace maps on extensions of finite fields, especially for the cases of quadratic and cubic extensions. Our achievements are obtained by investigating absolute irreducibility of some polynomials in two indeterminates. [ABSTRACT FROM AUTHOR]

We study equations in groups G with unique mth roots for each positive integer m. A word equation in two letters is an expression of the form w(X, A)=B, where w is a finite word in the alphabet {X, A}. We think of A, B∈G as fixed coefficients, and X∈G as the unknown. Certain word equations, such as XAXAX=B, have solutions in terms of radicals: while others such as X2 A X=B do not. We obtain the first known infinite families of word equations not solvable by radicals, and conjecture a complete classification.To a word w, we associate a polynomial Pw∈ℤ[x, y] in two commuting variables, which factors whenever w is a composition of smaller words. We prove that if Pw(x2, y2) has an absolutely irreducible factor in ℤ[x, y], then the equation w(X, A)=B is not solvable in terms of radicals. [ABSTRACT FROM PUBLISHER]

For any field F, there is a relation between the factorization of a polynomial f ∈ F[x1, ..., xn] and the integral decomposition of the Newton polytope of f. We extended this result to polynomial rings R[x1, ..., xn] where R is any ring containing some elements which are not zero-divisors. Moreover, we have constructed some new families of integrally indecomposable polytopes in ℝn giving infinite families of absolutely irreducible multivariate polynomials over arbitrary fields. [ABSTRACT FROM AUTHOR]

Being motivated by some methods for construction of homothetically indecomposable polytopes, weobtain new methods for construction of families of integrally indecomposable polytopes. As a result, wefind new infinite families of absolutely irreducible multivariate polynomials over any field F. Moreover, wep rovide different proofs of some of the main results of Gao [2] [ABSTRACT FROM AUTHOR]

Consider a polynomial with parametric coefficients. We show that the variety of parameters can be represented as a union of strata. For values of the parameters from each stratum, the decomposition of this polynomial into absolutely irreducible factors is given by algebraic formulas depending only on the stratum. Each stratum is a quasiprojective algebraic variety. This variety and the corresponding output are given by polynomials of degrees at most D with D = d′ d where d′, d are bounds on the degrees of the input polynomials. The number of strata is polynomial in the size of the input data. Thus, here we avoid double exponential upper bounds for the degrees and solve a long-standing problem. [ABSTRACT FROM AUTHOR]

In this paper we investigate the factorization behaviour of the binomial polynomials ( x n ) = x (x − 1) ⋯ (x − n + 1) n ! and their powers in the ring of integer-valued polynomials Int (Z). While it is well-known that the binomial polynomials are irreducible elements in Int (Z) , the factorization behaviour of their powers has not yet been fully understood. We fill this gap and show that the binomial polynomials are absolutely irreducible in Int (Z) , that is, ( x n ) m factors uniquely into irreducible elements in Int (Z) for all m ∈ N. By reformulating the problem in terms of linear algebra and number theory, we show that the question can be reduced to determining the rank of, what we call, the valuation matrix of n. A main ingredient in computing this rank is the following number-theoretical result for which we also provide a proof: If n > 10 and n , n − 1 , ..., n − (k − 1) are composite integers, then there exists a prime number p > 2 k that divides one of these integers. [ABSTRACT FROM AUTHOR]

Constructing permutation polynomials is a hot topic in the area of finite fields, and permutation polynomials have many applications in different areas. Recently, several classes of permutation trinomials were constructed. In 2015, Hou surveyed the achievements of permutation polynomials and novel methods. But, very few were known at that time. Recently, many permutation binomials and trinomials have been constructed. Here we survey the significant contribution made to the construction of permutation trinomials over finite fields in recent years. Emphasis is placed on significant results and novel methods. The covered material is split into three aspects: the existence of permutation trinomials of the respective forms xrh(xs), λ1xa + λ2xb + λ3xc and x + xs(qm-1)+1 + xt(qm-1)+1, with Niho-type exponents s, t. [ABSTRACT FROM AUTHOR]

We consider vectorial maps F (x 1 , ... , x n) = (f 1 (x 1 , ... , x n) , ... , f n (x 1 , ... , x n)) : F q n ↦ F q n , which induce permutations of F q n. We show that the degrees of the components f 1 , f 2 , ... , f n ∈ F q [ x 1 , ... , x n ] are at least 2 when 2 ≤ deg (F) = d < q and d | (q − 1). Our proof uses an absolutely irreducible curve over F q and the number of rational points on it that we relate to the cardinality of the value set of a polynomial. [ABSTRACT FROM AUTHOR]

In this paper, we present three algorithms: the first one solves zero-dimensional parametric homogeneous polynomial systems within single exponential time in the number n of unknowns; it decomposes the parameter space into a finite number of constructible sets and computes the finite number of solutions by parametric rational representations uniformly in each constructible set. The second algorithm factirizes absolutely multivariate parametic polynomials within single exponential time in n and in the upper bound d on the degree of the factorized polynomials. The third algorithm decomposes algebraic varieties defined by parametric polynomial systems of positive dimension into absolutely irreducible components uniformly in the values of the parameters. The complexity bound for this algorithm is double exponential in n. On the other hand, the lower bound on the complexity of the problem of resolution of parametric polynomial systems is double exponential in n. Bibliography: 72 titles. [ABSTRACT FROM AUTHOR]

Abstract 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 F q where q is sufficiently large. To facilitate these applications, the absolute irreducibility of certain polynomials in F q [ X , Y ] is established. [ABSTRACT FROM AUTHOR]

Abstract: We present a Las Vegas algorithm for finding a nontrivial reduction of groups that are irreducible with m generators and either lie in the subfield class of matrix or projective groups or are semilinear or have non-absolutely irreducible derived group. Let denote the cost of producing a random element from a matrix algebra and denote the cost of producing a random element in the normal closure of a group H by a group G. Then the algorithm runs in finite field operations. We also characterise the absolutely irreducible groups G over arbitrary fields whose derived group consists only of scalars, and prove probabilistic generation results about matrix groups. [Copyright &y& Elsevier]

Dans cette note, nous etablissons le lien entre le nombre de facteurs absolument irreductibles d'un polynome et la dimension d'un espace vectoriel de formes differentielles fermees. Notre resultat generalise un theoreme de Gao. In this note, we establish the relationship between the number of absolutely irreducible factors of a polynomial and the dimension of a space of closed differentials forms. Our result generalizes a theorem of Gao. [ABSTRACT FROM AUTHOR]

Abstract: I present some results towards a complete classification of monomials that are Almost Perfect Nonlinear (APN), or equivalently differentially 2-uniform, over for infinitely many positive integers n. APN functions are useful in constructing S-boxes in AES-like cryptosystems. An application of a theorem by Weil [A. Weil, Sur les courbes algébriques et les variétés qui s''en déduisent, in: Actualités Sci. Ind., vol. 1041, Hermann, Paris, 1948] on absolutely irreducible curves shows that a monomial is not APN over for all sufficiently large n if a related two variable polynomial has an absolutely irreducible factor defined over . I will show that the latter polynomial''s singularities imply that except in three specific, narrowly defined cases, all monomials have such a factor over a finite field of characteristic 2. Two of these cases, those with exponents of the form or for any integer k, are already known to be APN for infinitely many fields. The last, relatively rare case when a certain gcd is maximal is still unproven; my method fails. Some specific, special cases of power functions have already been known to be APN over only finitely many fields, but they also follow from the results below. [Copyright &y& Elsevier]

An algorithm is constructed for the absolute factorization of polynomials with algebraically independent parametric coefficients. It divides the parameter space into pairwise disjoint pieces such that the absolute factorization of polynomials with coefficients in each piece is given uniformly. Namely, for each piece there exist a positive integer l ≤ d, l variables C1, ..., Cl algebraically independent over the ground field F, and rational functions bJ,j of the parameters and of the variables C1, ..., Cl such that for any parametric polynomial f with coefficients in this piece, there exist c1, ..., $$c_l \in \bar F$$ with f = Π j G j where G j = Σ| J| B J,j Z J is absolutely irreducible. Here Z = (Z0, ..., Zn) are the variables of f, each BJ,j is the value of bJ,j at the coefficients of f and c1, ..., cl, and $$\bar F$$ denotes the algebraic closure of F. The number of pieces does not exceed (2d2+1)2n+3d+5, and the algorithm performs $$d^{O(ndr^2 )} $$ arithmetic operations in F (thus the number of operations is exponential in the number r = () of coefficients of f), and its binary complexity is bounded by $$d^{O(ndr^2 )} $$ if F = ℚ and by $$\left( {pd^{ndr^2 } } \right)^{O(1)} $$ if $$F = \mathbb{F}_p $$ , where d is an upper bound on the degrees of polynomials. The techniques used include the Hensel lemma and the quantifier elimination in the theory of algebraically closed fields. Bibliography: 20 titles. [ABSTRACT FROM AUTHOR]

The concept of a composed product for univariate polynomials has been explored oextensively by Brawley, Brown, Carlitz, Gao, Mills et al. Starting with these fundamental ideas and using fractional power series representation of bivariate polynomials, the current authors Mills and Neuerburg [Mills, D., Neuerburg, K. M. A bivariate analogue to the composed product of polynomials. Algebra Colloquium (to appear)] generalize the univariate results by defining and investigating a bivariate composed sum, composed multiplication, and composed product (based on function composition) for certain classes of bivariate polynomials. In this, the sequel, we extend the generalizations to certain classes of multivariate polynomials, written as f(x1,…,xn), over an algebraically closed field of characteristic zero. In particular, we consider a composed sum and composed multiplication defined on the class of quasiordinary polynomials. We then consider what algebraic structure is imposed by these operations. Next, we consider a generalization of the geometry associated with the class of ν-quasiordinary polynomials. Finally, we utilize this geometry to show that an algorithm given by Gao and Lauder [Gao, S., Lauder, A. G. B. (2001). Decomposition of polytopes and polynomials. Discrete Comput. Geom. 26(1):89–104] to determine whether a given bivariate polynomial is absolutely irreducible over a given field can be used to ascertain whether a given trivariate polynomial that factors completely in one variable decomposes according to the composed multiplication operation. [ABSTRACT FROM AUTHOR]

The Stöhr–Voloch approach is used to obtain a new bound for the number of solutions in (Fq)2 of an equation f(X, Y) = 0, where f(X, Y) is an absolutely irreducible polynomial with coefficients in a finite field Fq. [ABSTRACT FROM PUBLISHER]

Let F q 2 be the finite field with q 2 elements. We provide a simple and effective method, using reciprocal polynomials, for the construction of algebraic curves over F q 2 with many rational points. The curves constructed are Kummer covers or fiber products of Kummer covers of the projective line. Further, we compute the exact number of rational points for some of the curves. [ABSTRACT FROM AUTHOR]

Fix an arbitrary finite group A of order a, and let X(n, q) denote the set of homomorphisms from A to the finite general linear group GLn(q). The size of X(n, q) is a polynomial in q. In this note, it is shown that generically this polynomial has degree n2(1 – a−1) − εr and leading coefficient mr, where εr and mr are constants depending only on r := n mod a. We also present an algorithm for explicitly determining these constants. [ABSTRACT FROM PUBLISHER]

A proof is given of several conjectures from a recent paper of Nakaya concerning the supersingular polynomial s s p (N ∗) (X) for the Fricke group Γ 0 ∗ (N) , for N ∈ { 2 , 3 , 5 , 7 }. One of these conjectures gives a formula for the square of s s p (N ∗) (X) (mod p) in terms of a certain resultant, and the other relates the primes p for which s s p (N ∗) (X) splits into linear factors (mod p) to the orders of certain sporadic simple groups. [ABSTRACT FROM AUTHOR]

We define and study a class of Reed–Muller type error-correcting codes obtained from elementary symmetric functions in finitely many variables. We determine the code parameters and higher weight spectra in the simplest cases. [ABSTRACT FROM AUTHOR]

We generalize two results in the papers [1] and [2] about sums of subsets of to the more general case in which the sum is replaced by , where is a rather general polynomial. In particular, a lower bound is obtained for the cardinality of the range of , where the variables and belong to a subgroup of the multiplicative group of the field . We also prove that if a subgroup can be represented as the range of a polynomial for and , then the cardinalities of and are close in order to . [ABSTRACT FROM AUTHOR]

We prove an analogue of Lang's conjecture on divisible groups for polynomial dynamical systems over number fields. In our setting, the role of the divisible group is taken by the small orbit of a point $\alpha$ where the small orbit by a polynomial $f$ is given by \begin{align*} \mathcal{S}_\alpha = \{\beta \in \mathbb{C}; f^{\circ n}(\beta) = f^{\circ n}(\alpha) \text{ for some } n \in \mathbb{Z}_{\geq 0}\}. \end{align*} Our main theorem is a classification of the algebraic relations that hold between infinitely many pairs of points in $\mathcal {S}_\alpha$ when everything is defined over the algebraic numbers and the degree $d$ of $f$ is at least 2. Our proof relies on a careful study of localisations of the dynamical system and follows an entirely different approach than previous proofs in this area. In particular, we introduce transcendence theory and Mahler functions into this field. Our methods also allow us to classify all algebraic relations that hold for infinitely many pairs of points in the grand orbit \begin{align*} \mathcal{G}_\alpha = \{\beta \in \mathbb{C}; f^{\circ n}(\beta) = f^{\circ m}(\alpha) \text{ for some } n ,m\in \mathbb{Z}_{\geq 0}\} \end{align*} of $\alpha$ if $|f^{\circ n}(\alpha)|_v \rightarrow \infty$ at a finite place $v$ of good reduction co-prime to $d$. [ABSTRACT FROM AUTHOR]

An argument is given to fill a gap in a proof in the author’s article On certain algebraic curves related to polynomial maps, Compositio Math. 103 (1996), 319–350, that the polynomial Φn(x,c), whose roots are the periodic points of period n of a certain polynomial map x→f(x,c), is absolutely irreducible over the finite field of p elements, provided that f(x,1) has distinct roots and that the multipliers of the orbits of period n are also distinct over $\mathbb { F}_p$. Assuming that Φn(x,c) is reducible in characteristic p, we show that Hensel’s lemma and Laurent series expansions of the roots can be used to obtain a factorization of Φn(x,c) in characteristic 0, contradicting the absolute irreducibility of this polynomial over the rational field. [ABSTRACT FROM AUTHOR]

The main goal of this work is to answer a question of Dèbes and Haran by relaxing the condition for Hilbertianity. Namely we prove that for a field K to be Hilbertian it suffices that K has the irreducible specialization property merely for absolutely irreducible polynomials. [ABSTRACT FROM PUBLISHER]

An introduction to the part III of the book "Foundations and Trends in Theoretical Computer Science" is presented which discusses various topics such as the definition of the "Identity Testing" problem of testing if the available polynomial C(X) is identically zero, the possibility that C(X) is absolutely irreducible, and the polynomial equivalence problem.

We prove a GL(n)×GL(n-1) local converse theorem for ℓ-adic families of smooth representations of GLn(F) where F is a finite extension of Qp and ℓ≠p. Along the way, we extend the theory of Rankin-Selberg integrals, first introduced in Jacquet et al. [20], to the setting of families, continuing previous work of the author [23] [ABSTRACT FROM AUTHOR]

We classify all permutation polynomials of the form x 3 g (x q - 1) of F q 2 where g (x) = x 3 + b x + c and b , c ∈ F q ∗ . Moreover we find new examples of permutation polynomials and we correct some contradictory statements in the recent literature. We assume that gcd (3 , q - 1) = 1 and we use a well known criterion due to Wan and Lidl, Park and Lee, Akbary and Wang, Wang, and Zieve. [ABSTRACT FROM AUTHOR]

Let G be a simple linear algebraic group over a field F , and V an absolutely irreducible representation of G . We show that under some mild hypotheses there exists an invariant homogeneous polynomial f for the action of G on V defined over F , such that twisted forms of f up to a scalar multiple classify twisted forms of G for which the representation V is defined over F . This result extends the classical case of a quadratic form q and its orthogonal group O ( q ) . [ABSTRACT FROM AUTHOR]

In this article, we prove several new uniform upper bounds on the number of points of bounded height on varieties over Fq[t]$\mathbb {F}_q[t]$. For projective curves, we prove the analogue of Walsh' result with polynomial dependence on q$q$ and the degree d$d$ of the curve. For affine curves, this yields an improvement to bounds by Sedunova, and Cluckers, Forey and Loeser. In higher dimensions, we prove a version of dimension growth for hypersurfaces of degree d⩾64$d\geqslant 64$, building on work by Castryck, Cluckers, Dittmann and Nguyen in characteristic zero. These bounds depend polynomially on q$q$ and d$d$, and it is this dependence which simplifies the treatment of the dimension growth conjecture. [ABSTRACT FROM AUTHOR]