• Corpus ID: 15865304

Characteristic 1 , entropy and the absolute point

@inproceedings{Connes1997Characteristic1,
  title={Characteristic 1 , entropy and the absolute point},
  author={Alain Connes and Caterina Consani},
  year={1997}
}
We show that the mathematical meaning of working in characteristic one is directly connected to the fields of idempotent analysis and tropical algebraic geometry and we relate this idea to the notion of the absolute point Spec F1. After introducing the notion of “perfect” semi-ring of characteristic one, we explain how to adapt the construction of the Witt ring in characteristic p > 1 to the limit case of characteristic one. This construction also unveils an interesting connection with entropy… 

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References

SHOWING 1-10 OF 20 REFERENCES
On numbers and games
  • R. Guy
  • Mathematics
    Proceedings of the IEEE
  • 1978
TLDR
The motivation for ONAG may have been, and perhaps was-and I would like to think that it was-the attempt to bridge the theory gap between nim-like and chess-like games.
Commutative Semigroup Rings
"Commutative Semigroup Rings" was the first exposition of the basic properties of semigroup rings. Gilmer concentrates on the interplay between semigroups and rings, thereby illuminating both of
Semigroups: An Introduction to the Structure Theory
Semigroups Green's relations constructions commutative semigroups finite semigroups regular semigroups inverse semigroups fundamental regular semigroups four classes of regular semigroups.
Commutative semigroup rings, University of Chicago
  • 1980
Cohomology determinants and reciprocity laws (prepubli- cation)
  • Cohomology determinants and reciprocity laws (prepubli- cation)
The Arithmetic of Elliptic Curves, Graduate Texts
  • 1986
American Journal of Mathematics 116
  • American Journal of Mathematics 116
  • 1994
Algèbre commutative, Eléments de Math
  • Algèbre commutative, Eléments de Math
  • 1961
...
...