Narrow Results

We give an algorithm to compute term-by-term multivariate Puiseux series expansions of series arising as local parametrizations of zeroes of systems of algebraic equations at singular points. The algorithm is an extension of Newton's method for plane algebraic curves, replacing the Newton polygon by the tropical variety of the ideal generated by the system. As a corollary we deduce a property of tropical varieties of quasi-ordinary singularities.

In this paper, we study polar quotients and Łojasiewicz exponents of plane curve singularities, which are not necessarily reduced. We first show that, for complex plane curve singularities, the set of polar quotients is a topological invariant. We next prove that the Łojasiewicz gradient exponent can be computed in terms of the polar quotients, and so it is also a topological invariant. For real plane curve singularities, we also give a formula computing the Łojasiewicz gradient exponent via real polar branches. As an application, we give effective estimates of the Łojasiewicz exponents in the gradient and classical inequalities of polynomials in two (real or complex) variables.

We show that the -character variety of the (-2, 3, n) pretzel knot consists of two (respectively three) algebraic curves when 3 ∤ n (respectively 3 | n) and given an explicit calculation of the Culler-Shalem seminorms of these curves. Using this calculation, we describe the fundamental polygon and Newton polygon for these knots and give a list of Dehn surgerise yielding a manifold with finite or cyclic fundamental group. This constitutes a new proof of property P for these knots.

This paper reviews the two variable polynomial invariant of knots defined using representations of the fundamental group of the knot complement into . The slopes of the sides of the Newton polygon of this polynomial are boundary slopes of incompressible surfaces in the knot complement. The polynomial also contains information about which surgeries are cyclic, and about the shape of the cusp when the knot is hyperbolic. We prove that at least some mutants have the same polynomial, and that most untwisted doubles have non-trivial polynomial. We include several open questions.

Defectless irreducible polynomials over a Henselian valued field (F, v) have been studied by means of strict systems of polynomial extensions and complete (also called "saturated") distinguished chains. Strong connections are developed here between these two approaches and applications made to both. In the tame case in which a root α of an irreducible polynomial f generates a tamely ramified extension of (F, v), simple formulas are given for the Krasner constant, the Brink separant and the diameter of f. In this case a (best possible) result is given showing that a sufficiently good approximation in an extension field K of F to a root of a defectless polynomial f over F guarantees the existence of an exact root of f in K. Also in the tame case a (best possible) result is given describing when a polynomial is sufficiently close to a defectless polynomial so as to guarantee that the roots of the two polynomials generate the same extension fields. Another application in the tame case gives a simple characterization of the minimal pairs (in the sense of N. Popescu et al.). A key technical result is a computation in the tame case of the Newton polygon of f(x+α). Invariants of defectless polynomials are discussed and the existence of defectless polynomials with given invariants is proven. Khanduja's characterization of the tame polynomials whose Krasner constants equal their diameters is generalized to arbitrary defectless polynomials. Much of the work described here will be seen not to require the hypothesis that (F, v) is Henselian.

Let p be a prime number. In this paper we use an old technique of Ø. Ore, based on Newton polygons, to construct in an efficient way p-integral bases of number fields defined by p-regular equations. To illustrate the potential applications of this construction, we derive from this result an explicit description of a p-integral basis of an arbitrary quartic field in terms of a defining equation.

Let $(K,\nu )$ be a valued field, where $\nu$ is a rank-one discrete valuation, with valuation ring $R$. The goal of this paper is to investigate some basic concepts of Newton polygon techniques of a monic polynomial $f(x)\in R[x]$; namely, theorem of the product, of the polygon, and of the residual polynomial, in such way that improves that given in [D. Cohen, A. Movahhedi and A. Salinier, Factorization over local fields and the irreducibility of generalized difference polynomials, Mathematika47 (2000) 173–196] and generalizes that given in [J. Guardia, J. Montes and E. Nart, Newton polygons of higher order in algebraic number theory, Trans. Amer. Math. Soc.364(1) (2012) 361–416] to any rank-one valued field.

Let $q$ be a prime number and $K=Q(𝜃)$ be an algebraic number field with $𝜃$ a root of an irreducible polynomial $x^{q^s}-ax-b$ 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 $s\ge 2$ and $32$ divides $b-1$, then $K$ is not monogenic. We illustrate our results through examples.

This paper looks at the L-function of the kth symmetric power of the -sheaf Ai_f over the affine line associated to the generalized Airy family of exponential sums. Using ℓ-adic techniques, we compute the degree of this rational function as well as the local factors at infinity. Using p-adic techniques, we study the q-adic Newton polygon of the L-function.

We provide irreducibility conditions for polynomials of the form f(X) + p^kg(X), with f and g relatively prime polynomials with integer coefficients, deg f < deg g, p a prime number and k a positive integer. In particular, we prove that if k is prime to deg g - deg f and p^k exceeds a certain bound depending on the coefficients of f and g, then f(X) + p^kg(X) is irreducible over ℚ.

Motivated by Schur’s result on computing the Galois groups of the exponential Taylor polynomials, this paper aims to compute the Galois groups of the Taylor polynomials of the elementary functions $1+\mathrm{\text{log}}(1-x)$ and $\mathrm{\text{cos}}x$. We first show that the Galois groups of the $n$th Taylor polynomials of $1+\mathrm{\text{log}}(1-x)$ are as large as possible, namely, $S_n$ (full symmetric group) or $A_n$ (alternating group), depending on the residue of the integer number $n$ modulo $4$. We then compute the Galois groups of the $n$th Taylor polynomials of $\mathrm{\text{cos}}(x)$ and show that these Galois groups essentially coincide with the Coexter groups of type $B_n$ (or an index 2 subgroup of the corresponding Coexter group).

Bateman and Duquette have initiated the study of Salem elements in positive characteristic. Later, Kerada introduced and studied j-Salem numbers ($j\ge 2$). To complete this circle of ideas we introduce 2-Salem elements over the field of formal Laurent series over a finite field ${𝔽}_q$. 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 $T_2^{\prime \ast }$ of 2-Salem series when $q\ne 2^r$.

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.

Many slope filtrations occur in algebraic geometry, asymptotic analysis, ramification theory, p-adic theories, geometry of numbers …. These functorial filtrations, which are indexed by rational (or sometimes real) numbers, have a lot of common properties.

We propose a unified abstract treatment of slope filtrations, and survey how new ties between different domains have been woven by dint of deep correspondences between different concrete slope filtrations.