Related papers: Generic twisted $T$-adic exponential sums of binom…

In this paper we construct a Dwork theory for general exponential sums over affinoids in Witt towers. Using this, we compute the degree of the $L$-function, its Hodge polygon and examine when the Hodge and Newton polygons coincide.

We study Lusztig's t-analog of weight multiplicities associated to level one representations of twisted affine Kac-Moody algebras. An explicit closed form expression is obtained for the corresponding t-string function using constant term…

The paper contains a combinatorial theorem (the sequence of Newton polygons of a reccurent sequence of polynomials is quasi-linear) and two applications of it in classical and quantum topology, namely in the behavior of the $A$-polynomial…

The theory of Touchard polynomials is generalized using a method based on the definition of exponential operators, which extend the notion of the shift operator. The proposed technique, along with the use of the relevant operational…

We define twisted Alexander polynomials of a complex hypersurface with arbitrary singularities. These generalize the classical Alexander polynomials of high dimensional hypersurfaces and the twisted Alexander polynomial of plane curves. We…

In order to give a formal treatment of differential equations in positive characteristic p, it is necessary to use divided powers. One runs into an analog problem in the theory of q-difference equations when q is a pth root of unity. We…

We use Reidemeister torsion to study a twisted Alexander polynomial, as defined by Turaev, for links in the projective space. Using sign-refined torsion we derive a skein relation for a normalized form of this polynomial.

Let X be a smooth algebraic variety over a field K containing the real numbers. We introduce the notion of twisted associative (resp. Poisson) deformation of the structure sheaf of X. These are stack-like versions of usual deformations. We…

The Newton-Girard Formula allows one to write any elementary symmetric polynomial as a sum of products of power sum symmetric polynomials and elementary symmetric polynomials of lesser degree. It has numerous applications. We have…

We calculate the twisted Reidemeister torsion of the complement of an iterated torus knot associated with a representation of its fundamental group to the complex special linear group of degree two. We also show that the twisted…

Some connections between quadratic forms over the field of two elements, Clifford algebras of quadratic forms over the real numbers, real graded division algebras, and twisted group algebras will be highlighted. This allows to revisit real…

This paper derives a way to express differentiable complex-valued functions as the sum of powers of $(1-e^{\lambda x})$, where $\lambda\in\mathbb{R}$, with an explicit formula for the remainder. This formulation is then used to associate an…

We establish a new bound for the exponential sum \begin{eqnarray*} \sum_{x\in\mathcal{X}}\Big|\sum_{y\in \mathcal{Y}}\gamma(y)\exp(2\pi i a \lambda^{xy}/p)\Big|, \end{eqnarray*} where $\lambda$ is an element of the residue ring modulo a…

We introduce and study arithmetic polygons. We show that these arithmetic polygons are connected to triples of square pyramidal numbers. For every odd $N\geq3$, we prove that there is at least one arithmetic polygon with $N$ sides. We also…

We fix a monic polynomial $f(x) \in \mathbb F_q[x]$ over a finite field of characteristic $p$ of degree relatively prime to $p$. Let $a\mapsto \omega(a)$ be the Teichm\"uller lift of $\mathbb F_q$, and let $\chi:\mathbb{Z}\to \mathbb…

In this paper we consider the Newton polygons of $L$-functions coming from additive exponential sums associated to a polynomial over a finite field $\F_q$. These polygons define a stratification of the space of polynomials of fixed degree.…

By polynomial (or extended binomial) coefficients, we mean the coefficients in the expansion of integral powers, positive and negative, of the polynomial $1+t +\cdots +t^{m}$; $m\geq 1$ being a fixed integer. We will establish several…

We develop new closed form representations of sums of (n + {\alpha})th shifted harmonic numbers and reciprocal binomial coefficients in terms of {\alpha}th shifted harmonic numbers. Some interesting new consequences and illustrative…

In this article, we use Robba's method to give an estimate of the Newton polygon for the L function on torus and we can draw the Newton polygon in some special cases.

In the work we have considered p-adic functional series with binomial coefficients and discussed its p-adic convergence. Then we have derived a recurrence relation following with a summation formula which is invariant for rational argument.…