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Miura :: Miura

Miura -- A package for computing objects for general nonsingular curves in the Miura form, an extension of the Weierstrass form

Description

Miura expresses any nonsingular curves. For example, hyper-elliptic curves (e.g., y^2=x^5+x+1), C_{ab} curves (e.g. y^3=x^4+2x+1), complete intersection (e.g. {y^2-x^3-1,z^2-x*y-1}). For the Miura form, the pole orders should be specified such as 2 and 3 for x and y of an elliptic curve. Currently, only divisor class group computation is available for the package. For the elliptic curves, [(P)-(O)]+[(Q)-(O)] = [(P+Q)-(O)] for two points P, Q and the point O at infinity. For the general nonsingular curves, any divisor class is uniquely expressed by E-g(O) with E a positive divisor of degree g (genus). This package reduces the divisor class group addition to ideal class group multiplication, and utilizes Groebner basis computation. See http://arxiv.org/pdf/1512.08040v1.pdf for the details.

Author

Version

This documentation describes version 0.1 of Miura.

Source code

The source code from which this documentation is derived is in the file Miura.m2.

Exports

  • Functions and commands
    • add -- A method for computing the reduced ideal of J*K
    • double -- A method for computing the squared ideal J*J
    • inv -- A method for computing the reduced inverse ideal
    • multi -- A method for computing the reduced ideal of an integral ideal J multimplied by integer m
    • pR -- A method for specifying the underlying polynomial ring
    • qR -- A method for specifying the underlying quotient ring
    • reduced -- A method for computing the reduced ideal