shh.gl4
Class PPT

Object
  PPT

All Implemented Interfaces:

Comparable

public class PPT

extends Object

implements Comparable

A point in the projective space P^n-1 over the ring Z/NZ. This is the space of primitive vectors in Z^N (vectors whose coordinates together have gcd 1), modulo N and modulo multiplication by the units of Z/NZ. If N is prime, this is the usual projective space over a finite field.

The constructor ensures that the point's coordinates always lie in 0, 1, ..., N-1 and are always lexicographically the least among points in its equivalence class.

Each instance has a reference to a ZModNData object, which stores the level N and helper data. You can reuse the same object for each point in a given projective space.

Author:

Mark McConnell

Constructor Summary

PPT(ZModNData data, int[] coords)
Constructor.

Method Summary

int	compareTo(Object o) Compares PPTs lexicographically.
boolean	equals(Object o) Tests for equality of the coordinates.
int	hashCode() Consistent with equals(java.lang.Object).
static void	main(String[] args) For testing.
static PPT[]	makeP3(ZModNData data, Hashtable h, boolean bPrint) Lists the points of P3 in lexicographic order.
PPT	times(MatChar mm) Returns a new PPT consisting of this as a row vector times the matrix in mm.
String	toString() Returns, e.g., (1 0 1 1).
String	toStringNoParen() Returns, e.g., 1 0 1 1.

Methods inherited from class Object

clone, finalize, getClass, notify, notifyAll, wait, wait, wait

Constructor Detail

PPT

public PPT(ZModNData data,
           int[] coords)

Constructor.

Parameters:

data - Holds the level N and helper data. You can reuse the same object for each point in a given projective space.

coords - Its length is n when we are in P^n-1. The constructor does not alter its contents.

Throws:

IllegalStateException - If the coordinates together don't have gcd 1.

Method Detail

toString

public String toString()

Returns, e.g., (1 0 1 1).

toStringNoParen

public String toStringNoParen()

Returns, e.g., 1 0 1 1.

compareTo

public int compareTo(Object o)

Compares PPTs lexicographically. For the sake of speed, assumes without checking that o is not null and is a PPT with the same N and n as this.

Specified by:

compareTo in interface Comparable

equals

public boolean equals(Object o)

Tests for equality of the coordinates. For the sake of speed, assumes without checking that o is not null and is a PPT with the same N and n as this.

hashCode

public final int hashCode()

Consistent with equals(java.lang.Object).

makeP3

public static PPT[] makeP3(ZModNData data,
                           Hashtable h,
                           boolean bPrint)

Lists the points of P³ in lexicographic order. The CPT ("coded point") corresponding to a given PPT ("projective point") is the PPT's index in this array.

Parameters:

h - If non-null, will be overwritten to hold the reverse mapping from PPT to (Integer)CPT.

times

public PPT times(MatChar mm)

Returns a new PPT consisting of this as a row vector times the matrix in mm. Only works on P³.

main

public static void main(String[] args)

For testing. Input: N v0 v1 v2 .... Output: v projectively-niced mod N. Or if the input is just N, enumerate all N⁴ points in the affine four-space, projectivize those that can be projectivized, count the resulting points, and see if it equals ZModNData.cardP3.