Algebraic systems of quadratic forms of number fields and function fields

1989 ◽  
Vol 65 (2) ◽  
pp. 225-243 ◽  
Author(s):  
Martin Kr�skemper
Keyword(s):  
Quadratic Forms ◽  
Function Fields ◽  
Number Fields ◽  
Algebraic Systems ◽  
And Function

On a certain analogy between algebraic number fields and function fields

2001 ◽  
pp. 512-515
Author(s):  
Ichiro Satake ◽  
Genjiro Fujisaki ◽  
Kazuya Kato ◽  
Masato Kurihara ◽  
Shoichi Nakajima
Keyword(s):  
Algebraic Number ◽  
Function Fields ◽  
Number Fields ◽  
Algebraic Number Fields ◽  
And Function

scholarly journals Number fields and function fields: coalescences, contrasts and emerging applications

2015 ◽  
Vol 373 (2040) ◽  
pp. 20140315 ◽  
Author(s):  
J. P. Keating ◽  
Z. Rudnick ◽  
T. D. Wooley
Keyword(s):  
Finite Field ◽  
Function Fields ◽  
Analytic Number Theory ◽  
Number Fields ◽  
Past Half Century ◽  
Recent Developments ◽  
Substantial Progress ◽  
And Function ◽  
Past Half ◽  
Field Setting

The similarity between the density of the primes and the density of irreducible polynomials defined over a finite field of q elements was first observed by Gauss. Since then, many other analogies have been uncovered between arithmetic in number fields and in function fields defined over a finite field. Although an active area of interaction for the past half century at least, the language and techniques used in analytic number theory and in the function field setting are quite different, and this has frustrated interchanges between the two areas. This situation is currently changing, and there has been substantial progress on a number of problems stimulated by bringing together ideas from each field. We here introduce the papers published in this Theo Murphy meeting issue, where some of the recent developments are explained.

scholarly journals Bounded gaps between primes in number fields and function fields

2015 ◽  
Vol 143 (7) ◽  
pp. 2841-2856 ◽  
Author(s):  
Abel Castillo ◽  
Chris Hall ◽  
Robert J. Lemke Oliver ◽  
Paul Pollack ◽  
Lola Thompson
Keyword(s):  
Function Fields ◽  
Number Fields ◽  
And Function ◽  
Bounded Gaps

scholarly journals Quantum algorithms for one-dimensional infrastructures

2014 ◽  
Vol 14 (1&2) ◽  
pp. 56-90
Author(s):  
Pradeep Sarvepalli ◽  
Pawel M. Wocjan
Keyword(s):  
Quantum Algorithms ◽  
Function Fields ◽  
Success Probability ◽  
Number Fields ◽  
Fourth Power ◽  
Quadratic Number ◽  
Problem Size ◽  
One Dimensional ◽  
Quadratic Number Fields ◽  
And Function

Infrastructures are group-like objects that make their appearance in arithmetic geometry in the study of computational problems related to number fields and function fields over finite fields. The most prominent computational tasks of infrastructures are the computation of the circumference of the infrastructure and the generalized discrete logarithms. Both these problems are not known to have efficient classical algorithms for an arbitrary infrastructure. Our main contributions are polynomial time quantum algorithms for one-dimensional infrastructures that satisfy certain conditions. For instance, these conditions are always fulfilled for infrastructures obtained from number fields and function fields, both of unit rank one. Since quadratic number fields give rise to such infrastructures, this algorithm can be used to solve Pell's equation and the principal ideal problem. In this sense we generalize Hallgren's quantum algorithms for quadratic number fields, while also providing a polynomial speedup over them. Our more general approach shows that these quantum algorithms can also be applied to infrastructures obtained from complex cubic and totally complex quartic number fields. Our improved way of analyzing the performance makes it possible to show that these algorithms succeed with constant probability independent of the problem size. In contrast, the lower bound on the success probability due to Hallgren decreases as the fourth power of the logarithm of the circumference. Our analysis also shows that fewer qubits are required. We also contribute to the study of infrastructures, and show how to compute efficiently within infrastructures.

scholarly journals Algorithms for quadratic forms over global function fields of odd characteristic

2021 ◽  
Vol 55 (3) ◽  
pp. 68-72
Author(s):  
Mawunyo Kofi Darkey-Mensah
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Function Fields ◽  
Number Fields ◽  
Global Function ◽  
Global Function Fields ◽  
Anisotropic Part ◽  
Pythagoras Number ◽  
Present Algorithm

This paper presents an adaptation of recently developed algorithms for quadratic forms over number fields in [4] to global function fields of odd characteristics. First, we present algorithm for checking if a given non-degenerate quadratic form is isotropic or hyperbolic. Next we devise a method for computing the dimension of the anisotropic part of a quadratic form. Finally we present algorithms computing two field invariants: the level and the Pythagoras number.

scholarly journals ON LOGARITHMIC DERIVATIVES OF ZETA FUNCTIONS IN FAMILIES OF GLOBAL FIELDS

2011 ◽  
Vol 07 (08) ◽  
pp. 2139-2156 ◽  
Author(s):  
PHILIPPE LEBACQUE ◽  
ALEXEY ZYKIN
Keyword(s):  
Error Term ◽  
Function Fields ◽  
Zeta Functions ◽  
Number Fields ◽  
Global Fields ◽  
And Function ◽  
Derivatives Of ◽  
Logarithmic Derivatives ◽  
Siegel Theorem

We prove a formula for the limit of logarithmic derivatives of zeta functions in families of global fields with an explicit error term. This can be regarded as a rather far reaching generalization of the explicit Brauer–Siegel theorem both for number fields and function fields.

Analogies between number fields and function fields

2001 ◽  
pp. 606-611
Author(s):  
Ichiro Satake ◽  
Genjiro Fujisaki ◽  
Kazuya Kato ◽  
Masato Kurihara ◽  
Shoichi Nakajima
Keyword(s):  
Function Fields ◽  
Number Fields ◽  
And Function
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