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Let n be a positive integer different from 8 and n + 1 ≠ 2 u for any integer u ≥ 2 . Let ϕ (x) belonging to Z [ x ] be a monic polynomial which is irreducible modulo all primes less than or equal to n + 1 . Let a j (x) with 0 ≤ j ≤ n - 1 belonging to Z [ x ] be polynomials having degree less than deg ϕ (x) . Assume that the content of a n a 0 (x) is not divisible by any prime less than or equal to n + 1 . We prove that the polynomial f (x) = a n ϕ (x) n (n + 1) ! + ∑ j = 0 n - 1 a j (x) ϕ (x) j (j + 1) ! is irreducible over the field Q of rational numbers. This generalises a well-known result of Schur which states that the polynomial ∑ j = 0 n a j x j (j + 1) ! with a j ∈ Z and | a 0 | = | a n | = 1 is irreducible over Q . For proving our results, we use the notion of ϕ -Newton polygons and a few results on primes from number theory. We illustrate our result through examples. [ABSTRACT FROM AUTHOR]

Given a planar polynomial vector field X with a fixed Newton polytope P , we prove (under some non-degeneracy conditions) that the monomials associated to the upper boundary of P determine (under topological equivalence) the phase portrait of X in a neighborhood of boundary of the Poincaré–Lyapunov disk. This result can be seen as a version of the well known result of Berezovskaya, Brunella and Miari [2,5] for the dynamics at infinity. We also discuss the non-existence of periodic orbits near infinity via Newton polytope, as well as the effect of the Poincaré–Lyapunov compactification on P. [ABSTRACT FROM AUTHOR]

AbstractLet K=Q(α), where α is a root of the monic irreducible polynomial x2·3s·qr−m with m≠±1 is a square-free integer, r and s are two positive integers, and q is a prime of the form 3k+2. In this article, we study the monogenity of the number field K and establish conditions on m for K to be monogenic. Further, we provide sufficient conditions on r, s and m for K to be not monogenic. Finally, we illustrate our results with examples. [ABSTRACT FROM AUTHOR]

In this paper, for any rational prime p and for a fixed positive integer ν p , we provide infinite families of number fields K satisfying ν p (i (K)) = ν p . As an application, if i (K) ≠ 1 , then K is not monogenic. We illustrate our results by some computational examples. [ABSTRACT FROM AUTHOR]

AbstractAssume that P and Q are elements of A1 satisfying [P,Q]=1. The Dixmier Conjecture for A1 says that they always generate A1. We show that if P is a sum of not more than 4 homogeneous elements of A1 then P and Q generate A1, which generalizes the main result in [10]. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we calculate the index of any septic number field K generated by a root α of a monic irreducible trinomial F(x) = x7 + ax4 + b ∈ ℤ[x]. Our approach is based on Engstrom’s results and the factorization of any rational prime in K. In such a way we give a complete answer of Problem 22 of Narkiewicz ([28]) for this family of number fields. As an application of our results, if i(K) ≠ 1, then K is not monogenic. Also, we give generators of power integral bases in some cases where i(K) = 1. Our results are illustrated by some computational examples. [ABSTRACT FROM AUTHOR]

There is a proposition due to Kollár as reported by Kollár (Proceedings of the summer research institute, Santa Cruz, CA, USA, July 9–29, 1995, American Mathematical Society, Providence, 1997) on computing log canonical thresholds of certain hypersurface germs using weighted blowups, which we extend to weighted blowups with non-negative weights. Using this, we show that the log canonical threshold of a convergent complex power series is at most 1/c, where (c , ... , c) is a point on a facet of its Newton polyhedron. Moreover, in the case n = 2 , if the power series is weakly normalised with respect to this facet or the point (c, c) belongs to two facets, then we have equality. This generalises a theorem of Varchenko 1982 to non-isolated singularities. [ABSTRACT FROM AUTHOR]

In this paper, for any nonic number field generated by a root of a monic irreducible trinomial and for every rational prime , we characterize when divides the index of . We also describe the prime power decomposition of the index . In such a way we give a partial answer of Problem of Narkiewicz [23] for this family of number fields. As an application of our results, if , then is not monogenic. We illustrate our results by some computational examples. [ABSTRACT FROM AUTHOR]

The first order eigenvalue perturbation theory for structure-preserving perturbations of real or complex symplectic matrices started in [21] is continued and completed. While in [21] all eigenvalues were considered that qualitatively behave similarly under structure-preserving and structure-ignoring perturbations, the focus of this paper is on one particular case when the behavior under structure-preserving perturbations differs significantly from the one under structure-ignoring perturbations. This case occurs for the exceptional eigenvalues ±1 and is caused by restrictions in the possible Jordan structure of symplectic matrices. The main result gives asymptotic expansions of specific perturbed eigenvalues that are not yet covered by the results from [21]. While the main theorem requires that the rank of the perturbation is equal to the rank of its first order term, it is also discussed what can be said if this hypothesis is not satisfied. [ABSTRACT FROM AUTHOR]

Let p and q, where p q − q p = 1 , be the standard generators of the first Weyl algebra A1 over a field of characteristic zero. Then the spectrum of the inner derivation ad(pq) on A1 are exactly the set of integers. The algebra A1 is a Z -graded algebra with each i-component being the i-eigenspace of ad(pq), where i ∈ Z . Assume that z and w are elements of A1 satisfying z w − w z = 1 . The Dixmier Conjecture for A1 says that they always generate A1. We show that if z possesses no component belonging to the negative spectrum of ad(pq), then z and w generate A1. We give some generalization of this result, and some other useful criterions for z and w to generate A1. It is shown that if z is a sum of not more than 2 homogeneous elements of A1 then z and w generate A1, which generalizes a known result due to Bavula and Levandovskyy. [ABSTRACT FROM AUTHOR]

Let p and q, where p q − q p = 1 , be the standard generators of the first Weyl algebra A1 over a field of characteristic zero. Then the spectrum of the inner derivation ad(pq) on A1 are exactly the set of integers. The algebra A1 is a Z -graded algebra with each i-component being the i-eigenspace of ad(pq), where i ∈ Z . Assume that z and w are elements of A1 satisfying z w − w z = 1 . The Dixmier Conjecture for A1 says that they always generate A1. We show that if z possesses no component belonging to the negative spectrum of ad(pq), then z and w generate A1. We give some generalization of this result, and some other useful criterions for z and w to generate A1. It is shown that if z is a sum of not more than 2 homogeneous elements of A1 then z and w generate A1, which generalizes a known result due to Bavula and Levandovskyy. [ABSTRACT FROM AUTHOR]

Let q be a prime number and K = Q () be an algebraic number field with a root of an irreducible polynomial x q s − a x − b having integer coefficients. In this paper, we provide some explicit conditions involving only s , q , a , b for which K is non-monogenic. As an application, in the special case of a = 0 and q = 2 , we show that if s ≥ 2 and 3 2 divides b − 1 , then K is not monogenic. We illustrate our results through examples. [ABSTRACT FROM AUTHOR]

Let K = Q(a) be a pure number field generated by a root a of a monic irreducible polynomial xpr - a Z[x], where p is a rational prime and r is a positive integer. Let ZK be the ring of integers of K. In this paper, we calculate an integral basis of ZK and study the monogenity of K in some particular cases. [ABSTRACT FROM AUTHOR]

In this paper, we study the Newton polygon of the L -function of a generalized Kloosterman polynomial with two variables over finite fields. We give the explicit form of the monomial basis of the top dimensional cohomology space of the p -adic complex associated to the L -function. One consequence of this result is a concrete method for computing the Hodge polygon. Using decomposition theorems of Wan, we determine when a generalized Kloosterman polynomial is ordinary and when its Newton polytope is generically ordinary. [ABSTRACT FROM AUTHOR]

In this paper, we study the monogenity of any number field defined by a monic irreducible trinomial F (x) = x 12 + a x m + b ∈ Z [ x ] with 1 ≤ m ≤ 11 an integer. For every integer m, we give sufficient conditions on a and b so that the field index i(K) is not trivial. In particular, if i (K) ≠ 1 , then K is not monogenic. For m = 1 , we give necessary and sufficient conditions on a and b, which characterize when a rational prime p divides the index i(K). For every prime divisor p of i(K), we also calculate the highest power p dividing i(K), in such a way we answer the problem 22 of Narkiewicz (Elementary and analytic theory of algebraic numbers, Springer Verlag, Auflag, 2004) for the number fields defined by trinomials x 12 + a x + b . [ABSTRACT FROM AUTHOR]

In this paper, for any rational prime p and for a fixed positive integer ip, we provide infinite families of number fields defined by irreducible quintinomials of type xn+axm+bx²+cx+d in Z[x] satisfying νp(i(K))=ip. We illustrate our results by some computational examples. [ABSTRACT FROM AUTHOR]

Let K be a number field generated by a complex root θ of an irreducible trinomial F(x) = x5 + ax³ + b ∈ Z[x], where ab = 0. In this paper, we provide some explicit conditions on a and b for which 2 divides the index of K. We give a necessary and sufficient condition for which 3 divides the index of K. As an application of our results, we provide some infinite parametric families of number fields which are non-monogenic. [ABSTRACT FROM AUTHOR]

In this paper, we give a new proof using Newton polygons for an irreducibility criterion established by Singh and Kumar. Then we prove a conjecture of Koley and Reddy. In addition, we give a new proof for the statement that x n ± c x n − 1 ± c with c ⩾ 2 is irreducible unless (n , c) = (2 , 4) , which is a conjecture of Harrington. [ABSTRACT FROM AUTHOR]

In this article, we deal with the problem of monogenity of quintic number fields defined by monic irreducible quadrinomials F (x) = x 5 + a x m + b x + c ∈ Z [ x ] with m = 2, 3, 4. We give sufficient conditions on a, b, c, and m so that the index of the field K is nontrivial and we evaluate ν p (i (K)) in each case. In particular, if i (K) ≠ 1 , then K is not monogenic. Our results are illustrated by computational examples. [ABSTRACT FROM AUTHOR]

In this article, we are interested in the Gevrey properties of the formal power series solutions in time of some inhomogeneous nonlinear partial differential equations with analytic coefficients at the origin of Cn+1. We systematically examine the cases where the inhomogeneity is s-Gevrey for any s ≥ 0, in order to carefully distinguish the influence of the data (and their degree of regularity) from that of the equation (and its structure). We thus prove that we have a noteworthy dichotomy with respect to a nonnegative rational number sc fully determined by the Newton polygon of a convenient associated linear partial differential equation: for any s ≥ sc, the formal solutions and the inhomogeneity are simultaneously s-Gevrey; for any s < sc, the formal solutions are generically sc-Gevrey. In the latter case, we give an explicit example in which the solution is s 0 -Gevrey for no s 0 < sc. As a practical illustration, we apply our results to the generalized Burgers-Korteweg-de Vries equation. [ABSTRACT FROM AUTHOR]

In this paper, for any septic number field K generated by a root α of a monic irreducible trinomial F (x) = x 7 + a x 3 + b ∈ Z [ x ] , we describe all prime power divisors of the index of K answering Problem 22 of Narkiewicz [ 26]. In particular, if i (K) ≠ 1 , then K is not mongenic. We illustrate our results by some computational examples. [ABSTRACT FROM AUTHOR]

A quadrature formula is a formula computing a definite integration by evaluation at finite points. The existence of certain quadrature formulas for orthogonal polynomials is related to interesting problems such as Waring's problem in number theory and spherical designs in algebraic combinatorics. Sawa and Uchida proved the existence and the non-existence of certain rational quadrature formulas for the weight functions of certain classical orthogonal polynomials. Classical orthogonal polynomials belong to the Askey-scheme, which is a hierarchy of hypergeometric orthogonal polynomials. Thus, it is natural to extend the work of Sawa and Uchida to other polynomials in the Askey-scheme. In this article, we extend the work of Sawa and Uchida to the weight function of the Bessel polynomials. In the proofs, we use the Riesz–Shohat theorem and Newton polygons. It is also of number theoretic interest that proofs of some results are reduced to determining the sets of rational points on elliptic curves. [ABSTRACT FROM AUTHOR]

We study connections between polynomials which are close to each other, i.e., whose respective coefficients are close in the topology induced by a valuation. This paper consists of both an overview on known root continuity theorems and new results on the subject. We present theorems which have been published over the years, correcting and improving some of their formulations and proofs. We also give a sketch of several approaches to root continuity, such as employing the Newton Polygon and induction on the degree of the polynomial. We study the behavior of the irreducible factors of a given polynomial, the extensions generated by its roots and invariants connected to that polynomial under transition to a second polynomial which is sufficiently close. Further, we present applications of root continuity to the study of valued field extensions and ramification theory, including the theory of the defect. [ABSTRACT FROM AUTHOR]

In this article we characterize the foliations that have the same Newton polygon that their union of formal separatrices, they are the foliations called of the second type. In the case of cuspidal foliations studied by Loray [Lo], we precise this characterization using the Poincaré-Hopf index. This index also characterizes the cuspidal foliations having the same process of singularity reduction that the union of its separatrices. Finally we give necessary and sufficient conditions when these cuspidal foliations are generalized curves, and a characterization when they have only one separatrix.

In this article, we investigate the summability of the formal power series solutions in time of a class of inhomogeneous nonlinear partial differential equations in two variables, whose corresponding Newton polygon admits a unique positive slope k , the latter being determined by the highest spatial-derivative order of the initial equation. We give in particular a necessary and sufficient condition for the k -summability of the solutions in a given direction, and we illustrate this result by some examples. This condition generalizes the ones already given by the second author in Remy (2016, 2020, 2021 [25,26], 2022, 2023). In addition, we present some technical results on the generalized binomial and multinomial coefficients, which are needed for the proof of our main result. [ABSTRACT FROM AUTHOR]

We introduce and develop (π 1 / c , p) -adic Dwork theory for L -functions of exponential sums associated to one-variable rational functions, interpolating p k -order exponential sums over affinoids. Namely, we prove a generalization of the Dwork-Monsky-Reich trace formula and apply it to establish an analytic continuation of the C -function C f (s , π). We compute the lower (π 1 / c , p) -adic bound, the Hodge polygon, for this C -function. Along the way, we also show why a strictly π -adic theory will not work in this case. [ABSTRACT FROM AUTHOR]

In this paper, we compute the q -adic slopes of the L-functions of an important class of exponential sums arising from analytic number theory. Our main tools include Adolphson-Sperber's work on toric exponential sums and Wan's decomposition theorems. This paper is a generalization of our previous paper. [ABSTRACT FROM AUTHOR]

Let be a pure number field generated by a root of a monic irreducible polynomial , where is a square free integer, , , and are three positive integers. In this paper, we study the monogenity of . We prove that if , , and , then is monogenic. But if and or , , and or and or , , and , then is not monogenic. Some illustrating examples will be given. [ABSTRACT FROM AUTHOR]

The zeroes of Goss polynomials G k , Λ (X) for Λ = A : = F q [ T ] and similar lattices Λ are studied. Generically, the zero distribution follows a simple pattern governed by the q -adic expansion of k − 1. However, if q = p f with f ≥ 2 is a proper power of the prime p , irregularities of the q -adic sum-of-digits function may lead to deviations from this pattern, to irregular zeroes, and an abundance of trivial zeroes of G k , Λ compared to the generic formula. These phenomena are related to properties of the Sheats compositions of natural numbers n divisible by q − 1. Among other things, we give a necessary and sufficient condition for the existence of irregular zeroes of G k , A and a formula for the vanishing order of G k , A at x = 0. [ABSTRACT FROM AUTHOR]

We prove that for certain polynomial differential equations in the plane arising from predator-prey type III models with generalized rational functional response, any algebraic solution should be a rational function. As a consequence, limit cycles, which are unique for these dynamical systems, are necessarily trascendental ovals. We exemplify these findings by showing a numerical simulation within a system arising from zooplankton-phytoplankton dynamics.

Let f be a non-degenerate Laurent polynomial over a finite field and L ⁎ (f , T) the associated L -function of the toric exponential sums of f. The Newton polygon can be used to study the p -adic valuations of roots or poles of L ⁎ (f , T) , which has a lower bound called the Hodge polygon that depends only on the Newton polytope of f. These two polygons coincide when the coefficients of f are not zeros of the certain Hasse polynomial. Using the Dwork trace formula, we find a formula for the Hasse polynomial of the slope one side for a class of Laurent polynomials, which generalizes the results of Zhang, Feng and Chen. [ABSTRACT FROM AUTHOR]

Bateman and Duquette have initiated the study of Salem elements in positive characteristic. Later, Kerada introduced and studied j-Salem numbers (j ≥ 2). To complete this circle of ideas we introduce 2-Salem elements over the field of formal Laurent series over a finite field q . We extend the above-cited studies and we provide some analog results. The main result of this note is Theorem 1.2 which characterizes the set T 2 ′ ∗ of 2-Salem series when q ≠ 2 r . [ABSTRACT FROM AUTHOR]

We show that for some recursive sequence (c m) m ≥ 1 of integers and for sufficiently large n, the Galois group of polynomial f n (x) = x n n ! + c n - 1 x n - 1 (n - 1) ! + ⋯ + c 1 x 1 ! + 1 , contains the alternating group A n . In case n is a prime number, this group is the full symmetric group S n . [ABSTRACT FROM AUTHOR]

In this paper we shall give another proof of a special case of Gao's theorem for generating integrally indecomposable polygons in the sense of Minkowski. The approach of proving this theorem will enable us to give an effective algorithm for construction integrally indecomposable convex integral polygons with arbitrary many vertices. In such a way, classes of absolute irreducible bivariate polynomials corresponding to those indecomposable Newton polygons are generated. [ABSTRACT FROM AUTHOR]

We consider a nonlinear ordinary differential equation of arbitrary order with coefficients in the form of power series that converge in a neighborhood of the origin. The methods created in power geometry in recent years make it possible to compute formal solutions to that equation in the form of Dulac series. We describe the corresponding algorithm and prove a sufficient convergence condition for such formal solutions. [ABSTRACT FROM AUTHOR]

Let p ≥ 3 be a prime number. In this article, we study the canonical expansion of the primitive p n -th root of unity ζ p n in p-adic Mal'cev–Neumann field L p for n ≥ 1 . More precisely, we give the explicit formula for the first ℵ 0 terms of the expansion of ζ p n , and as an application, we use it to construct a uniformizer of K 2 , m = Q p ζ p 2 , p 1 / p m with m ≥ 1 . [ABSTRACT FROM AUTHOR]

Assume that f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_0, a_0\neq 0 is a polynomial with rational coefficients and let p be a prime number whose highest power r_i dividing a_i (where r_i = \infty if a_i = 0) satisfies r_n = 0, r_i \geq 1 for 0\leq i\leq n-1. In this article, we explicitly construct a number d depending only on r_i's and show that each irreducible factor of f(x) has degree at least d over \mathbf {Q}. This result extends the famous Eisenstein-Dumas irreducibility criterion. In fact, we prove our result in a more general setup for polynomials with coefficients in an arbitrary valued field. [ABSTRACT FROM AUTHOR]

We consider a large class of nonlinear ordinary differential equations of arbitrary order with coefficients in the form of power series that converge in a neighborhood of the origin. There are known power-geometry methods and algorithms based on them for the computation of power-logarithmic series (Dulac series) that formally satisfy such equations. We prove a sufficient condition for the convergence of such formal solutions. [ABSTRACT FROM AUTHOR]

The zero locus of a bivariate polynomial P(x,y)=0 defines a compact Riemann surface Σ. The fundamental second kind differential is a symmetric 1 ⊗ 1-form on Σ × Σ that has a double pole at coinciding points and no other pole. As its name indicates, this is one of the most important geometric objects on a Riemann surface. Here we give a rational expression in terms of combinatorics of the Newton's polygon of P, involving only integer combinations of products of coefficients of P. Since the expression uses only combinatorics, the coefficients are in the same field as the coefficients of P. [ABSTRACT FROM AUTHOR]

In this article, we investigate the summability of the formal power series solutions in time of a class of inhomogeneous nonlinear partial differential equations in two variables, whose the attached Newton polygon admits a unique positive slope k , the latter being determined by the highest spatial-derivative order of the initial equation. We give in particular a necessary and sufficient condition for the k -summability of the solutions in a given direction, and we illustrate this result by some examples. This condition generalizes the ones already given by the author in the linear case [1,2] and, more recently, in the semilinear case [3,4]. In addition, we present some technical results on the generalized binomial and multinomial coefficients, which are needed for the proof of our main result. [ABSTRACT FROM AUTHOR]

We provide an irreducibility test in the ring K [ [ x ] ] [ y ] whose complexity is quasi-linear with respect to the discriminant valuation, assuming the input polynomial F is square-free and K is a perfect field of characteristic not dividing deg (F) . The algorithm uses the theory of approximate roots and may be seen as a generalisation of Abhyankar's irreducibility criterion to the case of non-algebraically closed residue fields. [ABSTRACT FROM AUTHOR]

In this paper, we completely determine the slopes and weights of the L-functions of an important class of exponential sums arising from analytic number theory. Our main tools include Adolphson–Sperber's work on toric exponential sums and Wan's decomposition theorems. One consequence of our main result is a sharp estimate of these exponential sums. Another consequence is to obtain an explicit counterexample of Adolphson–Sperber's conjecture on weights of toric exponential sums. [ABSTRACT FROM AUTHOR]

Let f(x) = xn -- axm -- b be a monic irreducible polynomial of degree nhaving integer coefficients. Let K = Q(θ) be an algebraic number field with θ a root of f(x). In this paper, we provide some explicit conditions involving only a,b,m,n for which K is not monogenic. Further, as an application, in a special case, we show that if p is a prime number of the form 32k +1, k ∈ Z and θ is a root of a monic polynomial x32n -- 64axm -- p with 2 + n p|a Q(θ) is not monogenic. [ABSTRACT FROM AUTHOR]