# Sebastian Pauli's research while affiliated with University of North Carolina at Greensboro and other places

## Publications (32)

Preprint
Full-text available
The goal of this note is to improve on the currently available bounds for Stieltjes constants using the method of steepest descent applied by Coffey and Knessl to approximate Stieltjes constants.
Chapter
We present a method for evaluating the reverse Grünwald-Letnikov fractional derivatives of the Riemann Zeta function $$\zeta (s)$$ and use it to explore the location of zeros of integral and fractional derivatives on the left half plane.
Article
We study the non-integral generalized Stieltjes constants γα(a) arising from the Laurent series expansions of fractional derivatives of the Hurwitz zeta functions ζ(α)(s,a), and we prove that if ha(s)≔ζ(s,a)−1∕(s−1)−1∕as and Cα(a)≔γα(a)−logα(a)a, then Cα(a)=(−1)−αha(α)(1),for all real α≥0, where h(α)(x) denotes the α-th Grünwald–Letnikov fractional...
Article
We present an algorithm that finds the splitting field of a polynomial over a local field. Our algorithm is an OM algorithm modified for this task.
Article
We describe new zero-free regions for the derivatives ζ(κ)(s) of the Riemann zeta function, which take the form of vertical strips in the right half-plane. We show that the zeros located in the narrow complements of these zero-free regions are simple and exhibit vertical periodicities that enable one to give exact formulas for their number.
Article
We give an algorithm that constructs a minimal set of polynomials defining all extension of a $(\pi)$-adic field with given, inertia degree, ramification index, discriminant, ramification polygon, and residual polynomials of the segments of the ramification polygon.
Article
We present the discovery of many previously unknown zeros of the derivatives, ζ (k)(σ + it), of the Riemann zeta function for k ≤ 28 with $$-10 <\sigma < \frac{1} {2}$$ and − 10 < t < 10. Each zero found was simple and our computations show an interesting behavior of the zeros of ζ (k), namely they seem to lie on curves which are extensions of cert...
Article
We give an overview of methods used to track moving objects in video and describe how information about animal behavior can be extracted from tracking data. We discuss how computer-aided observation can be used to identify and pre-select potentially interesting video sequences from large amounts of video data for further observation, as well as dir...
Article
In this paper, we report the computation and tabulation, using MAGMA, of all congruence subgroups of PSL(2, Z) of genus less than or equal to 24. We include full tables of the congruence groups of genus 0, 1, 2, and 3 and a summary of the remaining cases.
Article
Let $f(x)$ be a separable polynomial over a local field. Montes algorithm computes certain approximations to the different irreducible factors of $f(x)$, with strong arithmetic properties. In this paper we develop an algorithm to improve any one of these approximations, till a prescribed precision is attained. The most natural application of this "...
Conference Paper
We present an algorithm for factoring polynomials over local fields, in which the Montes algorithm is combined with elements from Zassenhaus Round Four algorithm. This algorithm avoids the computation of characteristic polynomials and the resulting precision problems that occur in the Round Four algorithm.
Article
The main aim of this paper is twofold. First we generalize, in a novel way, most of the known non-vanishing results for the derivatives of the Riemann zeta function by establishing the existence of an infinite sequence of regions in the right half-plane where these derivatives cannot have any zeros; and then, in the rare regions of the complex plan...
Article
Full-text available
We present an algorithm for computing the 2-group of the positive divisor classes of a number field F in case F has exceptional dyadic places. As an application, we compute the 2-rank of the wild kernel WK2(F) in K2(F) for such number fields.
Article
Full-text available
We present an algorithm for the computation of logarithmic l-class groups of number fields. Our principal motivation is the effective determination of the l-rank of the wild kernel in the K-theory of number fields.
Article
We present an algorithm for computing the 2-group of narrow logarithmic divisor classes of degree 0 for number fields F. As an application, we compute in some cases the 2-rank of the wild kernel WK2(F).
Conference Paper
Full-text available
In recent years the computer algebra system KASH/KANT for number theory has evolved considerably. We present its new features and introduce the related components, QaoS (Querying Algebraic Objects System) and GiANT (Graphical Algebraic Number Theory).
Article
Let K be a p-adic field. We give an explicit charac- terization of the abelian extensions ofK of degreep by relating the coecients of the generating polynomials of extensions L/K of de- greep to the exponents of generators of the norm groupNL/K(L ). This is applied in an algorithm for the construction of class fields of degree pm, which yields an a...
Book
Elliptic curves have a well-known and explicit theory for the construction and application of endomorphisms, which can be applied to improve performance in scalar multiplication. Recent work has extended these techniques to hyperelliptic Jacobians, but one obstruction is the lack of explicit models of curves together with an efficiently computable...
Article
Let K be a global field and O be an order of K. We develop algorithms for the computation of the unit group of residue class rings for ideals in O. As an application we show how to compute the unit group and the Picard group of O provided that we are able to compute the unit group and class group of the maximal order O˜ of K.
Article
Full-text available
We present an algorithm for the computation of logarithmic $\l$-class groups of number fields. Our principal motivation is the effective determination of the $\l$-rank of the wild kernel in the {\ASI K}-theory of number fields.
Conference Paper
We present an algorithm for the computation of the discrete logarithm in logarithmic l-Class Groups. This is applied to the calculation of the l-rank of the wild kernel WK 2(F) of a number field F. In certain cases it can also be used to determine generators of the l-part of WK 2(F).
Article
Full-text available
Let k be a global eld with maximal order ok and let m0 be an ideal of ok. We present algorithms for the computation of the multiplicative group (ok=m0) of the residue class ring ok=m0 and the discrete logarithm therein based on the explicit representation of the group of principal units (Has80). We show how these algorithms can be combined with oth...
Article
Ecient Enumeration of Extensions of Local Fields Sebastian Pauli, Ph.D.
Article
Full-text available
We present an algorithm that returns a proper factor of a polynomial (x) over the p-adic integers Z p (if (x) is reducible over Qp ) orreturns a power basis of the ring of integers of Qp [x]=(x)Qp [x] (if (x) is irreducibleover Qp ). Our algorithm is based on the Round Four maximal orderalgorithm. Experimental results show that the new algorithm is...
Article
E#cient Enumeration of Extensions of Local Fields with Bounded Discriminant Sebastian Pauli, Ph.D. Concordia University, 2001 Let k be a p-adic field. It is well-known that k has only finitely many extensions of a given finite degree. Krasner [1966] gives formulae for the number of extensions of a given degree and discriminant. Following his work,...
Article
Full-text available
. Let k be a p-adic field. It is well-known that k has only finitely many extensions of a given finite degree. In [Kr66], Krasner gives formulae for the number of extensions of a given degree and discriminant. Following his work, we present an algorithm for the computation of generating polynomials for all extensions K=k of a given degree and discr...
Article
Introduction. We consider the problem of factoring polynomials with p-adic coefficients. Restricting our attention to monic, square-free polynomials in Z p [x], we present a method to compute the complete factorization of such polynomials into irreducible factors in Z p [x]. Our algorithm has its origins in the Round Four algorithm of Zassenhaus, b...
Article
Introduction. We consider the problem of factoring polynomials with p-adic coefficients. Restricting our attention to monic, square-free polynomials in Z p [x], we present a method to compute the complete factorization of such polynomials into irreducible factors in Z p [x]. Our algorithm has its origins in the Round Four algorithm of Zassenhaus, b...
Article
. Let k be a p-adic field. It is well-known that k has only finitely many extension of a given finite degree. In [Kr66], Krasner gives formulae for the number of extensions of a given degree and discriminant. Following his work, we present an algorithm for the computation of generating polynomials for all extensions K=k of a given degree and discri...

## Citations

... These constants have several interesting and unexpected applications in the zeta function theory, as was shown recently in [4], [3], [8], and [9]. Moreover, the classical Euler-Maclaurin Summation can be used to prove (see [10]) that, if we set C α (a) = γ α (a) − log α (a) ...
... In this paper we consider the related Hurwitz zeta function ζ(s, a) (see Hurwitz [11] of 1882), which for 0 < a ≤ 1 has the Laurent series expansion: where, for non-negative integers n, the coefficients γ n (a) are known as the Stieltjes constants ( [20]), which were generalized to fractional values α ∈ R + by Kreminski [12] in 2003. These constants have several interesting and unexpected applications in the zeta function theory, as was shown recently in [4], [3], [8], and [9]. Moreover, the classical Euler-Maclaurin Summation can be used to prove (see [10]) that, if we set C α (a) = γ α (a) − log α (a) ...
... And when Re → 0, namely a Stokes flow, the zeros of the second derivative form the lattice of points Γ Re=0 . Thus the properties of the [1] [15],[1] [13] zeroes of ζ (z) help us understand the behavior of this complex flow at the limit, when the Reynolds number is infinite. In [9] it was proven that a positive proportion of the zeroes of ζ (z) lie in the region (on the right): as T → ∞, where T is the height in the critical line. ...
... Recall (e.g. [19] or [8]) that to an extension of p-adic fields, we can attach a ramification polygon, which is an invariant of the extension. By attaching further residual information such as the residual polynomials of each face of the ramification polygon, we can form a finer invariant. ...
... At the end of the twentieth century, Ogg enumerated and studied elliptic [18] and hyperelliptic [19] modular curves; the resulting Diophantine study [20] informed Mazur's classification of rational isogenies of elliptic curves [16], where the curves of genus 0 are precisely the ones with infinitely many rational points. This explicit study continues today, extended to include all quotients of the upper half-plane by congruence subgroups of SL 2 (Z); the list up to genus 24 was computed by Cummins-Pauli [7]. Recent papers have studied curves with infinitely many rational points in the context of Mazur's Program B-see Rouse-Sutherland-Zureick-Brown [21] for further references and recent results in this direction. ...
... Classical implemented algorithms for factoring polynomials over Q (see e.g. [4,6,26,27]) are based on the Zassenhaus Round Four algorithm, suffering from loss of precision in computing characteristic polynomials. In [13], the authors introduced a new technique as a combination of the Montes algorithm [9,10] which exploits the Newton polygons of higher order (as initiated in [27]), and a Newton-like single factor lifting. ...
... For abelian extensions, local class field theory gives a one-to-one correspondence between the abelian extensions of K and the open subgroups of the unit group K × of K. An algorithm that constructs the wildly ramified part of the class field as towers of extensions of degree p was given in [Pau06]. Recently Monge [Mon14] has published an algorithm that, given a subgroup of K × of finite index, directly constructs the generating polynomial of the corresponding totally ramified extension. ...
... Even though many questions are still unanswered, efficient algorithms to compute them have been developed. From the class group, thanks to the results contained in [KP05], we can efficiently deduce the group of classes of invertible ideals for any order in a number field. ...
... An implementation using local fields could be given using Theorem 0.2 and the factoring algorithm of Cantor and Gordon ([8]), which runs in probabilistic polynomial time and provides explicit error estimates for the precision needed. See also the factoring algorithms of Pauli ([21], [22]) and the references therein. Take K = C p , and normalize the valuation ord(·) on C p so it extends the valuation ord p (·) on Q. ...
Citing conference paper
... The arithmetic of this logarithmic group can give some information on the wild kernels of number fields. One can see [4,89101112 and [16] for details. Especially, Pauli and Soriano-Gafiuk can describe the p-rank of the wild kernel of quadratic number field Q( √ d) by the logarithmic p-class group of the quadratic number field Q( √ −3d) without assuming Q( √ d) contains a primitive pth root of unity in [16]. ...