Comp_lib 1.1

This library aims at calculating the complexity of a basis that is a candidate for the lowest complexity basis for Fp^d/Fp, where p and d are prime numbers (Pickett and Vinatier, 2017). It enables also to determine the points multiplicities distribution in Fd^2 of the associated minimal Besicovitch arrangement.

It could be installed using pip:

pip install Comp_lib

Usage:

>>> from Comp_lib import Complexity, IncMatComp, P, axes, E

List of the functions:

• E(j,k,d)Create a matrix containing d^2-1 zeros and a single

  one at the (j+1)^th row and (k+1)^th column of the matrix

  Example:

  >>> E(0,4,7)
  

• axes(d)Create a matrix containing d^2-3d+2 zeros and 3d-2

  one at the first row, first column and diagonal of the matrix

  Example:

  >>> axes(7)
  

• P(x,d)The polynomial that determines lines equations of

  the minimal Besicovitch arrangment associated with the basis whose complexity is wanted; x is in Fd

  Example:

  >>> P(2,7)
  

• IncMatComp(d)Part of the matrix necessary for the calculation of

  the complexity of the considered basis; it contains the multiplicities of the points in Fd^2 determined by the associated minimal Besicovitch arrangement column of the matrix

  Example:

  >>> IncMatComp(7)
  

• IncMat(d)The matrix containing the multiplicities of the points

  in Fd^2 determined by the minimal Besicovitch arrangement associated with the basis whose complexity is wanted column of the matrix

  Example:

  >>> IncMat(7)
  

• Complexity(d)The value of the complexity i.e. the number of simple

  points in the matrix containing the multiplicities of the points in Fd^2 determined by a well-chosen Besicovitch arrangement (Pickett and Vinatier, 2017)

  Example:

  >>> Complexity(7)
  

• MulDist(d,n)The distribution of multiplicities in the matrix

  containing the multiplicities of the points in Fd^2 determined by a well-chosen Besicovitch arrangement (Pickett and Vinatier, 2017) The first value of the vector stands for the points of multiplicity zero, the second value stands for the simple points, etc. n is lenght of the vector returned; if this value is not large enought (hight multiplicity(ies) not yet counted) the following message is returned: IndexError: list index out of range column of the matrix

  Example:

  >>> MulDist(7,6)
  

• MulDistT(d,n)The distribution of multiplicities in the T zone of the

  matrix containing the multiplicities of the points in Fd^2 determined by a well-chosen Besicovitch arrangement (Pickett and Vinatier, 2017) The first value of the vector stands for the points of multiplicity zero, the second value stands for the simple points, etc. n is lenght of the vector returned; if this value is not large enought (hight multiplicity(ies) not yet counted) the following message is returned: IndexError: list index out of range column of the matrix

  Example:

  >>> MulDistT(7,3)
  

• MulDistDiag(d,n)The distribution of multiplicities on the diagonal of the

  matrix containing the multiplicities of the points in Fd^2 determined by a well-chosen Besicovitch arrangement (Pickett and Vinatier, 2017) The first value of the vector stands for the points of multiplicity zero, the second value stands for the simple points, etc. n is lenght of the vector returned; if this value is not large enought (hight multiplicity(ies) not yet counted) the following message is returned: IndexError: list index out of range column of the matrix

  Example:

  >>> MulDistDiag(7,6)
  

Licence: GPLv3.