Loading CoCoA packages from directory:
  /usr/local/cocoa-5.1/packages
   ______      ______      ___         ______
  / ____/___  / ____/___  /   |       / ____/
 / /   / __ `/ /   / __ `/ /| |______/___ `  
/ /___/ /_/ / /___/ /_/ / ___ /_____/___/ /  
`____/`____/`____/`____/_/  |_|    /_____/   
With CoCoALib and external libraries FROBBY READLINE 
indent(VersionInfo()); -- for information about this version

# indent(VersionInfo());
record[
  CoCoALibVersion := "0.99539",
  CoCoAVersion := "5.1.3",
  CompilationDate := "18 Feb 2016 at 09:13:03",
  CompilationDefines := "-DCoCoA_ULONG2LONG=1  -DCoCoA_WITH_BOOST  -DCoCoA_WITH_FROBBY    -DCoCoA_WITH_READLINE ",
  CompilationFlags := "-Wall -pedantic -fPIC -m64 -mtune=core2 -march=core2 -O2",
  Compiler := "g++",
  MachineIntNumBits := 32,
  MachineLongNumBits := 64
]

# // [CLO, pag. 90]

# use R ::= QQ[x,y], DegLex;

# f1 := x^3 - 2*x*y;

# f2 := x^2*y - 2*y^2 + x;

# I := ideal(f1,f2);

# G := GBasis(I);

# G;
[x^2, -2*y^2 +x, -x*y]

# ReducedGBasis(I);
[y^2 +(-1/2)*x, x*y, x^2]

# Define spoly(f,g);

Define # xgamma := lcm(LT(f),LT(g));

Define # Return (xgamma/LM(f))*f - (xgamma/LM(g))*g;

Define # EndDefine

# LT(f2);
x^2*y

# LM(f2);
x^2*y

# LM(f1);
x^3

# LM(3*x);
3*x

# LT(3*x);
x

# spoly(f1,f2);
-x^2

# // resto dell'algoritmo di divisione: NR

# f3 = NR(spoly(f1,f2),[f1,f2]);
--> ERROR: Cannot find a variable named "f3" in scope.
Similarly named variables (that you might need to import) are: "f1", "f2"
--> f3 = NR(spoly(f1,f2),[f1,f2]);
--> ^^

# f3 := NR(spoly(f1,f2),[f1,f2]);

# f3;
-x^2

# f4 := NR(spoly(f1,f3),[f1,f2,f3]); f4;
-2*x*y

# f5 := NR(spoly(f1,f4),[f1,f2,f3,f4]); f5;
0

# f5 := NR(spoly(f2,f3),[f1,f2,f3,f4]); f5;
-2*y^2 +x

# // Verifichiamo che [f1,f2,f3,f4,f5] e' base di Groebner:

# NR(spoly(f1,f2),[f1,f2,f3,f4,f5]);
0

# NR(spoly(f1,f3),[f1,f2,f3,f4,f5]);
0

# NR(spoly(f1,f4),[f1,f2,f3,f4,f5]);
0

# NR(spoly(f1,f5),[f1,f2,f3,f4,f5]);
0

# NR(spoly(f2,f3),[f1,f2,f3,f4,f5]);
0

# NR(spoly(f2,f4),[f1,f2,f3,f4,f5]);
0

# NR(spoly(f2,f5),[f1,f2,f3,f4,f5]);
0

# NR(spoly(f3,f4),[f1,f2,f3,f4,f5]);
0

# NR(spoly(f3,f5),[f1,f2,f3,f4,f5]);
0

# NR(spoly(f4,f5),[f1,f2,f3,f4,f5]);
0

# // Sfida: consultare il manuale e scrivere un ciclo per

# // verificare l'annullamento di tutti i resti della divisone

# // degli S-polinomi S(f_i,f_j) per [f1,f2,f3,f4,f5].

# ciao;
Bye.