Changeset 3f4e52 – Singular

Singular/LIB/algebra.lib

-                      rae9fd9
+                      r3f4e52
 //new proc  nonZeroEntry(id), used to fix a bug in proc finitenessTest
 ///////////////////////////////////////////////////////////////////////////////
 version="$Id: algebra.lib,v 1.23 2009-04-15 11:07:30 seelisch Exp $";
+version="$Id: algebra.lib,v 1.24 2009-07-28 10:34:54 Singular Exp $";
 category="Commutative Algebra";
 info="
 …
 NOTE:    the proc algebra_containment tests the same using a different
          algorithm, which is often faster
          if l[1] == 0 then l[2] may contain more than one relation h(y(0),y(1),...,y(k)),
+         if l[1] == 0 then l[2] may contain more than one relation h(y(0),y(1),...,y(k)),
          separated by comma
 EXAMPLE: example inSubring; shows an example

Singular/LIB/bfun.lib

-                      rae9fd9
+                      r3f4e52
 //////////////////////////////////////////////////////////////////////////////
 version="$Id: bfun.lib,v 1.11 2009-07-28 10:19:59 Singular Exp $";
+version="$Id: bfun.lib,v 1.12 2009-07-28 10:34:54 Singular Exp $";
 category="Noncommutative";
 info="
 …
 @*      one is interested in the global b-function (also known as Bernstein-Sato
 @*      polynomial) b(s) in K[s], defined to be the non-zero monic polynomial of minimal
 @*      degree, satisfying a functional identity L * F^{s+1} = b(s) F^s,
+@*      degree, satisfying a functional identity L * F^{s+1} = b(s) F^s,
 @*      for some operator L in D[s] (* stands for the action of differential operator)
 @*      By D one denotes the n-th Weyl algebra
 …
 @*   (that is, an ideal I in a Weyl algebra D, such that D/I is holonomic module)
 @*   with respect to the given weight vector w: For a polynomial p in D, its initial
 @*   form w.r.t. (-w,w) is defined as the sum of all terms of p which have
+@*   form w.r.t. (-w,w) is defined as the sum of all terms of p which have
 @*   maximal weighted total degree where the weight of x_i is -w_i and the weight
 @*   of d_i is w_i. Let J be the initial ideal of I w.r.t. (-w,w), i.e. the
+@*   of d_i is w_i. Let J be the initial ideal of I w.r.t. (-w,w), i.e. the
 @*   K-vector space generated by all initial forms w.r.t (-w,w) of elements of I.
 @*   Put s = w_1 x_1 d_1 + ... + w_n x_n d_n. Then the monic generator b_w(s) of
 …
 @*   arbitrary weights need not have rational roots at all. However, b-function
 @*  of a holonomic ideal is known to be non-zero as well.
 @*
 @*      References: [SST] Saito, Sturmfels, Takayama: Groebner Deformations of
+@*
+@*      References: [SST] Saito, Sturmfels, Takayama: Groebner Deformations of
 @*      Hypergeometric Differential Equations (2000),
 @*      Noro: An Efficient Modular Algorithm for Computing the Global b-function,
 …
 AUXILIARY PROCEDURES:
 allPositive(v);  checks whether all entries of an intvec are positive
+allPositive(v);  checks whether all entries of an intvec are positive
 scalarProd(v,w); computes the standard scalar product of two intvecs
 vec2poly(v[,i]); constructs an univariate polynomial with given coefficients
 …
       for (i=1; i<=ncols(I); i++) { M[i] = gen(i);  }
+    }
+  }
+  }
   dbprint(ppl-1,"// initially sorted ideal:", I);
   if (remembercoeffs <> 0) { dbprint(ppl-1,"// used permutations:", M); }
 …
 @*       If t<>0 (and by default) all monomials are reduced (if possible),
 @*       otherwise, only leading monomials are reduced.
 @*       If u<>0 (and by default), the ideal is linearly pre-reduced, i.e.
+@*       If u<>0 (and by default), the ideal is linearly pre-reduced, i.e.
 @*       instead of the given ideal, the output of @code{linReduceIdeal} is used.
 @*       If u is set to 0 and the given ideal does not equal the output of
 …
 NOTE:    If the intersection is zero, this procedure might not terminate.
 @*       If p>0 is given, this proc computes the generator of the intersection in
 @*       char p first and then only searches for a generator of the obtained
 @*       degree in the basering. Otherwise, it searches for all degrees by
+@*       char p first and then only searches for a generator of the obtained
+@*       degree in the basering. Otherwise, it searches for all degrees by
 @*       computing syzygies.
 @*       If s<>0, @code{std} is used for Groebner basis computations in char 0,
 @*       otherwise, and by default, @code{slimgb} is used.
 @*       If t<>0 and by default, @code{std} is used for Groebner basis
+@*       If t<>0 and by default, @code{std} is used for Groebner basis
 @*       computations in char >0, otherwise, @code{slimgb} is used.
 DISPLAY: If printlevel=1, progress debug messages will be printed,
 …
+  }
   dbprint(ppl,"// pIntersectSyz finished");
   if (solveincharp) { short = shortSave; }
+  if (solveincharp) { short = shortSave; }
   return(v);
+}
 …
   if (size(L) == 3) // irreducible factor
+  {
     if (L[3] <> "0" && L[3] <> "1")
+    if (L[3] <> "0" && L[3] <> "1")
+    {
       LL = LL + list(L[3]);
 …
 RETURN:  list of ideal and intvec
 PURPOSE: computes the roots of the Bernstein-Sato polynomial b(s)
 @*       for the hypersurface defined by f.
+@*       for the hypersurface defined by f.
 ASSUME:  The basering is commutative and of characteristic 0.
 BACKGROUND: In this proc, the initial Malgrange ideal is computed according to
 @*       the algorithm by Masayuki Noro and then a system of linear equations is
 @*       solved by linear reductions.
 NOTE:    In the output list, the ideal contains all the roots
+NOTE:    In the output list, the ideal contains all the roots
 @*       and the intvec their multiplicities.
 @*       If s<>0, @code{std} is used for GB computations,
 @*       otherwise, and by default, @code{slimgb} is used.
+@*       otherwise, and by default, @code{slimgb} is used.
 @*       If t<>0, a matrix ordering is used for Groebner basis computations,
 @*       otherwise, and by default, a block ordering is used.
 …
 @*       the algorithm by Masayuki Noro and then a system of linear equations is
 @*       solved by computing syzygies.
 NOTE:    In the output list, the ideal contains all the roots and the intvec
+NOTE:    In the output list, the ideal contains all the roots and the intvec
 @*       their multiplicities.
 @*       If r<>0, @code{std} is used for GB computations in characteristic 0,
 …
 ASSUME:  The basering is the n-th Weyl algebra in characteristic 0 and for all
 @*       1<=i<=n the identity var(i+n)*var(i)=var(i)*var(i+1)+1 holds, i.e. the
 @*       sequence of variables is given by x(1),...,x(n),D(1),...,D(n),
+@*       sequence of variables is given by x(1),...,x(n),D(1),...,D(n),
 @*       where D(i) is the differential operator belonging to x(i).
 @*       Further we assume that I is holonomic.
 …
 @*  L[3] is 1 (for nonzero constant) or 0 (for zero b-function).
 @*       If s<>0, @code{std} is used for GB computations in characteristic 0,
 @*       otherwise, and by default, @code{slimgb} is used.
+@*       otherwise, and by default, @code{slimgb} is used.
 @*       If t<>0, a matrix ordering is used for GB computations, otherwise,
 @*       and by default, a block ordering is used.
 …
 @*       their multiplicities.
 @*       If s<>0, @code{std} is used for the GB computation, otherwise,
 @*       and by default, @code{slimgb} is used.
+@*       and by default, @code{slimgb} is used.
 @*       If t<>0, a matrix ordering is used for GB computations,
 @*       otherwise, and by default, a block ordering is used.
 …
   def save = basering;
   int n = nvars(save);
   if (char(save) <> 0)
+  if (char(save) <> 0)
+  {
     ERROR("characteristic of basering has to be 0");
 …
   // create names for vars
   list Lvar;
   Lvar[1]   = safeVarName("t");
+  Lvar[1]   = safeVarName("t");
   Lvar[2]   = safeVarName("s");
   Lvar[n+3] = safeVarName("D"+Lvar[1]);
 …
   intvec @a = 1:N; @a[2] = 2;
   intvec @a2 = @a; @a2[2] = 0; @a2[2*n+4] = 0;
   list Lord;
+  list Lord;
   Lord[1] = list("a",@a); Lord[2] = list("a",@a2);
   if (methodord == 0) // default: block ordering
 …
 @*  If a<>0, only f is appended to Ann(f^s), otherwise, and by default,
 @*  f and all its partial derivatives are appended.
 @*  If b<>0, @code{std} is used for GB computations, otherwise, and by
+@*  If b<>0, @code{std} is used for GB computations, otherwise, and by
 @*  default, @code{slimgb} is used.
 @*  If c<>0, @code{std} is used for Groebner basis computations of ideals

Singular/LIB/crypto.lib

-                      rae9fd9
+                      r3f4e52
 //GP, last modified 28.6.06
 ///////////////////////////////////////////////////////////////////////////////
 version="$Id: crypto.lib,v 1.9 2009-04-15 11:11:20 seelisch Exp $";
+version="$Id: crypto.lib,v 1.10 2009-07-28 10:34:54 Singular Exp $";
 category="Teaching";
 info="
 …
 "USAGE:  generateG(a,b,m);
 RETURN: m-th division polynomial
 NOTE: generate the so-called division polynomials, i.e., the recursively defined
+NOTE: generate the so-called division polynomials, i.e., the recursively defined
 polynomials p_m=generateG(a,b,m) in Z[x, y] such that, for a point (x:y:1) on the
 elliptic curve defined by y^2=x^3+a*x+b  over Z/N the point@*

Singular/LIB/dmod.lib

-                      rae9fd9
+                      r3f4e52
 //////////////////////////////////////////////////////////////////////////////
 version="$Id: dmod.lib,v 1.46 2009-07-27 07:49:53 Singular Exp $";
+version="$Id: dmod.lib,v 1.47 2009-07-28 10:34:54 Singular Exp $";
 category="Noncommutative";
 info="
 …
 @*             Jorge Martin Morales,    jorge@unizar.es
 THEORY: Let K be a field of characteristic 0. Given a polynomial ring
+THEORY: Let K be a field of characteristic 0. Given a polynomial ring
 @*      R = K[x_1,...,x_n] and a polynomial F in R,
 @*      one is interested in the R[1/F]-module of rank one, generated by
+@*      one is interested in the R[1/F]-module of rank one, generated by
 @*      the symbol F^s for a symbolic discrete variable s.
 @* In fact, the module R[1/F]*F^s has a structure of a D(R)[s]-module, where D(R)
 …
 @* One is interested in the following data:
 @* - Ann F^s = I = I(F^s) in D(R)[s], denoted by LD in the output
 @* - global Bernstein polynomial in K[s], denoted by bs,
+@* - global Bernstein polynomial in K[s], denoted by bs,
 @* - its minimal integer root s0, the list of all roots of bs, which are known
 @*   to be rational, with their multiplicities, which is denoted by BS
 @* - Ann F^s0 = I(F^s0) in D(R), denoted by LD0 in the output
+@* - Ann F^s0 = I(F^s0) in D(R), denoted by LD0 in the output
 @*   (LD0 is a holonomic ideal in D(R))
 @* - Ann^(1) F^s in D(R)[s], denoted by LD1 (logarithmic derivations)
 …
 @*     PS*F^(s+1) = bs*F^s holds in K[x_1,...,x_n,1/F]*F^s.
 REFERENCES:
+REFERENCES:
 @* We provide the following implementations of algorithms:
 @* (OT) the classical Ann F^s algorithm from Oaku and Takayama (Journal of
+@* (OT) the classical Ann F^s algorithm from Oaku and Takayama (Journal of
 @* Pure and Applied Math., 1999),
 @* (LOT) Levandovskyy's modification of the Oaku-Takayama algorithm (ISSAC 2007)
 …
 @*        l'ideal de Bernstein associe a des polynomes, preprint, 2002)
 @* (LM08) V. Levandovskyy and J. Martin-Morales, ISSAC 2008
 @* (C) Countinho, A Primer of Algebraic D-Modules,
 @* (SST) Saito, Sturmfels, Takayama 'Groebner Deformations of Hypergeometric
+@* (C) Countinho, A Primer of Algebraic D-Modules,
+@* (SST) Saito, Sturmfels, Takayama 'Groebner Deformations of Hypergeometric
 @*         Differential Equations', Springer, 2000
 …
 "USAGE:  annfs(f [,S,eng]);  f a poly, S a string, eng an optional int
 RETURN:  ring
 PURPOSE: compute the D-module structure of basering[1/f]*f^s with the algorithm
+PURPOSE: compute the D-module structure of basering[1/f]*f^s with the algorithm
 @*  given in S and with the Groebner basis engine given in ''eng''
 NOTE:  activate the output ring with the @code{setring} command.
 …
 "USAGE:  Sannfs(f [,S,eng]);  f a poly, S a string, eng an optional int
 RETURN:  ring
 PURPOSE: compute the D-module structure of basering[f^s] with the algorithm
+PURPOSE: compute the D-module structure of basering[f^s] with the algorithm
 @*  given in S and with the Groebner basis engine given in eng
 NOTE:    activate the output ring with the @code{setring} command.
 …
 PURPOSE: compute the D-module structure of basering[1/f]*f^s
 NOTE:    activate the output ring with the @code{setring} command.
 @*   In the output ring D[s], the ideal LD1 is generated by the elements
+@*   In the output ring D[s], the ideal LD1 is generated by the elements
 @*   in Ann F^s in D[s], coming from logarithmic derivations.
 @*       If eng <>0, @code{std} is used for Groebner basis computations,
 …
 "USAGE:  ALTannfsBM(f [,eng]);  f a poly, eng an optional int
 RETURN:  ring
 PURPOSE: compute the annihilator ideal of f^s in D[s], where D is the Weyl
+PURPOSE: compute the annihilator ideal of f^s in D[s], where D is the Weyl
 @*   algebra, according to the algorithm by Briancon and Maisonobe
 NOTE:  activate the output ring with the @code{setring} command. In this ring,
 …
 "USAGE:  bernsteinBM(f [,eng]);  f a poly, eng an optional int
 RETURN:  list (of roots of the Bernstein polynomial b and their multiplicies)
 PURPOSE: compute the global Bernstein-Sato polynomial for a hypersurface,
+PURPOSE: compute the global Bernstein-Sato polynomial for a hypersurface,
 @* defined by f, according to the algorithm by Briancon and Maisonobe
 NOTE:    If eng <>0, @code{std} is used for Groebner basis computations,
 …
 "USAGE:  annfs2(I, F [,eng]);  I an ideal, F a poly, eng an optional int
 RETURN:  ring
 PURPOSE: compute the annihilator ideal of f^s in the Weyl Algebra,
+PURPOSE: compute the annihilator ideal of f^s in the Weyl Algebra,
 @*       based on the output of Sannfs-like procedure
 @*       annfs2 uses shorter expressions in the variable s (the idea of Noro).
 …
 "USAGE:  annfsRB(I, F [,eng]);  I an ideal, F a poly, eng an optional int
 RETURN:  ring
 PURPOSE: compute the annihilator ideal of f^s in the Weyl Algebra,
+PURPOSE: compute the annihilator ideal of f^s in the Weyl Algebra,
 @* based on the output of Sannfs like procedure
 NOTE:    activate the output ring with the @code{setring} command. In this ring,
 …
 "USAGE:  operatorBM(f [,eng]);  f a poly, eng an optional int
 RETURN:  ring
 PURPOSE: compute the B-operator and other relevant data for Ann F^s,
+PURPOSE: compute the B-operator and other relevant data for Ann F^s,
 @*  using e.g. algorithm by Briancon and Maisonobe for Ann F^s and BS.
 NOTE:    activate the output ring with the @code{setring} command. In the output ring D[s]
 …
 "USAGE:  operatorModulo(f,I,b);  f a poly, I an ideal, b a poly
 RETURN:  poly
 PURPOSE: compute the B-operator from the polynomial f,
 @*           ideal I = Ann f^s and Bernstein-Sato polynomial b
+PURPOSE: compute the B-operator from the polynomial f,
+@*           ideal I = Ann f^s and Bernstein-Sato polynomial b
 @*           using modulo i.e. kernel of module homomorphism
 NOTE:  The computations take place in the ring, similar to the one
+NOTE:  The computations take place in the ring, similar to the one
 @*       returned by Sannfs procedure.
 @*     Note, that operator is not completely reduced wrt Ann f^{s+1}.
+@*     Note, that operator is not completely reduced wrt Ann f^{s+1}.
 @*     If printlevel=1, progress debug messages will be printed,
 @*     if printlevel>=2, all the debug messages will be printed.
 …
 //     module GMT = transpose(G);
 //     GMT = GMT[2],GMT[3]; // modulo matrix
 //     module K = GMT[2];
+//     module K = GMT[2];
 //     GMT = transpose(GMT);
 //     K = transpose(K);
 …
       return(t);
+    }
     dbprint(ppl,"no explicit constant. Start one more GB computation");
+    dbprint(ppl,"no explicit constant. Start one more GB computation");
     // else: compute GB and do the same
     L = L[2..ncols(L)];
 …
   LD = groebner(LD);
   PS = NF(PS,subst(LD,s,s+1));; // reduction modulo Ann s^{s+1}
   size(PS);
+  size(PS);
   lead(PS);
   reduce(PS*F-bs,LD); // check the defining property of PS
 …
 "USAGE:  annfsBMI(F [,eng]);  F an ideal, eng an optional int
 RETURN:  ring
 PURPOSE: compute the D-module structure of basering[1/f]*f^s where
+PURPOSE: compute the D-module structure of basering[1/f]*f^s where
 @* f = F[1]*..*F[P], according to the algorithm by Briancon and Maisonobe.
 NOTE:    activate the output ring with the @code{setring} command. In this ring,
 …
 "USAGE:  annfsOT(f [,eng]);  f a poly, eng an optional int
 RETURN:  ring
 PURPOSE: compute the D-module structure of basering[1/f]*f^s,
+PURPOSE: compute the D-module structure of basering[1/f]*f^s,
 @* according to the algorithm by Oaku and Takayama
 NOTE:    activate the output ring with the @code{setring} command. In this ring,
 …
 "USAGE:  SannfsOT(f [,eng]);  f a poly, eng an optional int
 RETURN:  ring
 PURPOSE: compute the D-module structure of basering[1/f]*f^s, according to the
+PURPOSE: compute the D-module structure of basering[1/f]*f^s, according to the
 @* 1st step of the algorithm by Oaku and Takayama in the ring D[s]
 NOTE:    activate the output ring with the @code{setring} command.
 @*  In the output ring D[s], the ideal LD (which is NOT a Groebner basis)
+@*  In the output ring D[s], the ideal LD (which is NOT a Groebner basis)
 @*  is the needed D-module structure.
 @*  If eng <>0, @code{std} is used for Groebner basis computations,
 …
 "USAGE:  SannfsBM(f [,eng]);  f a poly, eng an optional int
 RETURN:  ring
 PURPOSE: compute the D-module structure of basering[1/f]*f^s, according to the
+PURPOSE: compute the D-module structure of basering[1/f]*f^s, according to the
 @* 1st step of the algorithm by Briancon and Maisonobe in the ring D[s].
 NOTE:  activate the output ring with the @code{setring} command.
 @*   In the output ring D[s], the ideal LD (which is NOT a Groebner basis) is
+@*   In the output ring D[s], the ideal LD (which is NOT a Groebner basis) is
 @*   the needed D-module structure.
 @*   If eng <>0, @code{std} is used for Groebner basis computations,
 …
   // ----- keep char, minpoly
   L[1] = RL[1];
   L[4] = RL[4];
+  L[4] = RL[4];
   // ----- create vars
   string newVar@t = safeVarName("t");
 …
+  {
     I = J,I;
     I = engine(I,eng);
+    I = engine(I,eng);
+  }
   kill J;
 …
+  {
     LD = LD,J;
     LD = engine(LD,eng);
+    LD = engine(LD,eng);
+  }
   if (addPD) { dbprint(ppl,"// -4-4- Finished GB computations for Ann f^s + <f, f_i>"); }
 …
 PURPOSE: compute Ann f^s and Groebner basis of Ann f^s+f in D[s]
 NOTE:    activate the output ring with the @code{setring} command.
 @*    This procedure, unlike SannfsBM, returns a ring with the degrevlex
+@*    This procedure, unlike SannfsBM, returns a ring with the degrevlex
 @*    ordering in all variables.
 @*    In the ring D[s], the ideal LD (which IS a Groebner basis) is the needed ideal.
 …
+"
+{
   // DEBUG INFO: ordering on the output ring = dp,
+  // DEBUG INFO: ordering on the output ring = dp,
   // use std(K,F); for reusing the std property of K
 …
 "USAGE:  SannfsLOT(f [,eng]);  f a poly, eng an optional int
 RETURN:  ring
 PURPOSE: compute the D-module structure of basering[1/f]*f^s, according to the
+PURPOSE: compute the D-module structure of basering[1/f]*f^s, according to the
 @* Levandovskyy's modification of the algorithm by Oaku and Takayama in D[s]
 NOTE:    activate the output ring with the @code{setring} command.
 @*    In the ring D[s], the ideal LD (which is NOT a Groebner basis) is
+@*    In the ring D[s], the ideal LD (which is NOT a Groebner basis) is
 @*    the needed D-module structure.
 @*    If eng <>0, @code{std} is used for Groebner basis computations,
 …
 "USAGE:  annfsLOT(F [,eng]);  F a poly, eng an optional int
 RETURN:  ring
 PURPOSE: compute the D-module structure of basering[1/f]*f^s, according to
+PURPOSE: compute the D-module structure of basering[1/f]*f^s, according to
 @* the Levandovskyy's modification of the algorithm by Oaku and Takayama
 NOTE:    activate the output ring with the @code{setring} command. In this ring,
 …
 "USAGE:  annfs0(I, F [,eng]);  I an ideal, F a poly, eng an optional int
 RETURN:  ring
 PURPOSE: compute the annihilator ideal of f^s in the Weyl Algebra, based
+PURPOSE: compute the annihilator ideal of f^s in the Weyl Algebra, based
 @* on the output of Sannfs-like procedure
 NOTE:    activate the output ring with the @code{setring} command. In this ring,
 …
 "USAGE:  annfspecial(I,F,mir,n);  I an ideal, F a poly, int mir, number n
 RETURN:  ideal
 PURPOSE: compute the annihilator ideal of F^n in the Weyl Algebra
+PURPOSE: compute the annihilator ideal of F^n in the Weyl Algebra
 @*           for the given rational number n
 ASSUME:  The basering is D[s] and contains 's' explicitly as a variable,
 …
 @*   the integer 'mir' is the minimal integer root of the BS polynomial of F,
 @*   and the number n is rational.
 NOTE: We compute the real annihilator for any rational value of n (both
 @*       generic and exceptional). The implementation goes along the lines of
 @*       the Algorithm 5.3.15 from Saito-Sturmfels-Takayama.
+NOTE: We compute the real annihilator for any rational value of n (both
+@*       generic and exceptional). The implementation goes along the lines of
+@*       the Algorithm 5.3.15 from Saito-Sturmfels-Takayama.
 DISPLAY: If printlevel=1, progress debug messages will be printed,
 @*       if printlevel>=2, all the debug messages will be printed.
 …
 RETURN:  int
 ASSUME: Basering is a commutative ring, alpha is a rational number.
 PURPOSE: check whether a rational number alpha is a root of the global
+PURPOSE: check whether a rational number alpha is a root of the global
 @* Bernstein-Sato polynomial of f and compute its multiplicity,
 @* with the algorithm given by S and with the Groebner basis engine given by eng.
 NOTE:  The annihilator of f^s in D[s] is needed and hence it is computed with the
+NOTE:  The annihilator of f^s in D[s] is needed and hence it is computed with the
 @*       algorithm by Briancon and Maisonobe. The value of a string S can be
 @*       'alg1' (default) - for the algorithm 1 of [LM08]
 …
 proc checkRoot2 (ideal I, poly F, number a, list #)
 "USAGE:  checkRoot2(I,f,a [,eng]);  I an ideal, f a poly, alpha a number, eng an optional int
 ASSUME:  I is the annihilator of f^s in D[s], basering is D[s],
+ASSUME:  I is the annihilator of f^s in D[s], basering is D[s],
 @* that is basering and I are the output os Sannfs-like procedure,
 @* f is a polynomial in K[_x] and alpha is a rational number.
 RETURN:  int, the multiplicity of -alpha as a root of the BS polynomial of f.
+RETURN:  int, the multiplicity of -alpha as a root of the BS polynomial of f.
 PURPOSE: check whether a rational number alpha is a root of the global Bernstein-
 @* Sato polynomial of f and compute its multiplicity from the known Ann F^s in D[s]
 …
+{
   // to check: alpha is rational (has char 0 check inside)
   if (!isRational(a))
 …
 ASSUME:  checkFactor is called from the basering, created by Sannfs-like proc,
 @* that is, from the Weyl algebra in x1,..,xN,d1,..,dN tensored with K[s].
 @* The ideal I is the annihilator of f^s in D[s], that is the ideal, computed
+@* The ideal I is the annihilator of f^s in D[s], that is the ideal, computed
 @* by Sannfs-like procedure (usually called LD there).
 @* Moreover, f is a polynomial in K[x1,..,xN] and qs is a polynomial in K[s].
 RETURN:  int, 1 if qs is a factor of the global Bernstein polynomial of f and 0 otherwise
 PURPOSE: check whether a univariate polynomial qs is a factor of the
+PURPOSE: check whether a univariate polynomial qs is a factor of the
 @*  Bernstein-Sato polynomial of f without explicit knowledge of the latter.
 NOTE:    If eng <>0, @code{std} is used for Groebner basis computations,
 …
 "USAGE:  indAR(L,n);  list L, int n
 RETURN:  list
 PURPOSE: computes arrangement inductively, using L and
+PURPOSE: computes arrangement inductively, using L and
 @* var(n) as the next variable
 ASSUME: L has a structure of an arrangement
 …
 proc isRational(number n)
 "USAGE:  isRational(n); n number
+"USAGE:  isRational(n); n number
 RETURN:  int
 PURPOSE: determine whether n is a rational number,

Singular/LIB/dmodapp.lib

-                      rae9fd9
+                      r3f4e52
 //////////////////////////////////////////////////////////////////////////////
 version="$Id: dmodapp.lib,v 1.32 2009-07-28 10:13:06 Singular Exp $";
+version="$Id: dmodapp.lib,v 1.33 2009-07-28 10:34:55 Singular Exp $";
 category="Noncommutative";
 info="
 …
 @*             Daniel Andres, daniel.andres@math.rwth-aachen.de
 GUIDE: Let K be a field of characteristic 0, R = K[x1,..xN] and
+GUIDE: Let K be a field of characteristic 0, R = K[x1,..xN] and
 @* D be the Weyl algebra in variables x1,..xN,d1,..dN.
 @* In this library there are the following procedures for algebraic D-modules:
 @* - localization of a holonomic module D/I with respect to a mult. closed set
 @* of all powers of a given polynomial F from R. Our aim is to compute an
 @* ideal L in D, such that D/L is a presentation of a localized module. Such L
 @* always exists, since such localizations are known to be holonomic and thus
+@* of all powers of a given polynomial F from R. Our aim is to compute an
+@* ideal L in D, such that D/L is a presentation of a localized module. Such L
+@* always exists, since such localizations are known to be holonomic and thus
 @* cyclic modules. The procedures for the localization are DLoc,SDLoc and DLoc0.
 @*
 @* - annihilator in D of a given polynomial F from R as well as
 @* of a given rational function G/F from Quot(R). These can be computed via
+@* of a given rational function G/F from Quot(R). These can be computed via
 @* procedures annPoly resp. annRat.
 @*
 @* - initial form and initial ideals in Weyl algebras with respect to a given
+@* - initial form and initial ideals in Weyl algebras with respect to a given
 @* weight vector can be computed with  inForm, initialMalgrange, initialIdealW.
 @*
 @* - appelF1, appelF2 and appelF4 return ideals in parametric Weyl algebras,
+@* - appelF1, appelF2 and appelF4 return ideals in parametric Weyl algebras,
 @* which annihilate corresponding Appel hypergeometric functions.
 REFERENCES:
 @* (SST) Saito, Sturmfels, Takayama 'Groebner Deformations of Hypergeometric
+REFERENCES:
+@* (SST) Saito, Sturmfels, Takayama 'Groebner Deformations of Hypergeometric
 @*         Differential Equations', Springer, 2000
 @* (ONW) Oaku, Takayama, Walther 'A Localization Algorithm for D-modules', 2000
 …
 "USAGE:  appelF1();
 RETURN:  ring  (and exports an ideal into it)
 PURPOSE: define the ideal in a parametric Weyl algebra,
+PURPOSE: define the ideal in a parametric Weyl algebra,
 @* which annihilates Appel F1 hypergeometric function
 NOTE: the ideal called  IAppel1 is exported to the output ring
 …
+}
 proc appelF2()
+proc appelF2()
 "USAGE:  appelF2();
 RETURN:  ring (and exports an ideal into it)
 PURPOSE: define the ideal in a parametric Weyl algebra,
+PURPOSE: define the ideal in a parametric Weyl algebra,
 @* which annihilates Appel F2 hypergeometric function
 NOTE: the ideal called  IAppel2 is exported to the output ring
 …
 "USAGE:  appelF4();
 RETURN:  ring  (and exports an ideal into it)
 PURPOSE: define the ideal in a parametric Weyl algebra,
+PURPOSE: define the ideal in a parametric Weyl algebra,
 @* which annihilates Appel F4 hypergeometric function
 NOTE: the ideal called  IAppel4 is exported to the output ring
 …
     ERROR("Basering is not a Weyl algebra");
+  }
   if (defined(LD0) || defined(BS))
+  if (defined(LD0) || defined(BS))
+  {
     ERROR("Reserved names LD0 and/or BS are used. Please rename the objects.");
 …
 "USAGE:  DLoc0(I, F);  I an ideal, F a poly
 RETURN:  ring
 PURPOSE: compute the presentation of the localization of D/I w.r.t. f^s,
+PURPOSE: compute the presentation of the localization of D/I w.r.t. f^s,
 @*           where D is a Weyl Algebra, based on the output of procedure SDLoc
 ASSUME: the basering is similar to the output ring of SDLoc procedure
 …
   gkdim(I); // 3
   def W = SDLoc(I,F);  setring W; // creates ideal LD in W = R[s]
   def U = DLoc0(LD, x2-y3);  setring U; // compute in R
+  def U = DLoc0(LD, x2-y3);  setring U; // compute in R
   LD0; // Groebner basis of the presentation of localization
   BS; // description of b-function for localization
 …
   setring R;
   poly F = x2-y3;
   ideal I = Dx*F, Dy*F;
+  ideal I = Dx*F, Dy*F;
   // note, that I is not holonomic, since it's dimension is not 2
   gkdim(I); // 3, while dim R = 4
 …
 RETURN:  poly
 PURPOSE: reconstruct a monic polynomial in one variable from its factorization
 ASSUME:  s is a string with the name of some variable and
+ASSUME:  s is a string with the name of some variable and
 @*         L is supposed to consist of two entries:
 @*         L[1] of the type ideal with the roots of a polynomial
 …
 ASSUME:  The basering is the n-th Weyl algebra in characteristic 0 and  for all
 @*       1<=i<=n the identity var(i+n)*var(i)=var(i)*var(i+1)+1 holds, i.e. the
 @*       sequence of variables is given by x(1),...,x(n),D(1),...,D(n),
+@*       sequence of variables is given by x(1),...,x(n),D(1),...,D(n),
 @*       where D(i) is the differential operator belonging to x(i).
 PURPOSE: computes the initial ideal with respect to given weights.
 …
       ERROR(string(var(i+n)) + " is not a differential operator for " + string(var(i)));
+    }
+  }
+  }
   // 1. create  homogenized Weyl algebra
   // 1.1 create ordering
 …
   for (i=1; i<=n; i++) { @relD[i,n+i] = var(N)^(homogweights[i]+homogweights[n+i]); }
   def Dh = nc_algebra(1,@relD);
   setring Dh; kill @Dh;
+  setring Dh; kill @Dh;
   dbprint(ppl-1,"// computing in ring",Dh);
   // 2. Compute the initial ideal
 …
+  {
     dbprint(ppl,"// found no roots");
+  }
+  }
   L = list(II,mm);
   if (ir <> 1)
 …
     L = L + list(string(ir));
+  }
   else
+  else
+  {
     dbprint(ppl,"// no irreducible factors found");
+  }
+  }
   setring save;
   L = imap(@S,L);
 …
   ring r = 0,(x,y),dp;
   bFactor((x^2-1)^2);
   bFactor((x^2+1)^2);
+  bFactor((x^2+1)^2);
   bFactor((y^2+1/2)*(y+9)*(y-7));
+}

Singular/LIB/elim.lib

-                      rae9fd9
+                      r3f4e52
 // $Id: elim.lib,v 1.34 2009-05-05 09:25:56 bulygin Exp $
+// $Id: elim.lib,v 1.35 2009-07-28 10:34:55 Singular Exp $
 // (GMG, modified 22.06.96)
 // GMG, last modified 30.10.08: new procedure elimRing;
 …
 // and can now choose as method slimgb or std.
 ///////////////////////////////////////////////////////////////////////////////
 version="$Id: elim.lib,v 1.34 2009-05-05 09:25:56 bulygin Exp $";
+version="$Id: elim.lib,v 1.35 2009-07-28 10:34:55 Singular Exp $";
 category="Commutative Algebra";
 info="
 …
        if ( #[2] == "std" ) { method = "std"; }
        if ( #[2] == "slimgb" ) { method = "slimgb"; }
+    }
     else
 …
       ERROR("expected `elim(ideal,intvec[,string])`");
+    }
     if (size(#) == 3)
+    {
 …
     if ( method == "" )
+    {
+      ERROR("expected \"std\" or \"slimgb\" or \"withWeights\" as the optional string parameters");
+    }
+      ERROR("expected \"std\" or \"slimgb\" or \"withWeights\" as the optional string parameters");
+    }
+  }

Singular/LIB/freegb.lib

-                      rae9fd9
+                      r3f4e52
 //////////////////////////////////////////////////////////////////////////////
 version="$Id: freegb.lib,v 1.24 2009-04-15 11:13:09 seelisch Exp $";
+version="$Id: freegb.lib,v 1.25 2009-07-28 10:34:55 Singular Exp $";
 category="Noncommutative";
 info="
 …
 PURPOSE: sets attributes for a letterplace ring:
 @*      'isLetterplaceRing' = true, 'uptodeg' = d, 'lV' = b, where
 @*      'uptodeg' stands for the degree bound,
+@*      'uptodeg' stands for the degree bound,
 @*      'lV' for the number of variables in the block 0.
 NOTE: Activate the resulting ring by using @code{setring}
 …
     // Set letterplace-specific attributes for the output ring!
   attrib(R, "uptodeg", uptodeg);
   attrib(R, "lV", lV);
   attrib(R, "isLetterplaceRing", 1);
   return (R);
+  attrib(R, "lV", lV);
+  attrib(R, "isLetterplaceRing", 1);
+  return (R);
+}
 example
 …
 "USAGE:  isVar(p);  poly p
 RETURN:  int
 PURPOSE: check, whether leading monomial of p is a power of a single variable
+PURPOSE: check, whether leading monomial of p is a power of a single variable
 @* from the basering. Returns the exponent or 0 if p is multivariate.
 EXAMPLE: example isVar; shows examples
 …
   def R = makeLetterplaceRing(d);
   setring R;
   int uptodeg = d;
   int lV = 2;
+  int uptodeg = d;
+  int lV = 2;
   def R = setLetterplaceAttributes(r,uptodeg,2); // supply R with letterplace structure
   setring R;

Singular/LIB/general.lib

-                      rae9fd9
+                      r3f4e52
 //eric, added absValue 11.04.2002
 ///////////////////////////////////////////////////////////////////////////////
 version="$Id: general.lib,v 1.64 2009-07-15 14:43:52 motsak Exp $";
+version="$Id: general.lib,v 1.65 2009-07-28 10:34:55 Singular Exp $";
 category="General purpose";
 info="
 …
    // Note that in general: lead(sort(M)) != sort(lead(M)), e.g:
    module M = [0, 1, 1, 0], [1, 0, 0, 1]; M;
    sort(lead(M), "c, dp")[1];
    lead(sort(M, "c, dp")[1]);
+   sort(lead(M), "c, dp")[1];
+   lead(sort(M, "c, dp")[1]);
    // In order to sort M wrt a NEW ordering by considering OLD leading
    // terms use one of the following equivalent commands:
    module( M[ sort(lead(M), "c,dp")[2] ] );
    sort( M, sort(lead(M), "c,dp")[2] )[1];
+   module( M[ sort(lead(M), "c,dp")[2] ] );
+   sort( M, sort(lead(M), "c,dp")[2] )[1];
+}
 ///////////////////////////////////////////////////////////////////////////////

Singular/LIB/gmssing.lib

-                      rae9fd9
+                      r3f4e52
 ///////////////////////////////////////////////////////////////////////////////
 version="$Id: gmssing.lib,v 1.14 2009-07-28 08:13:38 Singular Exp $";
+version="$Id: gmssing.lib,v 1.15 2009-07-28 10:34:55 Singular Exp $";
 category="Singularities";
 …
 KEYWORDS: singularities; Gauss-Manin system; Brieskorn lattice;
           mixed Hodge structure; V-filtration; weight filtration
           Bernstein-Sato polynomial; monodromy; spectrum; spectral pairs;
+          Bernstein-Sato polynomial; monodromy; spectrum; spectral pairs;
           good basis
 ";
 …
     int bs[2][i];  multiplicity of i-th root of b(s)
 @end format
 KEYWORDS: singularities; Gauss-Manin system; Brieskorn lattice;
+KEYWORDS: singularities; Gauss-Manin system; Brieskorn lattice;
           Bernstein-Sato polynomial
 EXAMPLE:  example bernstein; shows examples
 …
 @format
 list l;  Jordan data jordan(M) of monodromy matrix exp(-2*pi*i*M)
   ideal l[1];
+  ideal l[1];
     number l[1][i];  eigenvalue of i-th Jordan block of M
   intvec l[2];
+  intvec l[2];
     int l[2][i];  size of i-th Jordan block of M
   intvec l[3];
+  intvec l[3];
     int l[3][i];  multiplicity of i-th Jordan block of M
 @end format

Singular/LIB/ncdecomp.lib

-                      rae9fd9
+                      r3f4e52
 ///////////////////////////////////////////////////////////////////////////////
 version="$Id: ncdecomp.lib,v 1.15 2009-07-07 16:29:58 levandov Exp $";
+version="$Id: ncdecomp.lib,v 1.16 2009-07-28 10:34:55 Singular Exp $";
 category="Noncommutative";
 info="
 …
 OVERVIEW:
 @* This library presents algorithms for the central character decomposition of a module,
 @* i.e. a decomposition into generalized weight modules with respect to the center.
 @* Based on ideas of O. Khomenko and V. Levandovskyy (see the article [L2] in the
+@* This library presents algorithms for the central character decomposition of a module,
+@* i.e. a decomposition into generalized weight modules with respect to the center.
+@* Based on ideas of O. Khomenko and V. Levandovskyy (see the article [L2] in the
 @* References for details).
 …
 RETURN:  module
 PURPOSE: compute the central quotient M:G
 THEORY:  for an ideal G of the center of an algebra and a submodule M of A^n,
+THEORY:  for an ideal G of the center of an algebra and a submodule M of A^n,
 @* the central quotient of M by G is defined to be
 @* M:G  :=  { v in A^n | z*v in M, for all z in G }.

Singular/LIB/polymake.lib

-                      rae9fd9
+                      r3f4e52
 version="$Id: polymake.lib,v 1.17 2009-04-15 11:26:29 seelisch Exp $";
+version="$Id: polymake.lib,v 1.18 2009-07-28 10:34:55 Singular Exp $";
 category="Tropical Geometry";
 info="
 LIBRARY:   polymake.lib    Computations with polytopes and fans,
+LIBRARY:   polymake.lib    Computations with polytopes and fans,
                            interface to polymake and TOPCOM
 AUTHOR:    Thomas Markwig,  email: keilen@mathematik.uni-kl.de
 …
 WARNING:
    Most procedures will not work unless polymake or topcom is installed and
    if so, they will only work with the operating system LINUX!
+   if so, they will only work with the operating system LINUX!
    For more detailed information see the following note or consult the
    help string of the procedures.
 …
 NOTE:
    Even though this is a @sc{Singular} library for computing polytopes and fans
    such as the Newton polytope or the Groebner fan of a polynomial, most of
+   such as the Newton polytope or the Groebner fan of a polynomial, most of
    the hard computations are NOT done by @sc{Singular} but by the program
 @* - polymake by Ewgenij Gawrilow, TU Berlin and Michael Joswig, TU Darmstadt
 @*   (see http://www.math.tu-berlin.de/polymake/),
+@*   (see http://www.math.tu-berlin.de/polymake/),
 @* respectively (only in the procedure triangularions) by the program
 @* - topcom by Joerg Rambau, Universitaet Bayreuth (see http://www.uni-bayreuth.de/
      departments/wirtschaftsmathematik/rambau/TOPCOM);
 @*   This library should rather be seen as an interface which allows to use a
      (very limited) number of options which polymake respectively topcom offers
      to compute with polytopes and fans and to make the results available in
+     (very limited) number of options which polymake respectively topcom offers
+     to compute with polytopes and fans and to make the results available in
      @sc{Singular} for further computations;
      moreover, the user familiar with @sc{Singular} does not have to learn the syntax
      of polymake or topcom, if the options offered here are sufficient for his
+     of polymake or topcom, if the options offered here are sufficient for his
      purposes.
 @*   Note, though, that the procedures concerned with planar polygons are
+@*   Note, though, that the procedures concerned with planar polygons are
      independent of both, polymake and topcom.
 …
 proc polymakePolytope (intmat polytope,list #)
 "USAGE:  polymakePolytope(polytope[,#]);   polytope list, # string
 ASSUME:  each row of polytope gives the coordinates of a lattice point of a
          polytope with their affine coordinates as given by the output of
+ASSUME:  each row of polytope gives the coordinates of a lattice point of a
+         polytope with their affine coordinates as given by the output of
          secondaryPolytope
 PURPOSE: the procedure calls polymake to compute the vertices of the polytope
+PURPOSE: the procedure calls polymake to compute the vertices of the polytope
          as well as its dimension and information on its facets
 RETURN:  list L with four entries
 @*            L[1] : an integer matrix whose rows are the coordinates of vertices
                      of the polytope
+                     of the polytope
 @*            L[2] : the dimension of the polytope
 @*            L[3] : a list whose i-th entry explains to which vertices the
                      ith vertex of the Newton polytope is connected
                      -- i.e. L[3][i] is an integer vector and an entry k in
                         there means that the vertex L[1][i] is connected to the
+                     ith vertex of the Newton polytope is connected
+                     -- i.e. L[3][i] is an integer vector and an entry k in
+                        there means that the vertex L[1][i] is connected to the
                         vertex L[1][k]
 @*            L[4] : an integer matrix whose rows mulitplied by
                      (1,var(1),...,var(nvar)) give a linear system of equations
+@*            L[4] : an integer matrix whose rows mulitplied by
+                     (1,var(1),...,var(nvar)) give a linear system of equations
                      describing the affine hull of the polytope,
                      i.e. the smallest affine space containing the polytope
 NOTE: -  for its computations the procedure calls the program polymake by
          Ewgenij Gawrilow, TU Berlin and Michael Joswig, TU Darmstadt;
          it therefore is necessary that this program is installed in order
+NOTE: -  for its computations the procedure calls the program polymake by
+         Ewgenij Gawrilow, TU Berlin and Michael Joswig, TU Darmstadt;
+         it therefore is necessary that this program is installed in order
          to use this procedure;
          see http://www.math.tu-berlin.de/polymake/
 @*    -  note that in the vertex edge graph we have changed the polymake
          convention which starts indexing its vertices by zero while we start
+@*    -  note that in the vertex edge graph we have changed the polymake
+         convention which starts indexing its vertices by zero while we start
          with one !
 @*    -  the procedure creates the file  /tmp/polytope.polymake which contains
          the polytope in polymake format; if you wish to use this for further
          computations with polymake, you have to use the procedure
+@*    -  the procedure creates the file  /tmp/polytope.polymake which contains
+         the polytope in polymake format; if you wish to use this for further
+         computations with polymake, you have to use the procedure
          polymakeKeepTmpFiles in before
 @*    -  moreover, the procedure creates the file /tmp/polytope.output which
          it deletes again before ending
 @*    -  it is possible to provide an optional second argument a string
          which then will be used instead of 'polytope' in the name of the
+         which then will be used instead of 'polytope' in the name of the
          polymake output file
 EXAMPLE: example polymakePolytope;   shows an example"
 …
+      }
+    }
+  }
+  }
   newveg=newveg[1,size(newveg)-1];
   execute("list nveg="+newveg+";");
 …
    "EXAMPLE:";
    echo=2;
    // the lattice points of the unit square in the plane
+   // the lattice points of the unit square in the plane
    list points=intvec(0,0),intvec(0,1),intvec(1,0),intvec(1,1);
    // the secondary polytope of this lattice point configuration is computed
 …
                      of the Newton polytope of f
 @*            L[2] : the dimension of the Newton polytope of f
 @*            L[3] : a list whose ith entry explains to which vertices the
                      ith vertex of the Newton polytope is connected
                      -- i.e. L[3][i] is an integer vector and an entry k in
+@*            L[3] : a list whose ith entry explains to which vertices the
+                     ith vertex of the Newton polytope is connected
+                     -- i.e. L[3][i] is an integer vector and an entry k in
                         there means that the vertex L[1][i] is
                         connected to the vertex L[1][k]
 @*            L[4] : an integer matrix whose rows mulitplied by
                      (1,var(1),...,var(nvar)) give a linear system of equations
+@*            L[4] : an integer matrix whose rows mulitplied by
+                     (1,var(1),...,var(nvar)) give a linear system of equations
                      describing the affine hull of the Newton polytope, i.e. the
                      smallest affine space containing the Newton polytope
 NOTE: -  if we replace the first column of L[4] by zeros, i.e. if we move
          the affine hull to the origin, then we get the equations for the
          orthogonal comploment of the linearity space of the normal fan dual
+NOTE: -  if we replace the first column of L[4] by zeros, i.e. if we move
+         the affine hull to the origin, then we get the equations for the
+         orthogonal comploment of the linearity space of the normal fan dual
          to the Newton polytope, i.e. we get the EQUATIONS that
          we need as input for polymake when computing the normal fan
 …
          TU Berlin and Michael Joswig, so it only works if polymake is installed;
          see http://www.math.tu-berlin.de/polymake/
 @*    -  the procedure creates the file  /tmp/newtonPolytope.polymake which
          contains the polytope in polymake format and which can be used for
+@*    -  the procedure creates the file  /tmp/newtonPolytope.polymake which
+         contains the polytope in polymake format and which can be used for
          further computations with polymake
 @*    -  moreover, the procedure creates the file /tmp/newtonPolytope.output
+@*    -  moreover, the procedure creates the file /tmp/newtonPolytope.output
          and deletes it again before ending
 @*    -  it is possible to give as an optional second argument a string
          which then will be used instead of 'newtonPolytope' in the name of
+@*    -  it is possible to give as an optional second argument a string
+         which then will be used instead of 'newtonPolytope' in the name of
          the polymake output file
 EXAMPLE: example newtonPolytope;   shows an example"
+{
   int i,j;
   // compute the list of exponent vectors of the polynomial,
+  // compute the list of exponent vectors of the polynomial,
   // which are the lattice points
   // whose convex hull is the Newton polytope of f
 …
    np[2];
    // np[3] contains information how the vertices are connected to each other,
    // e.g. the first vertex (number 0) is connected to the second, third and
+   // e.g. the first vertex (number 0) is connected to the second, third and
    //      fourth vertex
    np[3][1];
 …
    np[1];
    // its dimension is
    np[2];
    // the Newton polytope is contained in the affine space given
+   np[2];
+   // the Newton polytope is contained in the affine space given
    //     by the equations
    np[4]*M;
 …
 proc newtonPolytopeLP (poly f)
 "USAGE:  newtonPolytopeLP(f);  f poly
 RETURN: list, the exponent vectors of the monomials occuring in f,
+RETURN: list, the exponent vectors of the monomials occuring in f,
               i.e. the lattice points of the Newton polytope of f
 EXAMPLE: example normalFan;   shows an example"
 …
 proc normalFan (intmat vertices,intmat affinehull,list graph,int er,list #)
 "USAGE:  normalFan (vert,aff,graph,rays,[,#]);   vert,aff intmat,  graph list, rays int, # string
 ASSUME:  - vert is an integer matrix whose rows are the coordinate of
            the vertices of a convex lattice polytope;
+ASSUME:  - vert is an integer matrix whose rows are the coordinate of
+           the vertices of a convex lattice polytope;
 @*       - aff describes the affine hull of this polytope, i.e.
            the smallest affine space containing it, in the following sense:
            denote by n the number of columns of vert, then multiply aff by
            (1,x(1),...,x(n)) and set the resulting terms to zero in order to
+           the smallest affine space containing it, in the following sense:
+           denote by n the number of columns of vert, then multiply aff by
+           (1,x(1),...,x(n)) and set the resulting terms to zero in order to
            get the equations for the affine hull;
 @*       - the ith entry of graph is an integer vector describing to which
            vertices the ith vertex is connected, i.e. a k as entry means that
+@*       - the ith entry of graph is an integer vector describing to which
+           vertices the ith vertex is connected, i.e. a k as entry means that
            the vertex vert[i] is connected to vert[k];
 @*       - the integer rays is either one (if the extreme rays should be
+@*       - the integer rays is either one (if the extreme rays should be
            computed) or zero (otherwise)
 RETURN:  list, the ith entry of L[1] contains information about the cone in the
                normal fan dual to the ith vertex of the polytope
 @*             L[1][i][1] = integer matrix representing the inequalities which
+RETURN:  list, the ith entry of L[1] contains information about the cone in the
+               normal fan dual to the ith vertex of the polytope
+@*             L[1][i][1] = integer matrix representing the inequalities which
                             describe the cone dual to the ith vertex
 @*             L[1][i][2] = a list which contains the inequalities represented
                             by L[i][1] as a list of strings, where we use the
+@*             L[1][i][2] = a list which contains the inequalities represented
+                            by L[i][1] as a list of strings, where we use the
                             variables x(1),...,x(n)
 @*             L[1][i][3] = only present if 'er' is set to 1; in that case it is
                             an interger matrix whose rows are the extreme rays
+                            an interger matrix whose rows are the extreme rays
                             of the cone
 @*             L[2] = is an integer matrix whose rows span the linearity space
                       of the fan, i.e. the linear space which is contained in
+@*             L[2] = is an integer matrix whose rows span the linearity space
+                      of the fan, i.e. the linear space which is contained in
                       each cone
 NOTE:    - the procedure calls for its computation polymake by Ewgenij Gawrilow,
            TU Berlin and Michael Joswig, so it only works if polymake is
+           TU Berlin and Michael Joswig, so it only works if polymake is
            installed;
            see http://www.math.tu-berlin.de/polymake/
 @*       - in the optional argument # it is possible to hand over other names
+@*       - in the optional argument # it is possible to hand over other names
            for the variables to be used -- be careful, the format must be correct
            which is not tested, e.g. if you want the variable names to be
 …
   list ineq; // stores the inequalities of the cones
   int i,j,k;
   // we work over the following ring
+  // we work over the following ring
   execute("ring ineqring=0,x(1.."+string(ncols(vertices))+"),lp;");
   string greatersign=">";
 …
   for (i=1;i<=nrows(vertices);i++)
+  {
     // first we produce for each vertex in the polytope
+    // first we produce for each vertex in the polytope
     // the inequalities describing the dual cone in the normal fan
     list pp;  // contain strings representing the inequalities
+    list pp;  // contain strings representing the inequalities
               // describing the normal cone
     intmat ie[size(graph[i])][ncols(vertices)]; // contains the inequalities
+    intmat ie[size(graph[i])][ncols(vertices)]; // contains the inequalities
                                                 // as rows
     // consider all the vertices to which the ith vertex in the
+    // consider all the vertices to which the ith vertex in the
     // polytope is connected by an edge
     for (j=1;j<=size(graph[i]);j++)
+    {
       // produce the vector ie_j pointing from the jth vertex to the ith vertex;
       // this will be the jth inequality for the cone in the normal fan dual to
+      // this will be the jth inequality for the cone in the normal fan dual to
       // the ith vertex
       ie[j,1..ncols(vertices)]=vertices[i,1..ncols(vertices)]-vertices[graph[i][j],1..ncols(vertices)];
 …
       p=(VAR*EXP)[1,1];
       pl,pr=0,0;
       // separate the terms with positive coefficients in p from
+      // separate the terms with positive coefficients in p from
       // those with negative coefficients
       for (k=1;k<=size(p);k++)
 …
+        }
+      }
       // build the string which represents the jth inequality
+      // build the string which represents the jth inequality
       // for the cone dual to the ith vertex
       // as polynomial inequality of type string, and store this
+      // as polynomial inequality of type string, and store this
       // in the list pp as jth entry
       pp[j]=string(pl)+" "+greatersign+" "+string(pr);
 …
     // create the file ineq.output
     write(":w /tmp/ineq.output","");
     int dimension; // keeps the dimension of the intersection the
+    int dimension; // keeps the dimension of the intersection the
                    // bad cones with the u11tobeseencone
     for (i=1;i<=size(ineq);i++)
+    {
       i,". Cone of ",nrows(vertices); // indicate how many
+      i,". Cone of ",nrows(vertices); // indicate how many
                                       // vertices have been dealt with
       ungleichungen=intmatToPolymake(ineq[i][1],"rays");
 …
+  }
   // get the linearity space
   return(list(ineq,linearity));
+  return(list(ineq,linearity));
+}
 example
 …
 proc groebnerFan (poly f,list #)
 "USAGE:  groebnerFan(f[,#]);  f poly, # string
 RETURN:  list, the ith entry of L[1] contains information about the ith cone
                in the Groebner fan dual to the ith vertex in the Newton
+RETURN:  list, the ith entry of L[1] contains information about the ith cone
+               in the Groebner fan dual to the ith vertex in the Newton
                polytope of the f
 @*             L[1][i][1] = integer matrix representing the inequalities
                             which describe the cone
 @*             L[1][i][2] = a list which contains the inequalities represented
+@*             L[1][i][1] = integer matrix representing the inequalities
+                            which describe the cone
+@*             L[1][i][2] = a list which contains the inequalities represented
                             by L[1][i][1] as a list of strings
 @*             L[1][i][3] = an interger matrix whose rows are the extreme rays
+@*             L[1][i][3] = an interger matrix whose rows are the extreme rays
                             of the cone
 @*             L[2] = is an integer matrix whose rows span the linearity space
                       of the fan, i.e. the linear space which is contained
                       in each cone
 @*             L[3] = the Newton polytope of f in the format of the procedure
+@*             L[2] = is an integer matrix whose rows span the linearity space
+                      of the fan, i.e. the linear space which is contained
+                      in each cone
+@*             L[3] = the Newton polytope of f in the format of the procedure
                       newtonPolytope
 @*             L[4] = integer matrix where each row represents the exponet
+@*             L[4] = integer matrix where each row represents the exponet
                       vector of one monomial occuring in the input polynomial
 NOTE: - if you have already computed the Newton polytope of f then you might want
         to use the procedure normalFan instead in order to avoid doing costly
+        to use the procedure normalFan instead in order to avoid doing costly
         computation twice
 @*    - the procedure calls for its computation polymake by Ewgenij Gawrilow,
         TU Berlin and Michael Joswig, so it only works if polymake is installed;
         see http://www.math.tu-berlin.de/polymake/
 @*    - the procedure creates the file  /tmp/newtonPolytope.polymake which
         contains the Newton polytope of f in polymake format and which can
+@*    - the procedure creates the file  /tmp/newtonPolytope.polymake which
+        contains the Newton polytope of f in polymake format and which can
         be used for further computations with polymake
 @*    - it is possible to give as an optional second argument as string which
         then will be used instead of 'newtonPolytope' in the name of the
+@*    - it is possible to give as an optional second argument as string which
+        then will be used instead of 'newtonPolytope' in the name of the
         polymake output file
 EXAMPLE: example groebnerFan;   shows an example"
+{
   int i,j;
   // compute the list of exponent vectors of the polynomial, which are
+  // compute the list of exponent vectors of the polynomial, which are
   // the lattice points whose convex hull is the Newton polytope of f
   intmat exponents[size(f)][nvars(basering)];
 …
 proc intmatToPolymake (intmat M,string art)
 "USAGE:  intmatToPolymake(M,art);  M intmat, art string
 ASSUME:  - M is an integer matrix which should be transformed into polymake
+ASSUME:  - M is an integer matrix which should be transformed into polymake
            format;
 @*       - art is one of the following strings:
 @*           + 'rays'   : indicating that a first column of 0's should be added
 @*           + 'points' : indicating that a first column of 1's should be added
 RETURN:  string, the matrix is transformed in a string and a first column has
+@*           + 'points' : indicating that a first column of 1's should be added
+RETURN:  string, the matrix is transformed in a string and a first column has
                  been added
 EXAMPLE: example intmatToPolymake;   shows an example"
 …
     string anf="0 ";
+  }
   else
+  else
+  {
     string anf="1 ";
 …
 proc polymakeToIntmat (string pm,string art)
 "USAGE:  polymakeToIntmat(pm,art);  pm, art string
 ASSUME:  pm is the result of calling polymake with one 'argument' like
          VERTICES, AFFINE_HULL, etc., so that the first row of the string is
          the name of the corresponding 'argument' and the further rows contain
+ASSUME:  pm is the result of calling polymake with one 'argument' like
+         VERTICES, AFFINE_HULL, etc., so that the first row of the string is
+         the name of the corresponding 'argument' and the further rows contain
          the result which consists of vectors either over the integers
          or over the rationals
 …
                  from the second row, where each row has been multiplied with the
                  lowest common multiple of the denominators of its entries as if
                  it is an integer matrix; moreover, if art=='affine', then
                  the first column is omitted since we only want affine
+                 it is an integer matrix; moreover, if art=='affine', then
+                 the first column is omitted since we only want affine
                  coordinates
 EXAMPLE: example polymakeToIntmat;   shows an example"
 …
     pm=stringdelete(pm,1);
+  }
   pm=stringdelete(pm,1);
   // find out how many entries each vector has, namely one more
+  pm=stringdelete(pm,1);
+  // find out how many entries each vector has, namely one more
   // than 'spaces' in a row
   int i=1;
 …
   // if we want to have affine coordinates
   if (art=="affine")
+  {
+  {
     s--; // then there is one column less
     // and the entry of the first column (in the first row) has to be removed
 …
     pm=stringdelete(pm,1);
+  }
   // we add two line breaks at the end in order to have this as
+  // we add two line breaks at the end in order to have this as
   // a stopping criterion
   pm=pm+zeilenumbruch+zeilenumbruch;
 …
         z++;
         pm[i]=",";
         // if we want to have affine coordinates,
+        // if we want to have affine coordinates,
         // then we have to delete the first entry in each row
         if (art=="affine")
 …
       if (pm[i]==" ")
+      {
         pm[i]=",";
+        pm[i]=",";
+      }
+    }
 …
     pm=stringdelete(pm,size(pm));
+  }
   // since the matrix could be over the rationals,
+  // since the matrix could be over the rationals,
   // we need a ring with rational coefficients
   ring zwischering=0,x,lp;
+  ring zwischering=0,x,lp;
   // create the matrix with the elements of pm as entries
   execute("matrix mm["+string(z)+"]["+string(s)+"]="+pm+";");
 …
 proc triangulations (list polygon)
 "USAGE:  triangulations(polygon); list polygon
 ASSUME:  polygon is a list of integer vectors of the same size representing
          the affine coordinates of the lattice points
+ASSUME:  polygon is a list of integer vectors of the same size representing
+         the affine coordinates of the lattice points
 PURPOSE: the procedure considers the marked polytope given as the convex hull of
          the lattice points and with these lattice points as markings; it then
          computes all possible triangulations of this marked polytope
+         computes all possible triangulations of this marked polytope
 RETURN:  list, each entry corresponds to one triangulation and the ith entry is
                itself a list of integer vectors of size three, where each integer
 …
 NOTE:- the procedure calls for its computations the program points2triangs
        from the program topcom by Joerg Rambau, Universitaet Bayreuth; it
        therefore is necessary that this program is installed in order to use
+       therefore is necessary that this program is installed in order to use
        this  procedure; see
 @*     http://www.uni-bayreuth.de/departments/wirtschaftsmathematik/rambau/TOPCOM
 @*   - the procedure creates the files /tmp/triangulationsinput and
+@*   - the procedure creates the files /tmp/triangulationsinput and
        /tmp/triangulationsoutput;
        the former is used as input for points2triangs and the latter is its
        output containing the triangulations of corresponding to points in the
        format of points2triangs; if you wish to use this for further
        computations with topcom, you have to use the procedure
+       the former is used as input for points2triangs and the latter is its
+       output containing the triangulations of corresponding to points in the
+       format of points2triangs; if you wish to use this for further
+       computations with topcom, you have to use the procedure
        polymakeKeepTmpFiles in before
 @*   - note that an integer i in an integer vector representing a triangle
        refers to the ith lattice point, i.e. polygon[i]; this convention is
        different from TOPCOM's convention, where i would refer to the i-1st
+@*   - note that an integer i in an integer vector representing a triangle
+       refers to the ith lattice point, i.e. polygon[i]; this convention is
+       different from TOPCOM's convention, where i would refer to the i-1st
        lattice point
 EXAMPLE: example triangulations;   shows an example"
+{
   int i,j;
   // prepare the input for points2triangs by writing the input polygon in the
+  // prepare the input for points2triangs by writing the input polygon in the
   // necessary format
   string spi="[";
 …
     system("sh","cd /tmp; rm -f triangulationsinput; rm -f triangulationsoutput");
+  }
   // preprocessing of p2t if points2triangs is version >= 0.15
+  // preprocessing of p2t if points2triangs is version >= 0.15
   // brings p2t to the format of version 0.14
   string np2t; // takes the triangulations in Singular format
 …
+      }
       else
+      {
+      {
         np2t=np2t+p2t[i];
+      }
 …
+  {
     if (np2t[size(np2t)]!=";")
+    {
+    {
       np2t=np2t+p2t[size(p2t)-1]+p2t[size(p2t)];
+    }
 …
           np2t=np2t+"))";
           i++;
+        }
+        }
         else
+        {
 …
    "EXAMPLE:";
    echo=2;
    // the lattice points of the unit square in the plane
+   // the lattice points of the unit square in the plane
    list polygon=intvec(0,0),intvec(0,1),intvec(1,0),intvec(1,1);
    // the triangulations of this lattice point configuration are computed
 …
   int i,j,k,l;
   intmat N[2][2]; // is used to compute areas of triangles
   intvec vertex;  // stores a point in the secondary polytope as
+  intvec vertex;  // stores a point in the secondary polytope as
                   // intermediate result
   int eintrag;
   int halt;
   intmat secpoly[size(triangs)][size(polygon)];   // stores all lattice points
+  intmat secpoly[size(triangs)][size(polygon)];   // stores all lattice points
                                                   // of the secondary polytope
   // consider each triangulation and compute the corresponding point
+  // consider each triangulation and compute the corresponding point
   // in the secondary polytope
   for (i=1;i<=size(triangs);i++)
+  {
     // for each triangulation we have to compute the coordinates
+    // for each triangulation we have to compute the coordinates
     // corresponding to each marked point
     for (j=1;j<=size(polygon);j++)
+    {
       eintrag=0;
       // for each marked point we have to consider all triangles in the
+      // for each marked point we have to consider all triangles in the
       // triangulation which involve this particular point
       for (k=1;k<=size(triangs[i]);k++)
 …
     secpoly[i,1..size(polygon)]=vertex;
+  }
   return(list(secpoly,triangs));
+  return(list(secpoly,triangs));
+}
 example
 …
 proc secondaryFan (list polygon,list #)
 "USAGE:  secondaryFan(polygon[,#]); list polygon, list #
 ASSUME:  - polygon is a list of integer vectors of the same size representing
+ASSUME:  - polygon is a list of integer vectors of the same size representing
            the affine coordinates of lattice points
 @*       - if the triangulations of the corresponding polygon have already been
            computed with the procedure triangulations then these can be given
            as a second (optional) argument in order to avoid doing this
+@*       - if the triangulations of the corresponding polygon have already been
+           computed with the procedure triangulations then these can be given
+           as a second (optional) argument in order to avoid doing this
            computation again
 PURPOSE: the procedure considers the marked polytope given as the convex hull of
          the lattice points and with these lattice points as markings; it then
          computes the lattice points of the secondary polytope given by this
+         computes the lattice points of the secondary polytope given by this
          marked polytope which correspond to the triangulations computed by
          the procedure triangulations
 RETURN:  list, the ith entry of L[1] contains information about the ith cone in
                the secondary fan of the polygon, i.e. the cone dual to the
+RETURN:  list, the ith entry of L[1] contains information about the ith cone in
+               the secondary fan of the polygon, i.e. the cone dual to the
                ith vertex of the secondary polytope
 @*             L[1][i][1] = integer matrix representing the inequalities which
+@*             L[1][i][1] = integer matrix representing the inequalities which
                             describe the cone dual to the ith vertex
 @*             L[1][i][2] = a list which contains the inequalities represented
+@*             L[1][i][2] = a list which contains the inequalities represented
                             by L[1][i][1] as a list of strings, where we use the
                             variables x(1),...,x(n)
 @*             L[1][i][3] = only present if 'er' is set to 1; in that case it is
                             an interger matrix whose rows are the extreme rays
+                            an interger matrix whose rows are the extreme rays
                             of the cone
 @*             L[2] = is an integer matrix whose rows span the linearity space
                       of the fan, i.e. the linear space which is contained in
+@*             L[2] = is an integer matrix whose rows span the linearity space
+                      of the fan, i.e. the linear space which is contained in
                       each cone
 @*             L[3] = the secondary polytope in the format of the procedure
+@*             L[3] = the secondary polytope in the format of the procedure
                       polymakePolytope
 @*             L[4] = the list of triangulations corresponding to the vertices
+@*             L[4] = the list of triangulations corresponding to the vertices
                       of the secondary polytope
 NOTE:- the procedure calls for its computation polymake by Ewgenij Gawrilow,
        TU Berlin and Michael Joswig, so it only works if polymake is installed;
        see http://www.math.tu-berlin.de/polymake/
 @*   - in the optional argument # it is possible to hand over other names for
+@*   - in the optional argument # it is possible to hand over other names for
        the variables to be used -- be careful, the format must be correct
        which is not tested, e.g. if you want the variable names to be
        u00,u10,u01,u11 then you must hand over the string 'u11,u10,u01,u11'
 @*   - if the triangluations are not handed over as optional argument the
        procedure calls for its computation of these triangulations the program
        points2triangs from the program topcom by Joerg Rambau, Universitaet
        Bayreuth; it therefore is necessary that this program is installed in
+@*   - if the triangluations are not handed over as optional argument the
+       procedure calls for its computation of these triangulations the program
+       points2triangs from the program topcom by Joerg Rambau, Universitaet
+       Bayreuth; it therefore is necessary that this program is installed in
        order to use this procedure; see
 @*     http://www.uni-bayreuth.de/departments/wirtschaftsmathematik/rambau/TOPCOM
 …
 proc cycleLength (list boundary,intvec interior)
 "USAGE:  cycleLength(boundary,interior); list boundary, intvec interior
 ASSUME:  boundary is a list of integer vectors describing a cycle in some
          convex lattice polygon around the lattice point interior ordered
+ASSUME:  boundary is a list of integer vectors describing a cycle in some
+         convex lattice polygon around the lattice point interior ordered
          clock wise
 RETURN:  string, the cycle length of the corresponding cycle in the dual
 …
+{
   int j;
   // create a ring whose variables are indexed by the points in
+  // create a ring whose variables are indexed by the points in
   // boundary resp. by interior
   string rst="ring cyclering=0,(u"+string(interior[1])+string(interior[2]);
 …
    // interior is a lattice point in the interior of this lattice polygon
    intvec interior=1,1;
    // compute the general cycle length of a cycle of the corresponding cycle
+   // compute the general cycle length of a cycle of the corresponding cycle
    // in the dual tropical curve, note that (0,1) and (2,1) do not contribute
    cycleLength(boundary,interior);
 …
 proc splitPolygon (list markings)
 "USAGE:  splitPolygon (markings);  markings list
 ASSUME:  markings is a list of integer vectors representing lattice points in
          the plane which we consider as the marked points of the convex lattice
+ASSUME:  markings is a list of integer vectors representing lattice points in
+         the plane which we consider as the marked points of the convex lattice
          polytope spanned by them
 PURPOSE: split the marked points in the vertices, the points on the facets
+PURPOSE: split the marked points in the vertices, the points on the facets
          which are not vertices, and the interior points
 RETURN:  list, L consisting of three lists
 …
 @*                       L[1][i][1] = intvec, the coordinates of the ith vertex
 @*                       L[1][i][2] = int, the position of L[1][i][1] in markings
 @*             L[2][i] : represents the lattice points on the facet of the
                          polygon with endpoints L[1][i] and L[1][i+1]
+@*             L[2][i] : represents the lattice points on the facet of the
+                         polygon with endpoints L[1][i] and L[1][i+1]
                          (i considered modulo size(L[1]))
 @*                       L[2][i][j][1] = intvec, the coordinates of the jth
+@*                       L[2][i][j][1] = intvec, the coordinates of the jth
                                                  lattice point on that facet
 @*                       L[2][i][j][2] = int, the position of L[2][i][j][1]
+@*                       L[2][i][j][2] = int, the position of L[2][i][j][1]
                                               in markings
 @*             L[3]    : represents the interior lattice points of the polygon
+@*             L[3]    : represents the interior lattice points of the polygon
 @*                       L[3][i][1] = intvec, coordinates of ith interior point
 @*                       L[3][i][2] = int, the position of L[3][i][1] in markings
 …
   vert[1]=pb[2];
   int i,j,k;      // indices
   list boundary;  // stores the points on the facets of the
+  list boundary;  // stores the points on the facets of the
                   // polygon which are not vertices
   // append to the boundary points as well as to the vertices
+  // append to the boundary points as well as to the vertices
   // the first vertex a second time
   pb[1]=pb[1]+list(pb[1][1]);
 …
   // store the information on the boundary in vert[2]
   vert[2]=boundary;
   // find the remaining points in the input which are not on
+  // find the remaining points in the input which are not on
   // the boundary by checking
   // for each point in markings if it is contained in pb[1]
 …
   // store the interior points in vert[3]
   vert[3]=interior;
   // add to each point in vert the index which it gets from
+  // add to each point in vert the index which it gets from
   // its position in the input markings;
   // do so for ver[1]
 …
+    }
     vert[3][i]=list(vert[3][i],j);
+  }
+  }
   return(vert);
+}
 …
    "EXAMPLE:";
    echo=2;
    // the lattice polygon spanned by the points (0,0), (3,0) and (0,3)
+   // the lattice polygon spanned by the points (0,0), (3,0) and (0,3)
    // with all integer points as markings
    list polygon=intvec(1,1),intvec(3,0),intvec(2,0),intvec(1,0),
 …
 proc eta (list triang,list polygon)
 "USAGE:  eta(triang,polygon);  triang, polygon list
 ASSUME:  polygon has the format of the output of splitPolygon, i.e. it is a
          list with three entries describing a convex lattice polygon in the
+ASSUME:  polygon has the format of the output of splitPolygon, i.e. it is a
+         list with three entries describing a convex lattice polygon in the
          following way:
 @*       polygon[1] : is a list of lists; for each i the entry polygon[1][i][1]
                       is a lattice point which is a vertex of the lattice
+@*       polygon[1] : is a list of lists; for each i the entry polygon[1][i][1]
+                      is a lattice point which is a vertex of the lattice
                       polygon, and polygon[1][i][2] is an integer assigned to
                       this lattice point as identifying index
 @*       polygon[2] : is a list of lists; for each vertex of the polygon,
                       i.e. for each entry in polygon[1], it contains a list
                       polygon[2][i], which contains the lattice points on the
                       facet with endpoints polygon[1][i] and polygon[1][i+1]
+@*       polygon[2] : is a list of lists; for each vertex of the polygon,
+                      i.e. for each entry in polygon[1], it contains a list
+                      polygon[2][i], which contains the lattice points on the
+                      facet with endpoints polygon[1][i] and polygon[1][i+1]
                       - i considered mod size(polygon[1]);
                       each such lattice point contributes an entry
+                      each such lattice point contributes an entry
                       polygon[2][i][j][1] which is an integer
                       vector giving the coordinate of the lattice point and an
+                      vector giving the coordinate of the lattice point and an
                       entry polygon[2][i][j][2] which is the identifying index
 @*       polygon[3] : is a list of lists, where each entry corresponds to a
                       lattice point in the interior of the polygon, with
+@*       polygon[3] : is a list of lists, where each entry corresponds to a
+                      lattice point in the interior of the polygon, with
                       polygon[3][j][1] being the coordinates of the point
                       and polygon[3][j][2] being the identifying index;
 @*       triang is a list of integer vectors all of size three describing a
          triangulation of the polygon described by polygon; if an entry of
+@*       triang is a list of integer vectors all of size three describing a
+         triangulation of the polygon described by polygon; if an entry of
          triang is the vector (i,j,k) then the triangle is built from the vertices
          with indices i, j and k
 RETURN:  intvec, the integer vector eta describing that vertex of the Newton
                  polytope discriminant of the polygone whose dual cone in the
                  Groebner fan contains the cone of the secondary fan of the
+RETURN:  intvec, the integer vector eta describing that vertex of the Newton
+                 polytope discriminant of the polygone whose dual cone in the
+                 Groebner fan contains the cone of the secondary fan of the
                  polygon corresponding to the given triangulation
 NOTE:  for a better description of eta see Gelfand, Kapranov,
+NOTE:  for a better description of eta see Gelfand, Kapranov,
        Zelevinski: Discriminants, Resultants and multidimensional Determinants.
        Chapter 10.
 …
+{
   int i,j,k,l,m,n; // index variables
   list ordpolygon;   // stores the lattice points in the order
+  list ordpolygon;   // stores the lattice points in the order
                      // used in the triangulation
   list triangarea; // stores the areas of the triangulations
 …
   for (i=1;i<=size(triang);i++)
+  {
     // Note that the ith lattice point in orderedpolygon has the
+    // Note that the ith lattice point in orderedpolygon has the
     // number i-1 in the triangulation!
     N=ordpolygon[triang[i][1]]-ordpolygon[triang[i][2]],ordpolygon[triang[i][1]]-ordpolygon[triang[i][3]];
 …
+  }
   intvec ETA;        // stores the eta_ij
   int etaij;         // stores the part of eta_ij during computations
+  int etaij;         // stores the part of eta_ij during computations
                      // which comes from triangle areas
   int seitenlaenge;  // stores the part of eta_ij during computations
+  int seitenlaenge;  // stores the part of eta_ij during computations
                      // which comes from boundary facets
   list seiten;       // stores the lattice points on facets of the polygon
   intvec v;          // used to compute a facet length
   // 3) store first in seiten[i] all lattice points on the facet
+  // 3) store first in seiten[i] all lattice points on the facet
   //    connecting the ith vertex,
   //    i.e. polygon[1][i], with the i+1st vertex, i.e. polygon[1][i+1],
+  //    i.e. polygon[1][i], with the i+1st vertex, i.e. polygon[1][i+1],
   //    where we replace i+1
   //    1 if i=size(polygon[1]);
   //    then append the last entry of seiten once more at the very
+  //    then append the last entry of seiten once more at the very
   //    beginning of seiten, so
   //    that the index is shifted by one
 …
       if ((polygon[1][j][2]==triang[k][1]) or (polygon[1][j][2]==triang[k][2]) or (polygon[1][j][2]==triang[k][3]))
+      {
         // ... if so, add the area of the triangle to etaij
+        // ... if so, add the area of the triangle to etaij
         etaij=etaij+triangarea[k];
         // then check if that triangle has a facet which is contained
         // in one of the
+        // then check if that triangle has a facet which is contained
+        // in one of the
         // two facets of the polygon which are adjecent to the given vertex ...
         // these two facets are seiten[j] and seiten[j+1]
 …
               if ((seiten[n][l][2]==triang[k][m]) and (seiten[n][l][2]!=polygon[1][j][2]))
+              {
                 // if so, then compute the vector pointing from this
+                // if so, then compute the vector pointing from this
                 // lattice point to the vertex
                 v=polygon[1][j][1]-seiten[n][l][1];
                 // and the lattice length of this vector has to be
+                // and the lattice length of this vector has to be
                 // subtracted from etaij
                 etaij=etaij-abs(gcd(v[1],v[2]));
 …
     ETA[polygon[1][j][2]]=etaij;
+  }
   // 5) compute the eta_ij for all lattice points on the facets
+  // 5) compute the eta_ij for all lattice points on the facets
   //    of the polygon which are not vertices, these are the
   //    lattice points in polygon[2][1] to polygon[2][size(polygon[1])]
 …
+  {
     for (j=1;j<=size(polygon[2][i]);j++)
+    {
+    {
       // initialise etaij
       etaij=0;
 …
         if ((polygon[2][i][j][2]==triang[k][1]) or (polygon[2][i][j][2]==triang[k][2]) or (polygon[2][i][j][2]==triang[k][3]))
+        {
           // ... if so, add the area of the triangle to etaij
+          // ... if so, add the area of the triangle to etaij
           etaij=etaij+triangarea[k];
           // then check if that triangle has a facet which is contained in the
+          // then check if that triangle has a facet which is contained in the
           // facet of the polygon which contains the lattice point in question,
           // this is the facet seiten[i+1];
 …
             // ... and for each lattice point in the triangle ...
             for (m=1;m<=size(triang[k]);m++)
+            {
+            {
               // ... if they coincide and are not the vertex itself ...
               if ((seiten[i+1][l][2]==triang[k][m]) and (seiten[i+1][l][2]!=polygon[2][i][j][2]))
+              {
                 // if so, then compute the vector pointing from
+                // if so, then compute the vector pointing from
                 // this lattice point to the vertex
                 v=polygon[2][i][j][1]-seiten[i+1][l][1];
                 // and the lattice length of this vector contributes
+                // and the lattice length of this vector contributes
                 // to seitenlaenge
                 seitenlaenge=seitenlaenge+abs(gcd(v[1],v[2]));
 …
+        }
+      }
       // if the lattice point was a vertex of any triangle
+      // if the lattice point was a vertex of any triangle
       // in the triangulation ...
       if (etaij!=0)
 …
       if ((polygon[3][j][2]==triang[k][1]) or (polygon[3][j][2]==triang[k][2]) or (polygon[3][j][2]==triang[k][3]))
+      {
         // ... if so, add the area of the triangle to etaij
+        // ... if so, add the area of the triangle to etaij
         etaij=etaij+triangarea[k];
+      }
 …
    "EXAMPLE:";
    echo=2;
    // the lattice polygon spanned by the points (0,0), (3,0) and (0,3)
+   // the lattice polygon spanned by the points (0,0), (3,0) and (0,3)
    // with all integer points as markings
    list polygon=intvec(1,1),intvec(3,0),intvec(2,0),intvec(1,0),
 …
    // split the polygon in its vertices, its facets and its interior points
    list sp=splitPolygon(polygon);
    // define a triangulation by connecting the only interior point
+   // define a triangulation by connecting the only interior point
    //        with the vertices
    list triang=intvec(1,2,5),intvec(1,5,10),intvec(1,5,10);
 …
    eta(triang,sp);
+}
 /////////////////////////////////////////////////////////////////////////////
 …
+  }
   // check is the polygon is only a line segment given by more than two points;
   // for this first compute sum of the absolute values of the determinants
+  // for this first compute sum of the absolute values of the determinants
   // of the matrices whose
   // rows are the vectors pointing from the first to the second point
+  // rows are the vectors pointing from the first to the second point
   // and from the
   // the first point to the ith point for i=3,...,size(polygon);
+  // the first point to the ith point for i=3,...,size(polygon);
   // if this sum is zero
   // then the polygon is a line segment and we have to find its end points
 …
     intmat laenge[size(polygon)][size(polygon)];
     intvec mp;
     //   for this collect first all vectors pointing from one lattice
+    //   for this collect first all vectors pointing from one lattice
     //   point to the next,
     //   compute their pairwise angles and their lengths
     for (i=1;i<=size(polygon)-1;i++)
+    {
+    {
       for (j=i+1;j<=size(polygon);j++)
+      {
 …
     polygon=sortlistbyintvec(polygon,abstand);
     return(list(polygon,endpoints));
+  }
+  }
   ///////////////////////////////////////////////////////////////
   list orderedvertices;  // stores the vertices in an ordered way
   list minimisedorderedvertices;  // stores the vertices in an ordered way;
+  list minimisedorderedvertices;  // stores the vertices in an ordered way;
                                   // redundant ones removed
   list comparevertices; // stores vertices which should be compared to
+  list comparevertices; // stores vertices which should be compared to
                         // the testvertex
   orderedvertices[1]=polygon[1]; // set the starting vertex
   minimisedorderedvertices[1]=polygon[1]; // set the starting vertex
   intvec testvertex=polygon[1];  //vertex to which the others have to be compared
   intvec startvertex=polygon[1]; // keep the starting vertex to test,
+  intvec startvertex=polygon[1]; // keep the starting vertex to test,
                                  // when the end is reached
   int endtest;                   // is set to one, when the end is reached
   int startvertexfound;// is 1, once for some testvertex a candidate
                        // for the next vertex has been found
+  int startvertexfound;// is 1, once for some testvertex a candidate
+                       // for the next vertex has been found
   polygon=delete(polygon,1);    // delete the testvertex
   intvec v,w;
   int l=1;  // counts the vertices
   // the basic idea is that a vertex can be
+  // the basic idea is that a vertex can be
   // the next one on the boundary if all other vertices
   // lie to the right of the vector v pointing
+  // lie to the right of the vector v pointing
   // from the testvertex to this one; this can be tested
   // by checking if the determinant of the 2x2-matrix
+  // by checking if the determinant of the 2x2-matrix
   // with first column v and second column the vector w,
   // pointing from the testvertex to the new vertex,
+  // pointing from the testvertex to the new vertex,
   // is non-positive; if this is the case for all
   // new vertices, then the one in consideration is
+  // new vertices, then the one in consideration is
   // a possible choice for the next vertex on the boundary
   // and it is stored in naechste; we can then order
+  // and it is stored in naechste; we can then order
   // the candidates according to their distance from
   // the testvertex; then they occur on the boundary in that order!
 …
       v=polygon[i]-testvertex; // points from the testvertex to the ith vertex
       comparevertices=delete(polygon,i); // we needn't compare v to itself
       // we should compare v to the startvertex-testvertex;
+      // we should compare v to the startvertex-testvertex;
       // in the first calling of the loop
       // this is irrelevant since the difference will be zero;
+      // this is irrelevant since the difference will be zero;
       // however, later on it will
       // be vital, since we delete the vertices
+      // be vital, since we delete the vertices
       // which we have already tested from the list
       // of all vertices, and when all vertices
+      // of all vertices, and when all vertices
       // on the boundary have been found we would
       // therefore find a vertex in the interior
+      // therefore find a vertex in the interior
       // as candidate; but always testing against
       // the starting vertex, this cannot happen
       comparevertices[size(comparevertices)+1]=startvertex;
+      comparevertices[size(comparevertices)+1]=startvertex;
       for (j=1;(j<=size(comparevertices)) and (d<=0);j++)
+      {
 …
         d=det(D);
+      }
       if (d<=0) // if all determinants are non-positive,
+      if (d<=0) // if all determinants are non-positive,
       { // then the ith vertex is a candidate
         naechste[k]=list(polygon[i],i,scalarproduct(v,v));// we store the vertex,
 …
+    }
     if (size(naechste)>0) // then a candidate for the next vertex has been found
+    {
+    {
       startvertexfound=1; // at least once a candidate has been found
       naechste=sortlist(naechste,3);  // we order the candidates according
+      naechste=sortlist(naechste,3);  // we order the candidates according
                                       // to their distance from testvertex;
       for (j=1;j<=size(naechste);j++) // then we store them in this
+      for (j=1;j<=size(naechste);j++) // then we store them in this
       { // order in orderedvertices
         l++;
         orderedvertices[l]=naechste[j][1];
+      }
       testvertex=naechste[size(naechste)][1];  // we store the last one as
+      testvertex=naechste[size(naechste)][1];  // we store the last one as
                                                // next testvertex;
       // store the next corner of NSD
       minimisedorderedvertices[size(minimisedorderedvertices)+1]=testvertex;
       naechste=sortlist(naechste,2); // then we reorder the vertices
+      minimisedorderedvertices[size(minimisedorderedvertices)+1]=testvertex;
+      naechste=sortlist(naechste,2); // then we reorder the vertices
                                      // according to their position
       for (j=size(naechste);j>=1;j--) // and we delete them from the vertices
 …
+      }
+    }
     else // that means either that the vertex was inside the polygon,
     {    // or that we have reached the last vertex on the boundary
+    else // that means either that the vertex was inside the polygon,
+    {    // or that we have reached the last vertex on the boundary
          // of the polytope
       if (startvertexfound==0) // the vertex was in the interior;
+      if (startvertexfound==0) // the vertex was in the interior;
       { // we delete it and start all over again
         orderedvertices[1]=polygon[1];
         minimisedorderedvertices[1]=polygon[1];
+        orderedvertices[1]=polygon[1];
+        minimisedorderedvertices[1]=polygon[1];
         testvertex=polygon[1];
         startvertex=polygon[1];
         polygon=delete(polygon,1);
+      }
       else // we have reached the last vertex on the boundary of
+      else // we have reached the last vertex on the boundary of
       { // the polytope and can stop
         endtest=1;
 …
     kill naechste;
+  }
   // test if the first vertex in minimisedorderedvertices
+  // test if the first vertex in minimisedorderedvertices
   // is on the same line with the second and
   // the last, i.e. if we started our search in the
+  // the last, i.e. if we started our search in the
   // middle of a face; if so, delete it
   v=minimisedorderedvertices[2]-minimisedorderedvertices[1];
 …
     minimisedorderedvertices=delete(minimisedorderedvertices,1);
+  }
   // test if the first vertex in minimisedorderedvertices
+  // test if the first vertex in minimisedorderedvertices
   // is on the same line with the two
   // last ones, i.e. if we started our search at the end of a face;
+  // last ones, i.e. if we started our search at the end of a face;
   // if so, delete it
   v=minimisedorderedvertices[size(minimisedorderedvertices)-1]-minimisedorderedvertices[1];
 …
 proc cyclePoints (list triang,list points,int pt)
 "USAGE:      cyclePoints(triang,points,pt)  triang,points list, pt int
 ASSUME:      - points is a list of integer vectors describing the lattice
+ASSUME:      - points is a list of integer vectors describing the lattice
                points of a marked polygon;
 @*           - triang is a list of integer vectors describing a triangulation
                of the marked polygon in the sense that an integer vector of
                the form (i,j,k) describes the triangle formed by polygon[i],
+@*           - triang is a list of integer vectors describing a triangulation
+               of the marked polygon in the sense that an integer vector of
+               the form (i,j,k) describes the triangle formed by polygon[i],
                polygon[j] and polygon[k];
 @*           - pt is an integer between 1 and size(points), singling out a
+@*           - pt is an integer between 1 and size(points), singling out a
                lattice point among the marked points
 PURPOSE:     consider the convex lattice polygon, say P, spanned by all lattice
              points in points which in the triangulation triang are connected
              to the point points[pt]; the procedure computes all marked points
+PURPOSE:     consider the convex lattice polygon, say P, spanned by all lattice
+             points in points which in the triangulation triang are connected
+             to the point points[pt]; the procedure computes all marked points
              in points which lie on the boundary of that polygon, ordered
              clockwise
 RETURN:      list, of integer vectors which are the coordinates of the lattice
                    points on the boundary of the above mentioned polygon P, if
                    this polygon is not the empty set (that would be the case if
                    points[pt] is not a vertex of any triangle in the
+RETURN:      list, of integer vectors which are the coordinates of the lattice
+                   points on the boundary of the above mentioned polygon P, if
+                   this polygon is not the empty set (that would be the case if
+                   points[pt] is not a vertex of any triangle in the
                    triangulation); otherwise return the empty list
 EXAMPLE:     example cyclePoints;   shows an example"
+{
   int i,j; // indices
   list v;  // saves the indices of lattice points connected to the
+  list v;  // saves the indices of lattice points connected to the
            // interior point in the triangulation
   // save all points in triangulations containing pt in v
 …
     pts[i]=points[v[i]];
+  }
   // consider the convex polytope spanned by the points in pts,
+  // consider the convex polytope spanned by the points in pts,
   // find the points on the
   // boundary and order them clockwise
 …
    "EXAMPLE:";
    echo=2;
    // the lattice polygon spanned by the points (0,0), (3,0) and (0,3)
+   // the lattice polygon spanned by the points (0,0), (3,0) and (0,3)
    // with all integer points as markings
    list points=intvec(1,1),intvec(3,0),intvec(2,0),intvec(1,0),
                intvec(0,0),intvec(2,1),intvec(0,1),intvec(1,2),
                intvec(0,2),intvec(0,3);
    // define a triangulation
+   // define a triangulation
    list triang=intvec(1,2,5),intvec(1,5,7),intvec(1,7,9),intvec(8,9,10),
                intvec(1,8,9),intvec(1,2,8);
 …
 "USAGE:  latticeArea(polygon);   polygon list
 ASSUME:  polygon is a list of integer vectors in the plane
 RETURN:  int, the lattice area of the convex hull of the lattice points in
+RETURN:  int, the lattice area of the convex hull of the lattice points in
               polygon, i.e. twice the Euclidean area
 EXAMPLE: example polygonlatticeArea;   shows an example"
 …
 proc picksFormula (list polygon)
 "USAGE:  picksFormula(polygon);   polygon list
 ASSUME:  polygon is a list of integer vectors in the plane and consider their
          convex hull C
 RETURN:  list, L of three integersthe
+ASSUME:  polygon is a list of integer vectors in the plane and consider their
+         convex hull C
+RETURN:  list, L of three integersthe
 @*             L[1] : the lattice area of C, i.e. twice the Euclidean area
 @*             L[2] : the number of lattice points on the boundary of C
 …
     bdpts=bdpts+abs(gcd(edge[1],edge[2]));
+  }
   // Pick's formula says that the lattice area A, the number g of interior
+  // Pick's formula says that the lattice area A, the number g of interior
   // points and
   // the number b of boundary points are connected by the formula: A=b+2g-2
 …
 "USAGE:  ellipticNF(polygon);   polygon list
 ASSUME:  polygon is a list of integer vectors in the plane such that their
          convex hull C has precisely one interior lattice point, i.e. C is the
+         convex hull C has precisely one interior lattice point, i.e. C is the
          Newton polygon of an elliptic curve
 PURPOSE: compute the normal form of the polygon with respect to the unimodular
+PURPOSE: compute the normal form of the polygon with respect to the unimodular
          affine transformations T=A*x+v; there are sixteen different normal forms
          (see e.g. Bjorn Poonen, Fernando Rodriguez-Villegas: Lattice Polygons
                    and the number 12.  Amer. Math. Monthly  107  (2000),  no. 3,
+         (see e.g. Bjorn Poonen, Fernando Rodriguez-Villegas: Lattice Polygons
+                   and the number 12.  Amer. Math. Monthly  107  (2000),  no. 3,
 --250.)
 RETURN:  list, L such that
 @*             L[1] : list whose entries are the vertices of the normal form of
+@*             L[1] : list whose entries are the vertices of the normal form of
                       the polygon
 @*             L[2] : the matrix A of the unimodular transformation
 @*             L[3] : the translation vector v of the unimodular transformation
 @*             L[4] : list such that the ith entry is the image of polygon[i]
+@*             L[4] : list such that the ith entry is the image of polygon[i]
                       under the unimodular transformation T
 EXAMPLE: example ellipticNF;   shows an example"
 …
   intvec trans;    // stores the vector by which we have to translate the polygon
   intmat A[2][2];  // stores the matrix by which we have to transform the polygon
   matrix M[3][3];  // stores the projective coordinates of the points
+  matrix M[3][3];  // stores the projective coordinates of the points
                    // which are to be transformed
   matrix N[3][3];  // stores the projective coordinates of the points to
+  matrix N[3][3];  // stores the projective coordinates of the points to
                    // which M is to be transformed
   intmat T[3][3];  // stores the unimodular affine transformation in
+  intmat T[3][3];  // stores the unimodular affine transformation in
                    // projective form
   // add the second point of pg once again at the end
   pg=insert(pg,pg[2],size(pg));
   // if there is only one edge which has the maximal number of lattice points,
+  // if there is only one edge which has the maximal number of lattice points,
   // then M should be:
   M=pg[max],1,pg[max+1],1,pg[max+2],1;
 …
     M=pg[max],1,pg[max+1],1,pg[max+2],1;
     // the orientation of the polygon matters
     A=pg[max-1]-pg[max],pg[max+1]-pg[max];
+    A=pg[max-1]-pg[max],pg[max+1]-pg[max];
     if (det(A)==4)
+    {
 …
+    {
       max++;
+    }
+    }
     M=pg[max],1,pg[max+1],1,pg[max+2],1;
     N=0,1,1,1,2,1,2,1,1;
 …
    // the vertices of the normal form are
    nf[1];
    // it has been transformed by the unimodular affine transformation A*x+v
+   // it has been transformed by the unimodular affine transformation A*x+v
    // with matrix A
    nf[2];
 …
 "USAGE:  ellipticNFDB(n[,#]);   n int, # list
 ASSUME:  n is an integer between 1 and 16
 PURPOSE: this is a database storing the 16 normal forms of planar polygons with
+PURPOSE: this is a database storing the 16 normal forms of planar polygons with
          precisely one interior point up to unimodular affine transformations
 @*       (see e.g. Bjorn Poonen, Fernando Rodriguez-Villegas: Lattice Polygons
+@*       (see e.g. Bjorn Poonen, Fernando Rodriguez-Villegas: Lattice Polygons
                    and the number 12.  Amer. Math. Monthly  107  (2000),  no. 3,
 --250.)
 RETURN:  list, L such that
 @*             L[1] : list whose entries are the vertices of the nth normal form
 @*             L[2] : list whose entries are all the lattice points of the
                       nth normal form
 @*             L[3] : only present if the optional parameter # is present, and
                       then it is a polynomial in the variables (x,y) whose
+@*             L[1] : list whose entries are the vertices of the nth normal form
+@*             L[2] : list whose entries are all the lattice points of the
+                      nth normal form
+@*             L[3] : only present if the optional parameter # is present, and
+                      then it is a polynomial in the variables (x,y) whose
                       Newton polygon is the nth normal form
 NOTE:    the optional parameter is only allowed if the basering has the
+NOTE:    the optional parameter is only allowed if the basering has the
          variables x and y
 EXAMPLE: example ellipticNFDB;   shows an example"
 …
 proc polymakeKeepTmpFiles (int i)
 "USAGE:  polymakeKeepTmpFiles(int i);   i int
 PURPOSE: some procedures create files in the directory /tmp which are used for
+PURPOSE: some procedures create files in the directory /tmp which are used for
          computations with polymake respectively topcom; these will be removed
          when the corresponding procedure is left; however, it might be
+         when the corresponding procedure is left; however, it might be
          desireable to keep them for further computations with either polymake or
          topcom; this can be achieved by this procedure; call the procedure as:
 …
 static proc scalarproduct (intvec w,intvec v)
 "USAGE:      scalarproduct(w,v); w,v intvec
 ASSUME:      w and v are integer vectors of the same length
+ASSUME:      w and v are integer vectors of the same length
 RETURN:      int, the scalarproduct of v and w
 NOTE:        the procedure is called by findOrientedBoundary"
 …
+  {
     int m=nrows(M);
+  }
   else
 …
+  {
     return("");
+  }
   if (i==1)
 …
         k++;
+      }
       else
+      else
+      {
         stop=1;
 …
         k++;
+      }
       else
+      else
+      {
         stop=1;
 …
 static proc polygonToCoordinates (list points)
 "USAGE:      polygonToCoordinates(points);   points list
 ASSUME:      points is a list of integer vectors each of size two describing the
              marked points of a convex lattice polygon like the output of
+ASSUME:      points is a list of integer vectors each of size two describing the
+             marked points of a convex lattice polygon like the output of
              polygonDB
 RETURN:      list, the first entry is a string representing the coordinates
+RETURN:      list, the first entry is a string representing the coordinates
                    corresponding to the latticpoints seperated by commata
                    the second entry is a list where the ith entry is a string
                    representing the coordinate of corresponding to the ith
                    lattice point the third entry is the latex format of the
+                   the second entry is a list where the ith entry is a string
+                   representing the coordinate of corresponding to the ith
+                   lattice point the third entry is the latex format of the
                    first entry
 NOTE:        the procedure is called by fan"

Singular/LIB/primdec.lib

-                      rae9fd9
+                      r3f4e52
 ///////////////////////////////////////////////////////////////////////////////
 version="$Id: primdec.lib,v 1.147 2009-04-15 08:10:22 Singular Exp $";
+version="$Id: primdec.lib,v 1.148 2009-07-28 10:34:55 Singular Exp $";
 category="Commutative Algebra";
 info="
 …
     They work in any characteristic.@*
     Baserings must have a global ordering and no quotient ideal.
 PROCEDURES:

Singular/LIB/random.lib

-                      rae9fd9
+                      r3f4e52
 //(GMG/BM, last modified 22.06.96)
 ///////////////////////////////////////////////////////////////////////////////
 version="$Id: random.lib,v 1.20 2009-04-15 11:18:27 seelisch Exp $";
+version="$Id: random.lib,v 1.21 2009-07-28 10:34:56 Singular Exp $";
 category="General purpose";
 info="
 …
 proc sparseHomogIdeal (int k, int u, list #)
 "USAGE:   sparseid(k,u[,o,p,b]);  k,u,o,p,b integers
 RETURN:  ideal having k homogeneous generators, each of random degree in the
          interval [u,o], p percent of terms in degree d are 0, the remaining
          have random coefficients in the interval [1,b], (default: o=u, p=75,
+RETURN:  ideal having k homogeneous generators, each of random degree in the
+         interval [u,o], p percent of terms in degree d are 0, the remaining
+         have random coefficients in the interval [1,b], (default: o=u, p=75,
          b=30000)
 EXAMPLE: example sparseid; shows an example
 …
+   {
        id = maxideal(random(u, o)); // monomial basis of some degree
        m = sparsemat(size(id),1,p,b); // random coefficients
+       m = sparsemat(size(id),1,p,b); // random coefficients
        i[ii] = (matrix(id)*m)[1,1];
+   }

Singular/LIB/ratgb.lib

-                      rae9fd9
+                      r3f4e52
 //////////////////////////////////////////////////////////////////////////////
 version="$Id: ratgb.lib,v 1.16 2009-04-15 11:15:40 seelisch Exp $";
+version="$Id: ratgb.lib,v 1.17 2009-07-28 10:34:56 Singular Exp $";
 category="Noncommutative";
 info="
 …
       va = L3[w][2];
       for(z=1;z<=nvars(save)-is;z++)
+      {
         vb[z] = va[is+z];
+      {
+        vb[z] = va[is+z];
+      }
       tmp3[1] = "a";
 …
   setring A;
   IAppel1;
   def F1 = ratstd(IAppel1,2);
   lead(pGBid);
+  def F1 = ratstd(IAppel1,2);
+  lead(pGBid);
   setring F1; rGBid;
+}
 …
   setring A;
   IAppel2;
   def F1 = ratstd(IAppel2,2);
   lead(pGBid);
+  def F1 = ratstd(IAppel2,2);
+  lead(pGBid);
   setring F1; rGBid;
+}
 …
   setring A;
   IAppel4;
   def F1 = ratstd(IAppel4,2);
   lead(pGBid);
+  def F1 = ratstd(IAppel4,2);
+  lead(pGBid);
   setring F1; rGBid;
+}

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Changeset 3f4e52 in git

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Singular/LIB/algebra.lib

Singular/LIB/bfun.lib

Singular/LIB/crypto.lib

Singular/LIB/dmod.lib

Singular/LIB/dmodapp.lib

Singular/LIB/elim.lib

Singular/LIB/freegb.lib

Singular/LIB/general.lib

Singular/LIB/gmssing.lib

Singular/LIB/ncdecomp.lib

Singular/LIB/polymake.lib

Singular/LIB/primdec.lib

Singular/LIB/random.lib

Singular/LIB/ratgb.lib

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