source: git/Singular/LIB/phindex.lib @ 6fe9a5

//////////////////////////////////////////////////////////////////////////////
version="$Id$";
category=" ";
info="
LIBRARY : phindex.lib Procedures to compute the index of real analytic vector fields
AUTHOR: Victor Castellanos

NOTE: To compute the Poincare-Hopf index of a real analytic vector field
      with an algebraically isolated singularity at 0 (w. an a. i. s),
      we use the algebraic formula for the degree of the real analytic map
      germ  found by Eisenbud-Levine in 1997. This result was also proved by
      Khimshiashvili. If the isolated singularity is non algebraically
      isolated  and the vector field has similar reduced complex zeroes of
      codimension 1, we use a formula as the Eisenbud-Levine found by Victor
      Castellanos, in both cases is necessary to use a local order (ds,...).
      To compute the signature of a quadratic form (or symmetric matrix)
      we use the method of Lagrange.

PROCEDURES:
 signatureL(M[,n]);   signature of symmetric matrix M, method of Lagrange.
 signatureLqf(h[,n]); signature of quadratic form h, method of Lagrange.
 PH_ais(I)            P-H index of real analytic vector field I w. an a. i. s.
 PH_nais(I)           P-H index of real analytic vector field I w. a non a. i. s
";

LIB "primdec.lib";
LIB "zeroset.lib";

/////////////////////////////////////////////////////////////////////////////
proc signatureL(M,int #)
"USAGE:    signatureL(M[,r]); M symmetric matrix, r int (optional).
RETURN:   the signature of M of type int or if r is given and !=0 then
          intvec with (signature, nr. of +, nr. of -) is returned.
THEORY:   Given the matrix M, we construct the quadratic form associated. Afterwards
          we use the method of Lagrange to compute the signature. The law of
          inertia for a real quadratic form A(x,x) says that in a
          representation of A(x,x) as a sum of independent squares
                            A(x,x)=sum_{i=1}^r a_iX_i^2.
          The number of positive and the number of negative squares are
          independent of the choice of representation. The signature -s- of
          A(x,x) is the difference between the number -pi- of positive squares
          and the number -nu- of negative squares in the representation of
          A(x,x). The rank -r- of M (or A(x,x)) and the signature -s-
          determine the numbers -pi- and -nu- uniquely, since
                            r=pi+nu,   s=pi-nu.
          The method of Lagrange is a procedure to reduce any real quadratic
          form to a sum of squares.
          Ref. Gantmacher, The theory of matrices, Vol. I, Chelsea Publishing
               Company, NY 1960, page 299.
EXAMPLE:  example signatureL; shows an example
"
{
  if(typeof(M)!="matrix")
  {
    ERROR("** The argument is not a matrix type");
  }
  option(noprot);
  option(noredefine);
  int nv1=ncols(M);
  matrix zero[nv1][nv1]=0;
  if (transpose(M)!=M)
    {
      ERROR("** The matrix is non symmetric");
    }
  if (M==0)
    {
      ERROR("** The matrix is zero");
    }
  option(noprot);
  option(noredefine);
  def h=basering;
  int chr=char(h);
  ring signLagrange=chr,(x(1..nv1)), lp; //ring to compute the quadratic form associated to M
  matrix Ma=fetch(h,M);
  int nv=ncols(Ma);
  matrix X[1][nv]=maxideal(1);
  matrix Ax=X*Ma*transpose(X);
  poly Axx=Ax[1,1]; //quadratic form associated to matrix M
  if (size(#)==0)
    {
      def sal=SigntL(Axx);
      return(sal[1]);
    }
  else
    {
      return(SigntL(Axx));
    }
}
example
{ "EXAMPLE:"; echo = 2;
  ring r=0,(x),ds;
  matrix M[5][5]=0,0,0,1,0,0,1,0,0,-1,0,0,1,0,0,1,0,0,3,0,0,-1,0,0,1;
  signatureL(M,1); //The rank of M is 3+1=4
  matrix H[5][5]=0,-7,0,1,0,-7,1,0,0,-1,0,0,1,0,0,1,0,0,-3,5,0,-1,0,5,1;
  signatureL(H);
}
////////////////////////////////////////////////////////////////////////
proc signatureLqf(h,int #)
"USAGE:    signatureLqf(h); h quadratic form (poly type).
RETURN:   the signature of h of type int or if r is given and !=0 then
          intvec with (signature, nr. of +, nr. of -) is returned.
THEORY:   To compute the signature we use the method of Lagrange. The law of
          inertia for a real quadratic form h(x,x) says that in a
          representation  of h(x,x) as a sum of independent squares
          h(x,x)=sum_{i=1}^r a_i*X_i^2 the number of positive and the number of negative squares are
          independent of the choice of representation. The signature -s- of
          h(x,x) is the difference between the number -pi- of positive squares
          and the number -nu- of negative squares in the representation of
          h(x,x). The rank -r- of h(x,x) and the signature -s- determine the
          numbers -pi- and -nu- uniquely, since
                             r=pi+nu,   s=pi-nu.
          The method of Lagrange is a procedure to reduce any real quadratic
          form to a sum of squares.
          Ref. Gantmacher, The theory of matrices, Vol. I, Chelsea Publishing
               Company, NY 1960, page 299.
EXAMPLE:  example signatureLqf; shows an example
"
{
  if(typeof(h)!="poly")
  {
    ERROR("** The argument is not a poly type");
  }
  option(noprot);
  option(noredefine);
  poly M=h;
  int nv1=nvars(basering);
  if (M==0)
   {
     ERROR("** The quadratic form is zero");
   }
  poly Axx=M;
  poly Bxx;
  poly bxx1;
  poly bxx2;
  def coe1;
  int i;
  int jb;
  int k;
  int haycuadrados;
  int haycruzados;
  int positivo=0;
  int negativo=0;
  int lAxx;
  while (Axx<>0) //Lagrange method to compute the signature
    {
      haycruzados=1;
      haycuadrados=1;
      lAxx=size(Axx);
      i=1;
      while (i<=lAxx and haycuadrados)
    {
      jb=1;
      while (jb<=nv1 and haycuadrados)
        {
          if (leadmonom(Axx[i])/(x(jb)^2)==1) //there is squares
        {
          Bxx=Axx;
          if (leadcoef(Axx[i])>0)
            {
              positivo=positivo+1;
            }
          else
            {
              negativo=negativo+1;
            }
          coe1=1/(4*leadcoef(Bxx[i]));
          Axx=Bxx-coe1*(diff(Bxx,x(jb)))^2;
          haycuadrados=0;
        }
          jb=jb+1;
        }
      i=i+1;
    }
      if (haycruzados) //there is no squares
    {
      int ia=1;
      int ja=1;
      int ka=1;
      while (ia<=nv1 and haycruzados)
        {
          while (ja<=nv1 and haycruzados)
        {
          ka=ja+1;
          while (ka<=nv1 and haycruzados)
            {
              if (leadmonom(Axx[ia])/leadmonom(x(ja)*x(ka))==1)
            {
              Bxx=Axx;
              bxx1=diff(Bxx,x(ja))+diff(Bxx,x(ka));
              bxx2=diff(Bxx,x(ja))-diff(Bxx,x(ka));
              coe1=1/(4*leadcoef(Bxx[ia]));
              Axx=Bxx-coe1*(bxx1^2-bxx2^2);
              positivo=positivo+1;
              negativo=negativo+1;
              haycruzados=0;
            }
              ka=ka+1;
            }
          ja=ja+1;
        }
          ia=ia+1;
        }
    }
    }
 if (size(#)==0)
    {
      def sal=positivo-negativo;
      return(sal);
    }
  else
    {
      int sig=positivo-negativo;
      intvec dat=sig,positivo,negativo;
      return(dat);
    }
}
example
{ "EXAMPLE:"; echo = 2;
  ring r=0,(x(1..4)),ds;
  poly Ax=4*x(1)^2+x(2)^2+x(3)^2+x(4)^2-4*x(1)*x(2)-4*x(1)*x(3)+4*x(1)*x(4)+4*x(2)*x(3)-4*x(2)*x(4);
  signatureLqf(Ax,1); //The rank of Ax is 3+1=4
  poly Bx=2*x(1)*x(4)+x(2)^2+x(3)^2;
  signatureLqf(Bx);
}
/////////////////////////////////////////////////////////////////////////////
proc PH_ais(I)
"USAGE:    PH_ais(I); I ideal of coordinates of the vector field.
RETURN:   the Poincare-Hopf index of type int.
NOTE:     the isolated singularity must be algebraically isolated.
THEORY:   The Poincare-Hopf index of a real vector field X at the isolated
          singularity 0 is the degree of the map (X/|X|) : S_epsilon ---> S,
          where S is the unit sphere, and the spheres are oriented as
          (n-1)-spheres in R^n. The degree depends only on the germ, X, of X
          at 0. If the vector field X is real analytic, then an invariant of
          the germ is its local ring
                            Qx=R[[x1..xn]]/Ix
          where R[[x1,..,xn]] is the ring of germs at 0 of real-valued analytic
          functions on R^n, and Ix is the ideal generated by the components
          of X. The isolated singularity of X is algebraically isolated if the
          algebra Qx is finite dimensional as real vector space, geometrically
          this mean that 0 is also an isolated singularity for the
          complexified vector field. In this case the Poincare-Hopf index is
          the signature of the non degenerate bilinear form <,> obtained by
          composition of the product in the algebra Qx with a linear
          functional map
                       <,> : (Qx)x(Qx) ---(.)--> Qx ---(L)--> R
          with L(Jo)>0, where Jo is the residue class of the Jacobian
          determinant in Qx. Here, we use a natural linear functional defined
          as follows. Suppose that E={E_1,..E_r} is a basis of Qx, then Jo can
          be written as
                      Jo=a_1E_{j1}+...+a_kE_{jk},  js\in {1...r}, s=1..k, k<=r,
          where a_s are constant. The linear functional L:Qx--->R is defined as
                                  L(E_{j1})=(a_1)/|a_1|=sign of a_1,
          the other elements of the base are sent to 0.
          Refs. -Eisenbud & Levine, An algebraic formula for the degree of
                 a C^\infty map germ, Ann. Math., 106, (1977), 19-38.
                -Khimshiashvili, On a local degree of a smooth map, trudi
                 Tbilisi Math. Inst., (1980), 105-124.
EXAMPLE:  example  PH_ais; shows an example.
"
{
  if(typeof(I)!="ideal")
  {
    ERROR("** The argument is not of ideal type");
  }
  ideal A=I;
  ideal qI=std(A);
  int siono=vdim(qI);
  int l;
  if (siono==-1)
    {
      ERROR("** The vector field does not have an algebraically isolated singularity");
    }
  if (siono!=0)
    {
      option(noredefine);
      option(noprot);
      def oldr=basering;
      def chr1=char(oldr);
      int n=nvars(oldr);
      ideal E=kbase(qI);
      int m=size(E);
      poly Jx=det(jacob(A));
      poly Jo=reduce(Jx,qI);
      ring newr=chr1,(x(1..m)),ds; //ring to compute the quadratic form
      int nv=nvars(basering);
      ideal E=fetch(oldr,E);
      ideal qI=fetch(oldr,qI);
      poly Jo=fetch(oldr,Jo);
      attrib(qI,"isSB",1);
      int scoef=1;
      int multby;
      poly Eik;
      poly Axx=0;
      int tEik;
      int stEik;
      def lcEik;
      if (leadcoef(Jo[1])<0)
    {
      scoef=-1;
    }
      for (int si=1; si<=nv; si++)
        {
          for (int sk=si; sk<=nv; sk++)
        {
          Eik=reduce(E[si]*E[sk],qI);
          tEik=size(Eik);
          for(int stEik=1; stEik<=tEik; stEik++)
        {
          if (leadmonom(Eik[stEik])==leadmonom(Jo[1]))
            {
              if (si==sk)
            {
              multby=1;
            }
              else
            {
              multby=2;
            }
              lcEik=leadcoef(Eik[stEik]);
              if (lcEik<0)
            {
              Axx=Axx-multby*scoef*lcEik*x(si)*x(sk);
            }
              else
            {
              Axx=Axx+multby*scoef*lcEik*x(si)*x(sk);
            }
            }
        }
        }
        }
      l=SignatLalt(Axx); //signature of billinear form
      kill newr;
    }
  else
    {
      l=0;
    }
  return(l);
}
example
{ "EXAMPLE"; echo = 2;
  ring r=0,(x,y,z),ds;
  ideal I=x3-3xy2,-y3+3yx2,z3;
  PH_ais(I);
}
///////////////////////////////////////////////////////////////////////////
proc PH_nais(I)
"USAGE:    PH_nais(I); I ideal of coordinates of the vector field.
RETURN:   the Poincare-Hopf index of type int.
NOTE:     the vector field must be a non algebraically isolated singularity
          at 0, with reduced complex zeros of codimension 1.
THEORY:   Suppose that 0 is an algebraically isolated singularity of the real
          analytic vector field X, geometrically this corresponds to the fact that the
          complexified vector field has positive dimension singular locus,
          algebraically this mean that the local ring Qx=R[[x1..xn]]/Ix
          where R[[x1,..,xn]] is the ring of germs at 0 of real-valued analytic
          functions on R^n, and Ix is the ideal generated by the components
          of X is infinite dimensional as real vector space. In the case that
          X has a reduced hypersurface as complex zeros we have the next.
          There exist a real analytic function f:R^n-->R, and a real analytic
          vector field Y s. t. X=fY. The function f does not change of sign
          out of 0 and
                      Mx=R[[x1..xn]]/(Ix : radical(Ix))
          is a finite dimensional sub-algebra of Qx. The Poincare-Hopf index
          of X at 0 is the sign of f times the signature of the non degenerate
          bilinear form <,> obtained by composition of the product in the
          algebra Mx with a linear functional map
                       <,> : (Mx)x(Mx) ---(.)--> Mx ---(L)--> R
          with L(Jp)>0, where Jp is the residue class of the Jacobian
          determinant of X, JX, over f^n, JX/(f^n) in Mx. Here, we use a
          natural linear functional defined as follows. Suppose that
          E={E_1,..E_r} is a basis of Mx, then Jp is writing as
                      Jp=a_1E_{j1}+...+a_kE_{jk},  js\in {1...r}, s=1..k, k<=r,
          where a_s are constant. The linear functional L:M--->R is defined as
                                  L(E_{j1})=(a_1)/|a_1|=sign of a_1,
          the other elements of the base are sent to 0.
          Refs. -Castellanos-Vargas, V., Una formula algebraica del indice de
                 Poincare-Hopf para campos vectoriales reales con una variedad
                 de ceros complejos, Ph. D. thesis CIMAT (2000), chapther 1,
                 Guanajuato Mexico.
                -Castellanos -Vargas, V. The index of non algebraically
                 isolated singularity, Bol. Soc. Mat. Mexicana, (3)
                 Vol. 8, 2002, 141-147.

EXAMPLE:  example  PH_nais; shows an example.
"
{
  if(typeof(I)!="ideal")
  {
    ERROR("** The argument is not of ideal type");
  }
  ideal A=I;
  int siono=vdim(std(A));
  int l;
  if (siono!=0)
    {
      if (siono!=-1)
    {
      ERROR("** The vector field has an algebraically isolated singularity, USE: PH_ais ");
    }
      option(noprot);
      option(noredefine);
      int n=nvars(basering);
      def oldr=basering;
      int chr1=char(oldr);
      ring newring=chr1,(x(1..n)), dp; //ring to compute the radical
      ideal A= fetch(oldr,A);
      ideal rI=radical(A);
      setring oldr;
      ideal rI=fetch(newring,rI);
      if (size(rI)!=1)
    {
      ERROR("** The vector field does not have a non algebraically isolated singularity of codimension 1");
    }
      ideal qI=std(quotient(A,rI));
      ideal E=kbase(qI);
      int m=size(E);
      poly Jx=det(jacob(A));
      poly Jy=Quotient(Jx,rI[1]^n)[1];
      poly Jo=reduce(Jy,qI);
      ring newr=chr1,(x(1..m)),ds; //ring to compute the quadratic form
      int nv=nvars(basering);
      ideal E=fetch(oldr,E);
      ideal qI=fetch(oldr,qI);
      poly Jo=fetch(oldr,Jo);
      attrib(qI,"isSB",1);
      int scoef=1;
      if (leadcoef(Jo[1])<0)
    {
      scoef=-1;
    }
      int multby;
      def lcEik;
      poly Eik;
      poly Axx=0;
      int si=1;
      int sk;
      int tEik;
      int stEik;
      while (si<=nv)
    {
      sk=si;
      while (sk<=nv)
        {
          Eik=reduce(E[si]*E[sk],qI);
          tEik=size(Eik);
          for(int stEik=1; stEik<=tEik; stEik++)
        {
          if (leadmonom(Eik[stEik])==leadmonom(Jo[1]))
            {
              if (si==sk)
            {
              multby=1;
            }
              else
            {
              multby=2;
            }
              lcEik=leadcoef(Eik[stEik]);
              if (lcEik<0)
            {
              Axx=Axx-multby*lcEik*scoef*x(si)*x(sk);
            }
              else
            {
              Axx=Axx+multby*lcEik*scoef*x(si)*x(sk);
            }
            }
        }
          sk=sk+1;
        }
      si=si+1;
    }
      l=SignatLalt(Axx); //signature of bilinear form
      return(l);
    }
  else
    {
      return(0);
    }
}
example
{"EXAMPLE:"; echo = 2;
  ring r=0,(x,y,z),ds;
  ideal I=x5-2x3y2-3xy4+x3z2-3xy2z2,-3x4y-2x2y3+y5-3x2yz2+y3z2,x2z3+y2z3+z5;
  PH_nais(I);
}
//////////////////////////////////////////////////////////////////////
static proc SigntL(poly M)  //static procedure to compute the signature of any quadratic form.
"USAGE:    SigntL(M); M is a quadratic form.
RETURN:   The signature of M of type int.
ASSUME:   M is a quadratic form (ply type).
"
{
  int nv1=nvars(basering);
  poly Axx=M;
  poly Bxx;
  poly bxx1;
  poly bxx2;
  def coe1;
  int i;
  int jb;
  int k;
  int haycuadrados;
  int haycruzados;
  int positivo=0;
  int negativo=0;
  int lAxx;
  while (Axx<>0)
    {
      haycruzados=1;
      haycuadrados=1;
      lAxx=size(Axx);
      i=1;
      while (i<=lAxx and haycuadrados)
    {
      jb=1;
      while (jb<=nv1 and haycuadrados)
        {
          if (leadmonom(Axx[i])/(x(jb)^2)==1)
        {
          Bxx=Axx;
          if (leadcoef(Axx[i])>0)
            {
              positivo=positivo+1;
            }
          else
            {
              negativo=negativo+1;
            }
          coe1=1/(4*leadcoef(Bxx[i]));
          Axx=Bxx-coe1*(diff(Bxx,x(jb)))^2;
          haycuadrados=0;
          haycruzados=0;
        }
          jb=jb+1;
        }
      i=i+1;
    }
      if (haycruzados)
    {
      int ia=1;
      int ja=1;
      int ka=1;
      while (ia<=nv1 and haycruzados)
        {
          while (ja<=nv1 and haycruzados)
        {
          ka=ja+1;
          while (ka<=nv1 and haycruzados)
            {
              if (leadmonom(Axx[ia])/leadmonom(x(ja)*x(ka))==1)
            {
              Bxx=Axx;
              bxx1=diff(Bxx,x(ja))+diff(Bxx,x(ka));
              bxx2=diff(Bxx,x(ja))-diff(Bxx,x(ka));
              coe1=1/(4*leadcoef(Bxx[ia]));
              Axx=Bxx-coe1*(bxx1^2-bxx2^2);
              positivo=positivo+1;
              negativo=negativo+1;
              haycruzados=0;
            }
              ka=ka+1;
            }
          ja=ja+1;
        }
          ia=ia+1;
        }
    }
    }
  int dat1=positivo-negativo;
  intvec dat=dat1,positivo,negativo;
  return(dat);
}
////////////////////////////////////////////////////////////////////////////
//NOTE: SignatLalt is a procedure to compute the signature of a special
//      bilinear form that is necessary to compute the Poincare-Hopf index.
static proc SignatLalt(poly M)
"USAGE:    SignatLalt(M); M is a quadratic form (a polynomial).
RETURN:   The signature of type int.
"
{
 int nv1=nvars(basering);
 if (M==0)
   {
     ERROR("** The quadratic form is zero");
   }
 poly Axx=M;
 poly Bxx;
 poly bxx1;
 poly bxx2;
 def coe1;
 int i;
 int jb;
 int k;
 int haycuadrados;
 int sihay=1;
 int positivo=0;
 int negativo=0;
 int variableactual=0;
 int posicion=1;
 int lAxx;
 while (Axx<>0 and sihay)
   {
     haycuadrados=1;
     lAxx=size(Axx);
     i=posicion;
     while (i<=lAxx and haycuadrados)
       {
     jb=variableactual+1;
     while (jb<=nv1 and haycuadrados)
       {
         if (leadmonom(Axx[i])/(x(jb)^2)==1)
           {
         Bxx=Axx;
         if (leadcoef(Axx[i])>0)
           {
             positivo=positivo+1;
           }
         else
           {
             negativo=negativo+1;
           }
         coe1=1/(4*leadcoef(Bxx[i]));
         Axx=Bxx-coe1*(diff(Bxx,x(jb)))^2;
         haycuadrados=0;
         variableactual=jb;
         posicion=i;
           }
         jb=jb+1;
       }
     if (i==lAxx and haycuadrados)
       {
         sihay=0;
       }
     i=i+1;
       }
   }
 return(positivo-negativo);
}