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

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

# I := ideal(x^2 + y,2*x*y + y^2);

# F := 2*x*y + y^2;

GB := GBasis(I); 

# GB;
[2*x*y +y^2, x^2 +y, -y^3 -4*y^2]

# ReducedGBasis(I);
[y^3 +4*y^2, x*y +(1/2)*y^2, x^2 +y]

use R ::= RR[x,y], Lex;   // DegRevLex, DegLex
--> ERROR: Cannot find a variable named "RR" in scope.
Similarly named variables (that you might need to import) are: "NR", "R", "rk"
--> use R ::= RR[x,y], Lex;   // DegRevLex, De ...
-->           ^^

# K := NewFractionField(NewPolyRing(QQ,["a"]));

# use R ::= K[x,y];

# LM(F);
2*x*y

# LT(F);
x*y

# LC(F);
2

# LPP(F);
x*y

# F;
2*x*y +y^2

# / Procedura (funzione) per calcolare s-polinomi ([CLO, pag. 85]):
--> ERROR: Invalid start of expression
--> / Procedura (funzione) per calc ...
--> ^

# 

# Define spoly(f,g);

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

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

Define # EndDefine;

# ciao; 

# ///////////////////////////////////////////////////////

# / [CLO, pag. 90]
--> ERROR: Invalid start of expression
--> / [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]

# 

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

#