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Concept
Infrastructure (number theory)
Résumé
In mathematics, an infrastructure is a group-like structure appearing in global fields.
In 1972, D. Shanks first discovered the infrastructure of a real quadratic number field and applied his baby-step giant-step algorithm to compute the regulator of such a field in binary operations (for every ), where is the discriminant of the quadratic field; previous methods required binary operations. Ten years later, H. W. Lenstra published a mathematical framework describing the infrastructure of a real quadratic number field in terms of "circular groups". It was also described by R. Schoof and H. C. Williams, and later extended by H. C. Williams, G. W. Dueck and B. K. Schmid to certain cubic number fields of unit rank one and by J. Buchmann and H. C. Williams to all number fields of unit rank one. In his habilitation thesis, J. Buchmann presented a baby-step giant-step algorithm to compute the regulator of a number field of arbitrary unit rank. The first description of infrastructures in number fields of arbitrary unit rank was given by R. Schoof using Arakelov divisors in 2008.
The infrastructure was also described for other global fields, namely for algebraic function fields over finite fields. This was done first by A. Stein and H. G. Zimmer in the case of real hyperelliptic function fields. It was extended to certain cubic function fields of unit rank one by R. Scheidler and A. Stein. In 1999, S. Paulus and H.-G. Rück related the infrastructure of a real quadratic function field to the divisor class group. This connection can be generalized to arbitrary function fields and, combining with R. Schoof's results, to all global fields.
A one-dimensional (abstract) infrastructure consists of a real number , a finite set together with an injective map . The map is often called the distance map.
By interpreting as a circle of circumference and by identifying with , one can see a one-dimensional infrastructure as a circle with a finite set of points on it.
A baby step is a unary operation on a one-dimensional infrastructure .
Source officielle
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