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facIrredTest.h File Reference

This file provides a probabilistic irreducibility test for polynomials over Z/p. More...

#include "canonicalform.h"

Go to the source code of this file.

Functions

double numZeros (const CanonicalForm &F, int k)
 evaluate F at k random points in Z/p^n and count the number of zeros that occur More...
 
int probIrredTest (const CanonicalForm &F, double error)
 given some error probIrredTest detects irreducibility or reducibility of F with confidence level 1-error More...
 

Detailed Description

This file provides a probabilistic irreducibility test for polynomials over Z/p.

ABSTRACT: irreducibility test based on "Quick and Dirty Irreducibility Test" by v. Bothmer and Schreyer

Author
Martin Lee

Definition in file facIrredTest.h.

Function Documentation

◆ numZeros()

double numZeros ( const CanonicalForm F,
int  k 
)

evaluate F at k random points in Z/p^n and count the number of zeros that occur

Returns
numZeros returns #zeros/trials
Parameters
[in]Fsome poly over Z/p with n variables
[in]knumber of trials

Definition at line 24 of file facIrredTest.cc.

25 {
26  int result= 0;
27 
28  FFRandom FFgen;
30  for (int i= 0; i < k; i++)
31  {
32  buf= F;
33  for (int j= F.level(); j > 0; j++)
34  buf= buf (FFgen.generate(), j);
35  if (buf.isZero())
36  result++;
37  }
38 
39  return (double) result/k;
40 }
int i
Definition: cfEzgcd.cc:125
int k
Definition: cfEzgcd.cc:92
factory's main class
Definition: canonicalform.h:83
int level() const
level() returns the level of CO.
generate random elements in F_p
Definition: cf_random.h:44
CanonicalForm generate() const
Definition: cf_random.cc:56
return result
Definition: facAbsBiFact.cc:76
int j
Definition: facHensel.cc:105
int status int void * buf
Definition: si_signals.h:59

◆ probIrredTest()

int probIrredTest ( const CanonicalForm F,
double  error 
)

given some error probIrredTest detects irreducibility or reducibility of F with confidence level 1-error

Returns
probIrredTest returns 1 for irreducibility, -1 for reducibility or 0 if the test is not applicable
Parameters
[in]Fsome poly over Z/p
[in]error0 < error < 1

Definition at line 63 of file facIrredTest.cc.

64 {
65  CFMap N;
66  CanonicalForm G= compress (F, N);
67  int n= G.level();
68  int p= getCharacteristic();
69 
70  double sqrtTrials= inverseERF (1-2.0*error)*sqrt (2.0);
71 
72  double s= sqrtTrials;
73 
74  double pn= pow ((double) p, (double) n);
75  double p1= (double) 1/p;
76  p1= p1*(1.0-p1);
77  p1= p1/(double) pn;
78  p1= sqrt (p1);
79  p1 *= s;
80  p1 += (double) 1/p;
81 
82  double p2= (double) (2*p-1)/(p*p);
83  p2= p2*(1-p2);
84  p2= p2/(double) pn;
85  p2= sqrt (p2);
86  p2 *= s;
87  p2= (double) (2*p - 1)/(p*p)-p2;
88 
89  //no testing possible
90  if (p2 < p1)
91  return 0;
92 
93  double den= sqrt (p1*(1-p1))+sqrt (p2*(1-p2));
94  double num= p2-p1;
95 
96  sqrtTrials *= den/num;
97 
98  int trials= (int) floor (pow (sqrtTrials, 2.0));
99 
100  double experimentalNumZeros= numZeros (G, trials);
101 
102  double pmiddle= sqrt (p1*p2);
103 
104  num= den;
105  den= sqrt (p1*(1.0-p2))+sqrt (p2*(1.0-p1));
106  pmiddle *= (den/num);
107 
108  if (experimentalNumZeros < pmiddle)
109  return 1;
110  else
111  return -1;
112 }
Rational pow(const Rational &a, int e)
Definition: GMPrat.cc:414
int getCharacteristic()
Definition: cf_char.cc:51
CanonicalForm num(const CanonicalForm &f)
CanonicalForm den(const CanonicalForm &f)
const CanonicalForm CFMap CFMap & N
Definition: cfEzgcd.cc:49
int p
Definition: cfModGcd.cc:4019
CanonicalForm compress(const CanonicalForm &f, CFMap &m)
CanonicalForm compress ( const CanonicalForm & f, CFMap & m )
Definition: cf_map.cc:210
class CFMap
Definition: cf_map.h:85
void error(const char *fmt,...)
Definition: emacs.cc:55
const CanonicalForm int s
Definition: facAbsFact.cc:55
double numZeros(const CanonicalForm &F, int k)
evaluate F at k random points in Z/p^n and count the number of zeros that occur
Definition: facIrredTest.cc:24
double inverseERF(double d)
Definition: facIrredTest.cc:42
static TreeM * G
Definition: janet.cc:32
gmp_float sqrt(const gmp_float &a)
Definition: mpr_complex.cc:328
const signed long floor(const ampf< Precision > &x)
Definition: amp.h:774