newPackage(
	"Sept19",
	Version => "1.0", 
    	Date => "Sept 19, 2016",
    	Authors => {Name => "Martin Helmer"},
    	Headline => "Examples for Class Sept 19",
    	DebuggingMode => false,
	Reload => true
    	);
export{"Spoly"
    }
Spoly = method(TypicalValue=>RingElement)
Spoly(RingElement,RingElement) := (f,g) -> (
    LcM:=lcm(leadMonomial f,leadMonomial g);
    Spol:=(LcM//leadTerm(f))*f-(LcM//leadTerm(g))*g;
    return Spol;
)


TEST ///
{*  
    restart
    needsPackage "Sept19"
*}
R=QQ[x,y,MonomialOrder=>GLex]
I=ideal(x^3-2*x*y,x^2*y-2*y^2+x)
J=ideal groebnerBasis I

--Now do Buchberger's algorithm 
F={I_0,I_1}
f3=Spoly(I_0,I_1)
--Append the new poly to our list F
F=append(F,f3)
f4=Spoly(f3,I_0)
F=append(F,f4)
--would still get non-zero remainder
--add another
f5=Spoly(f3,I_1)
F=append(F,f5)
--Now we finally get zero remainder
J
--Note that the last 3 we computed are infact enough...
J==ideal(F)

///

TEST ///
{*  
    restart
    --reduction with respect to a GB
*}
R=QQ[x,y,z]
f=z^2-x^4*y
I=ideal(-x^3+z,-x^2+y)
--does reduction with respect to a GB of I, even if you haven't given it one.
f%I
--how to find a GB in M2
I=ideal groebnerBasis I
f%I
///