cfNewtonPolygon.cc File Reference

This file provides functions to compute the Newton polygon of a bivariate polynomial. More...

#include "config.h"
#include "cf_assert.h"
#include <stdlib.h>
#include "canonicalform.h"
#include "cf_iter.h"
#include "cf_algorithm.h"
#include "cf_primes.h"
#include "cf_reval.h"
#include "cf_factory.h"
#include "gfops.h"
#include "cfNewtonPolygon.h"
#include "templates/ftmpl_functions.h"

Go to the source code of this file.

Functions
static void	translate (int **points, int *point, int sizePoints)
static int	smallestPointIndex (int **points, int sizePoints)
static void	swap (int **points, int i, int j)
static bool	isLess (int *point1, int *point2)
static void	quickSort (int lo, int hi, int **points)
static void	sort (int **points, int sizePoints)
static bool	isConvex (int *point1, int *point2, int *point3)
static bool	isConvex (int **points, int i)
int	grahamScan (int **points, int sizePoints)
int	polygon (int **points, int sizePoints)
	compute a polygon More...
static int *	getDegrees (const CanonicalForm &F, int &sizeOfOutput)
int **	getPoints (const CanonicalForm &F, int &n)
int **	merge (int **points1, int sizePoints1, int **points2, int sizePoints2, int &sizeResult)
int **	newtonPolygon (const CanonicalForm &F, int &sizeOfNewtonPoly)
	compute the Newton polygon of a bivariate polynomial More...
int **	newtonPolygon (const CanonicalForm &F, const CanonicalForm &G, int &sizeOfNewtonPoly)
	compute the convex hull of the support of two bivariate polynomials More...
bool	isInPolygon (int **points, int sizePoints, int *point)
	check if point is inside a polygon described by points More...
void	lambda (int **points, int sizePoints)
void	lambdaInverse (int **points, int sizePoints)
void	tau (int **points, int sizePoints, int k)
void	mu (int **points, int sizePoints)
void	getMaxMin (int **points, int sizePoints, int &minDiff, int &minSum, int &maxDiff, int &maxSum, int &maxX, int &maxY)
void	mpz_mat_mul (const mpz_t *N, mpz_t *&M)
void	mpz_mat_inv (mpz_t *&M)
void	convexDense (int **points, int sizePoints, mpz_t *&M, mpz_t *&A)
	Algorithm 5 as described in Convex-Dense Bivariate Polynomial Factorization by Berthomieu, Lecerf. More...
CanonicalForm	compress (const CanonicalForm &F, mpz_t *&M, mpz_t *&A, bool computeMA)
	compress a bivariate poly More...
CanonicalForm	decompress (const CanonicalForm &F, const mpz_t *inverseM, const mpz_t *A)
	decompress a bivariate poly More...
int *	getRightSide (int **polygon, int sizeOfPolygon, int &sizeOfOutput)
	get the y-direction slopes of all edges with positive slope in y-direction of a convex polygon with at least one point of the polygon lying on the x-axis and one lying on the y-axis More...
bool	irreducibilityTest (const CanonicalForm &F)
	computes the Newton polygon of F and checks if it satisfies the irreducibility criterion from S.Gao "Absolute irreducibility of polynomials via polytopes", Example 1 More...
bool	absIrredTest (const CanonicalForm &F)
	absolute irreducibility test as described in "Modular Las Vegas Algorithms for Polynomial Absolute Factorization" by C. Bertone, G. Cheze, A. Galligo More...
bool	modularIrredTest (const CanonicalForm &F)
	modular absolute irreducibility test as described in "Modular Las Vegas Algorithms for Polynomial Absolute Factorization" by C. Bertone, G. Cheze, A. Galligo More...
bool	modularIrredTestWithShift (const CanonicalForm &F)
	modular absolute irreducibility test with shift as described in "Modular Las Vegas Algorithms for Polynomial Absolute Factorization" by C. Bertone, G. Cheze, A. Galligo More...

Detailed Description

This file provides functions to compute the Newton polygon of a bivariate polynomial.

Author

Martin Lee

Definition in file cfNewtonPolygon.cc.

Function Documentation

absIrredTest()

bool absIrredTest

(

const CanonicalForm &

F

)

absolute irreducibility test as described in "Modular Las Vegas Algorithms for Polynomial Absolute Factorization" by C. Bertone, G. Cheze, A. Galligo

Returns

true if F satisfies condition (C) from the above paper and thus is absolutely irreducible, false otherwise

Parameters

[in]

F

a bivariate polynomial irreducible over ground field

Definition at line 1163 of file cfNewtonPolygon.cc.

{
  ASSERT (getNumVars (F) == 2, "expected bivariate polynomial");
  ASSERT (factorize (F).length() <= 2, " expected irreducible polynomial");
 
  int sizeOfNewtonPolygon;
  int ** newtonPolyg= newtonPolygon (F, sizeOfNewtonPolygon);
  bool isRat= isOn (SW_RATIONAL);
  if (isRat)
    Off (SW_RATIONAL);
  int p=getCharacteristic();
  int d=1;
  char bufGFName='Z';
  bool GF= (CFFactory::gettype()==GaloisFieldDomain);
  if (GF)
  {
    d= getGFDegree();
    bufGFName=gf_name;
  }
 
  setCharacteristic(0);
 
  CanonicalForm g= gcd (newtonPolyg[0][0], newtonPolyg[0][1]); //maybe it's better to use plain intgcd
 
  int i= 1;
  while (!g.isOne() && i < sizeOfNewtonPolygon)
  {
    g= gcd (g, newtonPolyg[i][0]);
    g= gcd (g, newtonPolyg[i][1]);
    i++;
  }
 
  bool result= g.isOne();
 
  if (GF)
    setCharacteristic (p, d, bufGFName);
  else
    setCharacteristic(p);
 
  if (isRat)
    On (SW_RATIONAL);
 
  for (int i= 0; i < sizeOfNewtonPolygon; i++)
    delete [] newtonPolyg[i];
 
  delete [] newtonPolyg;
 
  return result;
}

compress()

CanonicalForm compress	(	const CanonicalForm &	F,
		mpz_t *&	inverseM,
		mpz_t *&	A,
		bool	computeMA = true
	)

compress a bivariate poly

Returns

compress returns a compressed bivariate poly

See also

convexDense, decompress

Parameters

[in]	F	compressed, i.e. F.level()==2, bivariate poly
[in,out]	M	returns the inverse of M, if computeMA==true, M otherwise
[in,out]	A	returns translation
[in]	computeMA	whether to compute M and A

Definition at line 706 of file cfNewtonPolygon.cc.

{
  int n;
  int ** newtonPolyg= NULL;
  if (computeMA)
  {
    newtonPolyg= newtonPolygon (F, n);
    convexDense (newtonPolyg, n, M, A);
  }
 
  CanonicalForm result= 0;
  Variable x= Variable (1);
  Variable y= Variable (2);
 
  mpz_t expX, expY, minExpX, minExpY;
  mpz_init (expX);
  mpz_init (expY);
  mpz_init (minExpX);
  mpz_init (minExpY);
 
  int k= 0;
  Variable alpha;
  mpz_t * exps= new mpz_t [2*size (F)];
  int exps_maxcount=0;
  int count= 0;
  for (CFIterator i= F; i.hasTerms(); i++)
  {
    if (i.coeff().inCoeffDomain() && hasFirstAlgVar (i.coeff(), alpha))
    {
      mpz_set (expX, A[0]);
      mpz_set (expY, A[1]);
      mpz_addmul_ui (expX, M[1], i.exp());
      mpz_addmul_ui (expY, M[3], i.exp());
 
      if (k == 0)
      {
        mpz_set (minExpX, expX);
        mpz_set (minExpY, expY);
        k= 1;
      }
      else
      {
        if (mpz_cmp (minExpY, expY) > 0)
          mpz_set (minExpY, expY);
        if (mpz_cmp (minExpX, expX) > 0)
          mpz_set (minExpX, expX);
      }
      mpz_init_set (exps[count], expX);
      count++;
      mpz_init_set (exps[count], expY);
      exps_maxcount=count;
      count++;
      continue;
    }
    CFIterator j= i.coeff();
    if (k == 0)
    {
      mpz_set (expX, A[0]);
      mpz_addmul_ui (expX, M[1], i.exp());
      mpz_addmul_ui (expX, M[0], j.exp());
 
      mpz_set (expY, A[1]);
      mpz_addmul_ui (expY, M[3], i.exp());
      mpz_addmul_ui (expY, M[2], j.exp());
 
      mpz_set (minExpX, expX);
      mpz_set (minExpY, expY);
      mpz_init_set (exps[count], expX);
      count++;
      mpz_init_set (exps[count], expY);
      exps_maxcount=count;
      count++;
      j++;
      k= 1;
    }
 
    for (; j.hasTerms(); j++)
    {
      mpz_set (expX, A[0]);
      mpz_addmul_ui (expX, M[1], i.exp());
      mpz_addmul_ui (expX, M[0], j.exp());
 
      mpz_set (expY, A[1]);
      mpz_addmul_ui (expY, M[3], i.exp());
      mpz_addmul_ui (expY, M[2], j.exp());
 
      mpz_init_set (exps[count], expX);
      count++;
      mpz_init_set (exps[count], expY);
      exps_maxcount=count;
      count++;
      if (mpz_cmp (minExpY, expY) > 0)
        mpz_set (minExpY, expY);
      if (mpz_cmp (minExpX, expX) > 0)
        mpz_set (minExpX, expX);
    }
  }
 
  count= 0;
  int mExpX= mpz_get_si (minExpX);
  int mExpY= mpz_get_si (minExpY);
  for (CFIterator i= F; i.hasTerms(); i++)
  {
    if (i.coeff().inCoeffDomain() && hasFirstAlgVar (i.coeff(), alpha))
    {
      result += i.coeff()*power (x, mpz_get_si (exps[count])-mExpX)*
                power (y, mpz_get_si (exps[count+1])-mExpY);
      count += 2;
      continue;
    }
    CFIterator j= i.coeff();
    for (; j.hasTerms(); j++)
    {
      result += j.coeff()*power (x, mpz_get_si (exps[count])-mExpX)*
                power (y, mpz_get_si (exps[count+1])-mExpY);
      count += 2;
    }
  }
 
  CanonicalForm tmp= LC (result);
  if (tmp.inPolyDomain() && degree (tmp) <= 0)
  {
    int d= degree (result);
    Variable x= result.mvar();
    result -= tmp*power (x, d);
    result += Lc (tmp)*power (x, d);
  }
 
  if (computeMA)
  {
    for (int i= 0; i < n; i++)
      delete [] newtonPolyg [i];
    delete [] newtonPolyg;
    mpz_mat_inv (M);
  }
 
  for(int tt=exps_maxcount;tt>=0;tt--) mpz_clear(exps[tt]);
  delete [] exps;
  mpz_clear (expX);
  mpz_clear (expY);
  mpz_clear (minExpX);
  mpz_clear (minExpY);
 
  return result;
}

convexDense()

void convexDense	(	int **	points,
		int	sizePoints,
		mpz_t *&	M,
		mpz_t *&	A
	)

Algorithm 5 as described in Convex-Dense Bivariate Polynomial Factorization by Berthomieu, Lecerf.

Parameters

[in,out]	points	a set of points in Z^2, returns M (points)+A
[in]	sizePoints	size of points
[in,out]	M	returns an invertible 2x2 matrix
[in,out]	A	returns translation

Definition at line 558 of file cfNewtonPolygon.cc.

{
  if (sizePoints < 3)
  {
    if (sizePoints == 2)
    {
      mpz_t u,v,g,maxX,maxY;
      mpz_init (u);
      mpz_init (v);
      mpz_init (g);
      mpz_init_set_si (maxX,
                       (points[1][1] < points[0][1])?points[0][1]:points[1][1]);
      mpz_init_set_si (maxY,
                       (points[1][0] < points[0][0])?points[0][0]:points[1][0]);
      mpz_gcdext (g, u, v, maxX, maxY);
      if (points [0] [1] != points [0] [0] && points [1] [0] != points [1] [1])
      {
        mpz_set (A[0], u);
        mpz_mul (A[0], A[0], maxX);
        mpz_set (M[2], maxY);
        mpz_divexact (M[2], M[2], g);
        mpz_set (A[1], M[2]);
        mpz_neg (A[1], A[1]);
        mpz_mul (A[1], A[1], maxX);
        mpz_neg (u, u);
        mpz_set (M[0], u);
        mpz_set (M[1], v);
        mpz_set (M[3], maxX);
        mpz_divexact (M[3], M[3], g);
      }
      else
      {
        mpz_set (M[0], u);
        mpz_set (M[1], v);
        mpz_set (M[2], maxY);
        mpz_divexact (M[2], M[2], g);
        mpz_neg (M[2], M[2]);
        mpz_set (M[3], maxX);
        mpz_divexact (M[3], M[3], g);
      }
      mpz_clear (u);
      mpz_clear (v);
      mpz_clear (g);
      mpz_clear (maxX);
      mpz_clear (maxY);
    }
    else if (sizePoints == 1)
    {
      mpz_set_si (M[0], 1);
      mpz_set_si (M[3], 1);
    }
    return;
  }
  mpz_set_si (M[0], 1);
  mpz_set_si (M[3], 1);
 
  mpz_t * Mu= new mpz_t[4];
  mpz_init_set_si (Mu[1], 1);
  mpz_init_set_si (Mu[2], 1);
  mpz_init (Mu[0]);
  mpz_init (Mu[3]);
 
  mpz_t * Lambda= new mpz_t[4];
  mpz_init_set_si (Lambda[0], 1);
  mpz_init_set_si (Lambda[1], -1);
  mpz_init_set_si (Lambda[3], 1);
  mpz_init (Lambda[2]);
 
  mpz_t * InverseLambda= new mpz_t[4];
  mpz_init_set_si (InverseLambda[0], 1);
  mpz_init_set_si (InverseLambda[1], 1);
  mpz_init_set_si (InverseLambda[3], 1);
  mpz_init (InverseLambda[2]);
 
  mpz_t tmp;
  mpz_init (tmp);
  int minDiff, minSum, maxDiff, maxSum, maxX, maxY, b, d, f, h;
  getMaxMin(points, sizePoints, minDiff, minSum, maxDiff, maxSum, maxX, maxY);
  do
  {
    if (maxX < maxY)
    {
      mu (points, sizePoints);
 
      mpz_mat_mul (Mu, M);
 
      mpz_set (tmp, A[0]);
      mpz_set (A[0], A[1]);
      mpz_set (A[1], tmp);
    }
    getMaxMin(points, sizePoints, minDiff, minSum, maxDiff, maxSum, maxX, maxY);
    b= maxX - maxDiff;
    d= maxX + maxY - maxSum;
    f= maxY + minDiff;
    h= minSum;
    if (b + f > maxY)
    {
      lambda (points, sizePoints);
      tau (points, sizePoints, maxY - f);
 
      mpz_mat_mul (Lambda, M);
 
      if (maxY-f > 0)
        mpz_add_ui (A[0], A[0], maxY-f);
      else
        mpz_add_ui (A[0], A[0], f-maxY);
      maxX= maxX + maxY - b - f;
    }
    else if (d + h > maxY)
    {
      lambdaInverse (points, sizePoints);
      tau (points, sizePoints, -h);
 
      mpz_mat_mul (InverseLambda, M);
 
      if (h < 0)
        mpz_add_ui (A[0], A[0], -h);
      else
        mpz_sub_ui (A[0], A[0], h);
      maxX= maxX + maxY - d - h;
    }
    else
    {
      mpz_clear (tmp);
      mpz_clear (Mu[0]);
      mpz_clear (Mu[1]);
      mpz_clear (Mu[2]);
      mpz_clear (Mu[3]);
      delete [] Mu;
 
      mpz_clear (Lambda[0]);
      mpz_clear (Lambda[1]);
      mpz_clear (Lambda[2]);
      mpz_clear (Lambda[3]);
      delete [] Lambda;
 
      mpz_clear (InverseLambda[0]);
      mpz_clear (InverseLambda[1]);
      mpz_clear (InverseLambda[2]);
      mpz_clear (InverseLambda[3]);
      delete [] InverseLambda;
 
      return;
    }
  } while (1);
}

decompress()

CanonicalForm decompress	(	const CanonicalForm &	F,
		const mpz_t *	M,
		const mpz_t *	A
	)

decompress a bivariate poly

Returns

decompress returns a decompressed bivariate poly

See also

convexDense, decompress

Parameters

[in]	F	compressed, i.e. F.level()<= 2, uni- or bivariate poly
[in]	inverseM	matrix M obtained from compress
[in]	A	vector A obtained from compress

Definition at line 853 of file cfNewtonPolygon.cc.

{
  CanonicalForm result= 0;
  Variable x= Variable (1);
  Variable y= Variable (2);
 
  mpz_t expX, expY, minExpX, minExpY;
  mpz_init (expX);
  mpz_init (expY);
  mpz_init (minExpX);
  mpz_init (minExpY);
 
  mpz_t * exps= new mpz_t [2*size(F)];
  int max_exps=0;
  int count= 0;
  if (F.isUnivariate() && F.level() == 1)
  {
    CFIterator i= F;
 
    mpz_set_si (expX, i.exp());
    mpz_sub (expX, expX, A[0]);
    mpz_mul (expX, expX, inverseM[0]);
    mpz_submul (expX, inverseM[1], A[1]);
 
    mpz_set_si (expY, i.exp());
    mpz_sub (expY, expY, A[0]);
    mpz_mul (expY, expY, inverseM[2]);
    mpz_submul (expY, inverseM[3], A[1]);
 
    mpz_set (minExpX, expX);
    mpz_set (minExpY, expY);
 
    mpz_init_set (exps[count], expX);
    count++;
    mpz_init_set (exps[count], expY);
    count++;
    max_exps=count;
 
    i++;
    for (; i.hasTerms(); i++)
    {
      mpz_set_si (expX, i.exp());
      mpz_sub (expX, expX, A[0]);
      mpz_mul (expX, expX, inverseM[0]);
      mpz_submul (expX, inverseM[1], A[1]);
 
      mpz_set_si (expY, i.exp());
      mpz_sub (expY, expY, A[0]);
      mpz_mul (expY, expY, inverseM[2]);
      mpz_submul (expY, inverseM[3], A[1]);
 
      mpz_init_set (exps[count], expX);
      count++;
      mpz_init_set (exps[count], expY);
      max_exps=count;
      count++;
 
      if (mpz_cmp (minExpY, expY) > 0)
        mpz_set (minExpY, expY);
      if (mpz_cmp (minExpX, expX) > 0)
        mpz_set (minExpX, expX);
    }
 
    int mExpX= mpz_get_si (minExpX);
    int mExpY= mpz_get_si (minExpY);
    count= 0;
    for (i= F; i.hasTerms(); i++)
    {
      result += i.coeff()*power (x, mpz_get_si (exps[count])-mExpX)*
                power (y, mpz_get_si (exps[count+1])-mExpY);
      max_exps=count+1;
      count += 2;
    }
 
    mpz_clear (expX);
    mpz_clear (expY);
    mpz_clear (minExpX);
    mpz_clear (minExpY);
 
    for(int tt=max_exps;tt>=0;tt--) mpz_clear(exps[tt]);
    delete [] exps;
    return result/ Lc (result); //normalize
  }
 
  mpz_t tmp;
  mpz_init (tmp);
  int k= 0;
  Variable alpha;
  for (CFIterator i= F; i.hasTerms(); i++)
  {
    if (i.coeff().inCoeffDomain() && hasFirstAlgVar (i.coeff(), alpha))
    {
      mpz_set_si (expX, i.exp());
      mpz_sub (expX, expX, A[1]);
      mpz_mul (expX, expX, inverseM[1]);
      mpz_submul (expX, A[0], inverseM[0]);
 
      mpz_set_si (expY, i.exp());
      mpz_sub (expY, expY, A[1]);
      mpz_mul (expY, expY, inverseM[3]);
      mpz_submul (expY, A[0], inverseM[2]);
 
      if (k == 0)
      {
        mpz_set (minExpX, expX);
        mpz_set (minExpY, expY);
        k= 1;
      }
      else
      {
        if (mpz_cmp (minExpY, expY) > 0)
          mpz_set (minExpY, expY);
        if (mpz_cmp (minExpX, expX) > 0)
          mpz_set (minExpX, expX);
      }
      mpz_init_set (exps[count], expX);
      count++;
      mpz_init_set (exps[count], expY);
      max_exps=count;
      count++;
      continue;
    }
    CFIterator j= i.coeff();
    if (k == 0)
    {
      mpz_set_si (expX, j.exp());
      mpz_sub (expX, expX, A[0]);
      mpz_mul (expX, expX, inverseM[0]);
      mpz_set_si (tmp, i.exp());
      mpz_sub (tmp, tmp, A[1]);
      mpz_addmul (expX, tmp, inverseM[1]);
 
      mpz_set_si (expY, j.exp());
      mpz_sub (expY, expY, A[0]);
      mpz_mul (expY, expY, inverseM[2]);
      mpz_set_si (tmp, i.exp());
      mpz_sub (tmp, tmp, A[1]);
      mpz_addmul (expY, tmp, inverseM[3]);
 
      mpz_set (minExpX, expX);
      mpz_set (minExpY, expY);
 
      mpz_init_set (exps[count], expX);
      count++;
      mpz_init_set (exps[count], expY);
      max_exps=count;
      count++;
 
      j++;
      k= 1;
    }
 
    for (; j.hasTerms(); j++)
    {
      mpz_set_si (expX, j.exp());
      mpz_sub (expX, expX, A[0]);
      mpz_mul (expX, expX, inverseM[0]);
      mpz_set_si (tmp, i.exp());
      mpz_sub (tmp, tmp, A[1]);
      mpz_addmul (expX, tmp, inverseM[1]);
 
      mpz_set_si (expY, j.exp());
      mpz_sub (expY, expY, A[0]);
      mpz_mul (expY, expY, inverseM[2]);
      mpz_set_si (tmp, i.exp());
      mpz_sub (tmp, tmp, A[1]);
      mpz_addmul (expY, tmp, inverseM[3]);
 
      mpz_init_set (exps[count], expX);
      count++;
      mpz_init_set (exps[count], expY);
      max_exps=count;
      count++;
 
      if (mpz_cmp (minExpY, expY) > 0)
        mpz_set (minExpY, expY);
      if (mpz_cmp (minExpX, expX) > 0)
        mpz_set (minExpX, expX);
    }
  }
 
  int mExpX= mpz_get_si (minExpX);
  int mExpY= mpz_get_si (minExpY);
  count= 0;
  for (CFIterator i= F; i.hasTerms(); i++)
  {
    if (i.coeff().inCoeffDomain() && hasFirstAlgVar (i.coeff(), alpha))
    {
      result += i.coeff()*power (x, mpz_get_si (exps[count])-mExpX)*
                power (y, mpz_get_si (exps[count+1])-mExpY);
      max_exps=count+1;
      count += 2;
      continue;
    }
    for (CFIterator j= i.coeff(); j.hasTerms(); j++)
    {
      result += j.coeff()*power (x, mpz_get_si (exps[count])-mExpX)*
                power (y, mpz_get_si (exps[count+1])-mExpY);
      max_exps=count+1;
      count += 2;
    }
  }
 
  mpz_clear (expX);
  mpz_clear (expY);
  mpz_clear (minExpX);
  mpz_clear (minExpY);
  mpz_clear (tmp);
 
  for(int tt=max_exps;tt>=0;tt--) mpz_clear(exps[tt]);
  delete [] exps;
 
  return result/Lc (result); //normalize
}

getDegrees()

static int * getDegrees ( const CanonicalForm &  F, int &  sizeOfOutput  )

static

Definition at line 179 of file cfNewtonPolygon.cc.

{
  if (F.inCoeffDomain())
  {
    int* result= new int [1];
    result [0]= 0;
    sizeOfOutput= 1;
    return result;
  }
  sizeOfOutput= size (F);
  int* result= new int [sizeOfOutput];
  int j= 0;
  for (CFIterator i= F; i.hasTerms(); i++, j++)
    result [j]= i.exp();
  return result;
}

getMaxMin()

void getMaxMin	(	int **	points,
		int	sizePoints,
		int &	minDiff,
		int &	minSum,
		int &	maxDiff,
		int &	maxSum,
		int &	maxX,
		int &	maxY
	)

Definition at line 478 of file cfNewtonPolygon.cc.

{
  minDiff= points [0] [1] - points [0] [0];
  minSum= points [0] [1] + points [0] [0];
  maxDiff= points [0] [1] - points [0] [0];
  maxSum= points [0] [1] + points [0] [0];
  maxX= points [0] [1];
  maxY= points [0] [0];
  int diff, sum;
  for (int i= 1; i < sizePoints; i++)
  {
    diff= points [i] [1] - points [i] [0];
    sum= points [i] [1] + points [i] [0];
    minDiff= tmin (minDiff, diff);
    minSum= tmin (minSum, sum);
    maxDiff= tmax (maxDiff, diff);
    maxSum= tmax (maxSum, sum);
    maxX= tmax (maxX, points [i] [1]);
    maxY= tmax (maxY, points [i] [0]);
  }
}

getPoints()

int ** getPoints	(	const CanonicalForm &	F,
		int &	n
	)

Definition at line 197 of file cfNewtonPolygon.cc.

{
  n= size (F);
  int ** points= new int* [n];
  for (int i= 0; i < n; i++)
    points [i]= new int [2];
 
  int j= 0;
  int * buf;
  int bufSize;
  if (F.isUnivariate() && F.level() == 1)
  {
    for (CFIterator i= F; i.hasTerms(); i++, j++)
    {
      points [j] [0]= i.exp();
      points [j] [1]= 0;
    }
    return points;
  }
  for (CFIterator i= F; i.hasTerms(); i++)
  {
    buf= getDegrees (i.coeff(), bufSize);
    for (int k= 0; k < bufSize; k++, j++)
    {
      points [j] [0]= i.exp();
      points [j] [1]= buf [k];
    }
    delete [] buf;
  }
  return points;
}

getRightSide()

int * getRightSide	(	int **	polygon,
		int	sizeOfPolygon,
		int &	sizeOfOutput
	)

get the y-direction slopes of all edges with positive slope in y-direction of a convex polygon with at least one point of the polygon lying on the x-axis and one lying on the y-axis

Returns

an array containing the slopes as described above

Parameters

[in]	polygon	vertices of a polygon
[in]	sizeOfPolygon	number of vertices
[in,out]	sizeOfOutput	size of the output

Definition at line 1071 of file cfNewtonPolygon.cc.

{
  int maxY= polygon [0][0];
  int indexY= 0;
  for (int i= 1; i < sizeOfPolygon; i++)
  {
    if (maxY < polygon [i][0])
    {
      maxY= polygon [i][0];
      indexY= i;
    }
    else if (maxY == polygon [i][0])
    {
      if (polygon [indexY][1] < polygon[i][1])
        indexY= i;
    }
    if (maxY > polygon [i][0])
      break;
  }
 
  int count= -1;
  for (int i= indexY; i < sizeOfPolygon; i++)
  {
    if (polygon[i][0] == 0)
    {
      count= i - indexY;
      break;
    }
  }
 
  int * result;
  int index= 0;
  if (count < 0)
  {
    result= new int [sizeOfPolygon - indexY];
    sizeOfOutput= sizeOfPolygon - indexY;
    count= sizeOfPolygon - indexY - 1;
    result [0]= polygon[sizeOfPolygon - 1][0] - polygon [0] [0];
    index= 1;
  }
  else
  {
    sizeOfOutput= count;
    result= new int [count];
  }
 
  for (int i= indexY + count; i > indexY; i--, index++)
    result [index]= polygon [i - 1] [0] - polygon [i] [0];
 
  return result;
}

grahamScan()

int grahamScan	(	int **	points,
		int	sizePoints
	)

Definition at line 128 of file cfNewtonPolygon.cc.

{
  swap (points, 0, smallestPointIndex (points, sizePoints));
  int * minusPoint= new int [2];
  minusPoint [0]= points[0] [0];
  minusPoint [1]= points[0] [1];
  translate (points, minusPoint, sizePoints);
  sort (points, sizePoints);
  minusPoint[0]= - minusPoint[0];
  minusPoint[1]= - minusPoint[1];
  translate (points, minusPoint, sizePoints); //reverse translation
  delete [] minusPoint;
  int i= 3, k= 3;
  while (k < sizePoints)
  {
    swap (points, i, k);
    while (!isConvex (points, i - 1))
    {
      swap (points, i - 1, i);
      i--;
    }
    k++;
    i++;
  }
  if (i + 1 <= sizePoints || i == sizePoints)
  {
    long relArea=
    (points [i-2][0] - points [i-1][0])*(points [0][1] - points [i-1][1])-
    (points [i-2][1] - points [i-1][1])*(points [0][0] - points [i-1][0]);
    if (relArea == 0)
    {
      if (abs (points [i-2][0] - points [0][0]) +
          abs (points [i-2][1] - points [0][1]) >=
          abs (points [i-1][0] - points [i-2][0]) +
          abs (points [i-1][1] - points [i-2][1]) +
          abs (points [i-1][0] - points [0][0]) +
          abs (points [i-1][1] - points [0][1]))
          i--;
    }
  }
  return i;
}

irreducibilityTest()

bool irreducibilityTest

(

const CanonicalForm &

F

)

computes the Newton polygon of F and checks if it satisfies the irreducibility criterion from S.Gao "Absolute irreducibility of polynomials via polytopes", Example 1

Returns

true if it satisfies the above criterion, false otherwise

Parameters

[in]

F

a bivariate polynomial

Definition at line 1123 of file cfNewtonPolygon.cc.

{
  ASSERT (getNumVars (F) == 2, "expected bivariate polynomial");
  ASSERT (getCharacteristic() == 0, "expected polynomial over integers or rationals");
 
  int sizeOfNewtonPolygon;
  int ** newtonPolyg= newtonPolygon (F, sizeOfNewtonPolygon);
  if (sizeOfNewtonPolygon == 3)
  {
    bool check1=
        (newtonPolyg[0][0]==0 || newtonPolyg[1][0]==0 || newtonPolyg[2][0]==0);
    if (check1)
    {
      bool check2=
        (newtonPolyg[0][1]==0 || newtonPolyg[1][1]==0 || newtonPolyg[2][0]==0);
      if (check2)
      {
        bool isRat= isOn (SW_RATIONAL);
        if (isRat)
          Off (SW_RATIONAL);
        CanonicalForm tmp= gcd (newtonPolyg[0][0],newtonPolyg[0][1]); // maybe it's better to use plain intgcd
        tmp= gcd (tmp, newtonPolyg[1][0]);
        tmp= gcd (tmp, newtonPolyg[1][1]);
        tmp= gcd (tmp, newtonPolyg[2][0]);
        tmp= gcd (tmp, newtonPolyg[2][1]);
        if (isRat)
          On (SW_RATIONAL);
        for (int i= 0; i < sizeOfNewtonPolygon; i++)
          delete [] newtonPolyg [i];
        delete [] newtonPolyg;
        return (tmp==1);
      }
    }
  }
  for (int i= 0; i < sizeOfNewtonPolygon; i++)
    delete [] newtonPolyg [i];
  delete [] newtonPolyg;
  return false;
}

isConvex() [1/2]

static bool isConvex ( int **  points, int  i  )

static

Definition at line 123 of file cfNewtonPolygon.cc.

{
  return isConvex (points[i - 1], points [i], points [i + 1]);
}

isConvex() [2/2]

static bool isConvex ( int *  point1, int *  point2, int *  point3  )

static

Definition at line 107 of file cfNewtonPolygon.cc.

{
  long relArea= (point1[0] - point2[0])*(point3[1] - point2[1]) -
                (point1[1] - point2[1])*(point3[0] - point2[0]);
  if (relArea < 0)
    return true;
  if (relArea == 0)
  {
    return !(abs (point1[0] - point3[0]) + abs (point1[1] - point3[1]) >=
             (abs (point2[0] - point1[0]) + abs (point2[1] - point1[1]) +
             abs (point2[0] - point3[0]) + abs (point2[1] - point3[1])));
  }
  return false;
}

isInPolygon()

bool isInPolygon	(	int **	points,
		int	sizePoints,
		int *	point
	)

check if point is inside a polygon described by points

Returns

true if point is inside a polygon described by points

Parameters

[in]	points	an array of points in the plane describing a polygon
[in]	sizePoints	size of points@param point a point in the plane

Definition at line 383 of file cfNewtonPolygon.cc.

{
  int ** buf= new int* [sizePoints + 1];
  for (int i= 0; i < sizePoints; i++)
  {
    buf [i]= new int [2];
    buf [i] [0]= points [i] [0];
    buf [i] [1]= points [i] [1];
  }
  buf [sizePoints]= new int [2];
  buf [sizePoints] [0]= point [0];
  buf [sizePoints] [1]= point [1];
  int sizeBuf= sizePoints + 1;
 
  swap (buf, 0, smallestPointIndex (buf, sizeBuf));
  int * minusPoint= new int [2];
  minusPoint [0]= buf[0] [0];
  minusPoint [1]= buf[0] [1];
  translate (buf, minusPoint, sizeBuf);
  sort (buf, sizeBuf);
  minusPoint[0]= - minusPoint[0];
  minusPoint[1]= - minusPoint[1];
  translate (buf, minusPoint, sizeBuf); //reverse translation
  delete [] minusPoint;
 
  if (buf [0] [0] == point [0] && buf [0] [1] == point [1])
  {
    for (int i= 0; i < sizeBuf; i++)
      delete [] buf[i];
    delete [] buf;
    return false;
  }
 
  for (int i= 1; i < sizeBuf-1; i++)
  {
    if (buf [i] [0] == point [0] && buf [i] [1] == point [1])
    {
      bool result= !isConvex (buf, i);
      for (int i= 0; i < sizeBuf; i++)
        delete [] buf [i];
      delete [] buf;
      return result;
    }
  }
 
  if (buf [sizeBuf - 1] [0] == point [0] && buf [sizeBuf-1] [1] == point [1])
  {
    buf [1] [0]= point [0];
    buf [1] [1]= point [1];
    buf [2] [0]= buf [0] [0];
    buf [2] [1]= buf [0] [1];
    buf [0] [0]= buf [sizeBuf-2] [0];
    buf [0] [1]= buf [sizeBuf-2] [1];
    bool result= !isConvex (buf, 1);
    for (int i= 0; i < sizeBuf; i++)
      delete [] buf [i];
    delete [] buf;
    return result;
  }
  for (int i= 0; i < sizeBuf; i++)
    delete [] buf [i];
  delete [] buf;
 
  return false;
}

isLess()

static bool isLess ( int *  point1, int *  point2  )

static

Definition at line 64 of file cfNewtonPolygon.cc.

{
  long area= point1[0]*point2[1]- point1[1]*point2[0];
  if (area > 0) return true;
  if (area == 0)
  {
    return (abs (point1[0]) + abs (point1[1]) >
            abs (point2[0]) + abs (point2[1]));
  }
  return false;
}

lambda()

void lambda	(	int **	points,
		int	sizePoints
	)

Definition at line 449 of file cfNewtonPolygon.cc.

{
  for (int i= 0; i < sizePoints; i++)
    points [i] [1]= points [i] [1] - points [i] [0];
}

lambdaInverse()

void lambdaInverse	(	int **	points,
		int	sizePoints
	)

Definition at line 455 of file cfNewtonPolygon.cc.

{
  for (int i= 0; i < sizePoints; i++)
    points [i] [1]= points [i] [1] + points [i] [0];
}

merge()

int ** merge	(	int **	points1,
		int	sizePoints1,
		int **	points2,
		int	sizePoints2,
		int &	sizeResult
	)

Definition at line 230 of file cfNewtonPolygon.cc.

{
  int i, j;
  sizeResult= sizePoints1+sizePoints2;
  for (i= 0; i < sizePoints1; i++)
  {
    for (j= 0; j < sizePoints2; j++)
    {
      if (points1[i][0] != points2[j][0])
        continue;
      else
      {
        if (points1[i][1] != points2[j][1])
          continue;
        else
        {
          points2[j][0]= -1;
          points2[j][1]= -1;
          sizeResult--;
        }
      }
    }
  }
  if (sizeResult == 0)
    return points1;
 
  int ** result= new int *[sizeResult];
  for (i= 0; i < sizeResult; i++)
    result [i]= new int [2];
 
  int k= 0;
  for (i= 0; i < sizePoints1; i++, k++)
  {
    result[k][0]= points1[i][0];
    result[k][1]= points1[i][1];
  }
  for (i= 0; i < sizePoints2; i++)
  {
    if (points2[i][0] < 0)
      continue;
    else
    {
      result[k][0]= points2[i][0];
      result[k][1]= points2[i][1];
      k++;
    }
  }
  return result;
}

modularIrredTest()

bool modularIrredTest

(

const CanonicalForm &

F

)

modular absolute irreducibility test as described in "Modular Las Vegas Algorithms for Polynomial Absolute Factorization" by C. Bertone, G. Cheze, A. Galligo

Returns

true if F is absolutely irreducible, false otherwise

Parameters

[in]

F

a bivariate polynomial irreducible over Z

Definition at line 1213 of file cfNewtonPolygon.cc.

{
  ASSERT (getNumVars (F) == 2, "expected bivariate polynomial");
  ASSERT (factorize (F).length() <= 2, " expected irreducible polynomial");
 
  bool isRat= isOn (SW_RATIONAL);
  if (isRat)
    Off (SW_RATIONAL);
 
  if (isRat)
  {
    ASSERT (bCommonDen (F).isOne(), "poly over Z expected");
  }
 
  CanonicalForm Fp, N= maxNorm (F);
  int tdeg= totaldegree (F);
 
  int i= 0;
  //TODO: maybe it's better to choose the characteristic as large as possible
  //      as factorization over large finite field will be faster
  //TODO: handle those cases where our factory primes are not enough
  //TODO: factorize coefficients corresponding to the vertices of the Newton
  //      polygon and only try the obtained factors
  if (N < cf_getSmallPrime (cf_getNumSmallPrimes()-1))
  {
    while (i < cf_getNumSmallPrimes() && N > cf_getSmallPrime(i))
    {
      setCharacteristic (cf_getSmallPrime (i));
      Fp= F.mapinto();
      i++;
      if (totaldegree (Fp) == tdeg)
      {
        if (absIrredTest (Fp))
        {
          CFFList factors= factorize (Fp);
          if (factors.length() == 2 && factors.getLast().exp() == 1)
          {
            if (isRat)
              On (SW_RATIONAL);
            setCharacteristic (0);
            return true;
          }
        }
      }
      setCharacteristic (0);
    }
  }
  else
  {
    while (i < cf_getNumPrimes() && N > cf_getPrime (i))
    {
      setCharacteristic (cf_getPrime (i));
      Fp= F.mapinto();
      i++;
      if (totaldegree (Fp) == tdeg)
      {
        if (absIrredTest (Fp))
        {
          CFFList factors= factorize (Fp);
          if (factors.length() == 2 && factors.getLast().exp() == 1)
          {
            if (isRat)
              On (SW_RATIONAL);
            setCharacteristic (0);
            return true;
          }
        }
      }
      setCharacteristic (0);
    }
  }
 
  if (isRat)
    On (SW_RATIONAL);
 
  return false;
}

modularIrredTestWithShift()

bool modularIrredTestWithShift

(

const CanonicalForm &

F

)

modular absolute irreducibility test with shift as described in "Modular Las Vegas Algorithms for Polynomial Absolute Factorization" by C. Bertone, G. Cheze, A. Galligo

Returns

true if F is absolutely irreducible, false otherwise

Parameters

[in]

F

a bivariate polynomial irreducible over Z

Definition at line 1292 of file cfNewtonPolygon.cc.

{
  ASSERT (getNumVars (F) == 2, "expected bivariate polynomial");
  ASSERT (factorize (F).length() <= 2, " expected irreducible polynomial");
 
  bool isRat= isOn (SW_RATIONAL);
  if (isRat)
    Off (SW_RATIONAL);
 
  if (isRat)
  {
    ASSERT (bCommonDen (F).isOne(), "poly over Z expected");
  }
 
  Variable x= Variable (1);
  Variable y= Variable (2);
  CanonicalForm Fp;
  int tdeg= totaldegree (F);
 
  REvaluation E;
 
  setCharacteristic (2);
  Fp= F.mapinto();
 
  E= REvaluation (1,2, FFRandom());
 
  E.nextpoint();
 
  Fp= Fp (x+E[1], x);
  Fp= Fp (y+E[2], y);
 
  if (tdeg == totaldegree (Fp))
  {
    if (absIrredTest (Fp))
    {
      CFFList factors= factorize (Fp);
      if (factors.length() == 2 && factors.getLast().exp() == 1)
      {
        if (isRat)
          On (SW_RATIONAL);
        setCharacteristic (0);
        return true;
      }
    }
  }
 
  E.nextpoint();
 
  Fp= Fp (x+E[1], x);
  Fp= Fp (y+E[2], y);
 
  if (tdeg == totaldegree (Fp))
  {
    if (absIrredTest (Fp))
    {
      CFFList factors= factorize (Fp);
      if (factors.length() == 2 && factors.getLast().exp() == 1)
      {
        if (isRat)
          On (SW_RATIONAL);
        setCharacteristic (0);
        return true;
      }
    }
  }
 
  int i= 0;
  while (cf_getSmallPrime (i) < 102)
  {
    setCharacteristic (cf_getSmallPrime (i));
    i++;
    E= REvaluation (1, 2, FFRandom());
 
    for (int j= 0; j < 3; j++)
    {
      Fp= F.mapinto();
      E.nextpoint();
      Fp= Fp (x+E[1], x);
      Fp= Fp (y+E[2], y);
 
      if (tdeg == totaldegree (Fp))
      {
        if (absIrredTest (Fp))
        {
          CFFList factors= factorize (Fp);
          if (factors.length() == 2 && factors.getLast().exp() == 1)
          {
            if (isRat)
              On (SW_RATIONAL);
            setCharacteristic (0);
            return true;
          }
        }
      }
    }
  }
 
  setCharacteristic (0);
  if (isRat)
    On (SW_RATIONAL);
 
  return false;
}

mpz_mat_inv()

void mpz_mat_inv

(

mpz_t *&

M

)

Definition at line 535 of file cfNewtonPolygon.cc.

{
  mpz_t det;
  mpz_init_set (det, M[0]);
  mpz_mul (det, det, M[3]);
  mpz_submul (det, M[1], M[2]);
 
  mpz_t tmp;
  mpz_init_set (tmp, M[0]);
  mpz_divexact (tmp, tmp, det);
  mpz_set (M[0], M[3]);
  mpz_divexact (M[0], M[0], det);
  mpz_set (M[3], tmp);
 
  mpz_neg (M[1], M[1]);
  mpz_divexact (M[1], M[1], det);
  mpz_neg (M[2], M[2]);
  mpz_divexact (M[2], M[2], det);
 
  mpz_clear (det);
  mpz_clear (tmp);
}

mpz_mat_mul()

void mpz_mat_mul	(	const mpz_t *	N,
		mpz_t *&	M
	)

Definition at line 502 of file cfNewtonPolygon.cc.

{
  mpz_t * tmp= new mpz_t[4];
 
  mpz_init_set (tmp[0], N[0]);
  mpz_mul (tmp[0], tmp[0], M[0]);
  mpz_addmul (tmp[0], N[1], M[2]);
 
  mpz_init_set (tmp[1], N[0]);
  mpz_mul (tmp[1], tmp[1], M[1]);
  mpz_addmul (tmp[1], N[1], M[3]);
 
  mpz_init_set (tmp[2], N[2]);
  mpz_mul (tmp[2], tmp[2], M[0]);
  mpz_addmul (tmp[2], N[3], M[2]);
 
  mpz_init_set (tmp[3], N[2]);
  mpz_mul (tmp[3], tmp[3], M[1]);
  mpz_addmul (tmp[3], N[3], M[3]);
 
  mpz_set (M[0], tmp[0]);
  mpz_set (M[1], tmp[1]);
  mpz_set (M[2], tmp[2]);
  mpz_set (M[3], tmp[3]);
 
  mpz_clear (tmp[0]);
  mpz_clear (tmp[1]);
  mpz_clear (tmp[2]);
  mpz_clear (tmp[3]);
 
  delete [] tmp;
}

mu()

void mu	(	int **	points,
		int	sizePoints
	)

Definition at line 467 of file cfNewtonPolygon.cc.

{
  int tmp;
  for (int i= 0; i < sizePoints; i++)
  {
    tmp= points [i] [0];
    points [i] [0]= points [i] [1];
    points [i] [1]= tmp;
  }
}

newtonPolygon() [1/2]

int ** newtonPolygon	(	const CanonicalForm &	F,
		const CanonicalForm &	G,
		int &	sizeOfNewtonPoly
	)

compute the convex hull of the support of two bivariate polynomials

Returns

an array of points in the plane which are the vertices of the convex hull of the support of F and G

Parameters

[in]	F	a bivariate polynomial
[in]	G	a bivariate polynomial
[in,out]	sizeOfNewtonPoly	size of the result

Definition at line 321 of file cfNewtonPolygon.cc.

{
  int sizeF= size (F);
  int ** pointsF= new int* [sizeF];
  for (int i= 0; i < sizeF; i++)
    pointsF [i]= new int [2];
  int j= 0;
  int * buf;
  int bufSize;
  for (CFIterator i= F; i.hasTerms(); i++)
  {
    buf= getDegrees (i.coeff(), bufSize);
    for (int k= 0; k < bufSize; k++, j++)
    {
      pointsF [j] [0]= i.exp();
      pointsF [j] [1]= buf [k];
    }
    delete [] buf;
  }
 
  int sizeG= size (G);
  int ** pointsG= new int* [sizeG];
  for (int i= 0; i < sizeG; i++)
    pointsG [i]= new int [2];
  j= 0;
  for (CFIterator i= G; i.hasTerms(); i++)
  {
    buf= getDegrees (i.coeff(), bufSize);
    for (int k= 0; k < bufSize; k++, j++)
    {
      pointsG [j] [0]= i.exp();
      pointsG [j] [1]= buf [k];
    }
    delete [] buf;
  }
 
  int sizePoints;
  int ** points= merge (pointsF, sizeF, pointsG, sizeG, sizePoints);
 
  int n= polygon (points, sizePoints);
 
  int ** result= new int* [n];
  for (int i= 0; i < n; i++)
  {
    result [i]= new int [2];
    result [i] [0]= points [i] [0];
    result [i] [1]= points [i] [1];
  }
 
  sizeOfNewtonPoly= n;
  for (int i= 0; i < sizeF; i++)
    delete [] pointsF[i];
  delete [] pointsF;
  for (int i= 0; i < sizeG; i++)
    delete [] pointsG[i];
  delete [] pointsG;
 
  return result;
}

newtonPolygon() [2/2]

int ** newtonPolygon	(	const CanonicalForm &	F,
		int &	sizeOfNewtonPoly
	)

compute the Newton polygon of a bivariate polynomial

Returns

an array of points in the plane which are the vertices of the Newton polygon of F

Parameters

[in]	F	a bivariate polynomial
[in,out]	sizeOfNewtonPoly	size of the result

Definition at line 282 of file cfNewtonPolygon.cc.

{
  int sizeF= size (F);
  int ** points= new int* [sizeF];
  for (int i= 0; i < sizeF; i++)
    points [i]= new int [2];
  int j= 0;
  int * buf;
  int bufSize;
  for (CFIterator i= F; i.hasTerms(); i++)
  {
    buf= getDegrees (i.coeff(), bufSize);
    for (int k= 0; k < bufSize; k++, j++)
    {
      points [j] [0]= i.exp();
      points [j] [1]= buf [k];
    }
    delete [] buf;
  }
 
  int n= polygon (points, sizeF);
 
  int ** result= new int* [n];
  for (int i= 0; i < n; i++)
  {
    result [i]= new int [2];
    result [i] [0]= points [i] [0];
    result [i] [1]= points [i] [1];
  }
 
  sizeOfNewtonPoly= n;
  for (int i= 0; i < sizeF; i++)
    delete [] points[i];
  delete [] points;
 
  return result;
}

polygon()

int polygon	(	int **	points,
		int	sizePoints
	)

compute a polygon

Returns

an integer n such that the first n entries of points are the vertices of the convex hull of points

Parameters

[in,out]	points	an array of points in the plane
[in]	sizePoints	number of elements in points

Definition at line 172 of file cfNewtonPolygon.cc.

{
  if (sizePoints < 3) return sizePoints;
  return grahamScan (points, sizePoints);
}

quickSort()

static void quickSort ( int  lo, int  hi, int **  points  )

static

Definition at line 77 of file cfNewtonPolygon.cc.

{
  int i= lo, j= hi;
  int* point= new int [2];
  point [0]= points [(lo+hi)/2] [0];
  point [1]= points [(lo+hi)/2] [1];
  while (i <= j)
  {
    while (isLess (points [i], point) && i < hi) i++;
    while (isLess (point, points[j]) && j > lo) j--;
    if (i <= j)
    {
      swap (points, i, j);
      i++;
      j--;
    }
  }
  delete [] point;
  if (lo < j) quickSort (lo, j, points);
  if (i < hi) quickSort (i, hi, points);
}

smallestPointIndex()

static int smallestPointIndex ( int **  points, int  sizePoints  )

static

Definition at line 43 of file cfNewtonPolygon.cc.

{
  int min= 0;
  for (int i= 1; i < sizePoints; i++)
  {
    if (points[i][0] < points[min][0] ||
        (points[i] [0] == points[min] [0] && points[i] [1] < points[min] [1]))
      min= i;
  }
  return min;
}

sort()

static void sort ( int **  points, int  sizePoints  )

static

Definition at line 100 of file cfNewtonPolygon.cc.

{
  quickSort (1, sizePoints - 1, points);
}

swap()

static void swap ( int **  points, int  i, int  j  )

static

Definition at line 56 of file cfNewtonPolygon.cc.

{
  int* tmp= points[i];
  points[i]= points[j];
  points[j]= tmp;
}

tau()

void tau	(	int **	points,
		int	sizePoints,
		int	k
	)

Definition at line 461 of file cfNewtonPolygon.cc.

{
  for (int i= 0; i < sizePoints; i++)
    points [i] [1]= points [i] [1] + k;
}

translate()

static void translate ( int **  points, int *  point, int  sizePoints  )

static

Definition at line 33 of file cfNewtonPolygon.cc.

{
  for (int i= 0; i < sizePoints; i++)
  {
    points[i] [0] -= point [0];
    points[i] [1] -= point [1];
  }
}