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\begin{center}{\Large Local monomial orderings for integral closures of ideals }\end{center}
\section{Simple examples of integral closures of ideals}
The simplest type of example that gets across what integral closures of ideals are about is:
\[ I_m:=\langle y^m,\ x^m\rangle\subset \mathbf{F}[y,x]=:R,\]
having integral closure
\[ C(I_m,R):=\langle y^m,\ y^{m-1}x,\ \ldots,\ yx^{m-1},\ x^m\rangle\]
generated by all the monomials lying ``between'' $y^m$ and $x^m$.
But such monomial ideals rarely give insight into any monomial ordering,
so try binomial ideals such as:
\[ I_{m,n}:=\langle y^m-y^n,\ x^m-x^n\rangle\subset \mathbf{F}[y,x]=:R\]
with $m>n$.
Probably one can guess that the integral closure is related to either
$C(I_m,R)$ or to $C(I_n,R)$.
To see which try out {\tt normalI} in {\sc Singular} and {\tt integralClosure}
in {\sc Macaulay2}.
\begin{verbatim}
SINGULAR /
A Computer Algebra System for Polynomial Computations / version 3-1-5
0<
by: W. Decker, G.-M. Greuel, G. Pfister, H. Schoenemann \ Jul 2012
FB Mathematik der Universitaet, D-67653 Kaiserslautern \
> LIB "normal.lib";
> ring r=0,(y,x),dp;
> ideal i=y3-y2,x3-x2;
> list nor=normalI(i);nor;
[1]:
_[1]=y3-y2
_[2]=x3-x2
_[3]=-y2x2+y2x+yx2-yx
> ideal s=std(nor[1]);s;
s[1]=x3-x2
s[2]=y3-y2
s[3]=y2x2-y2x-yx2+yx
\end{verbatim}
\begin{verbatim}
Macaulay2, version 1.4
with packages: ConwayPolynomials, Elimination, IntegralClosure, LLLBases,
PrimaryDecomposition, ReesAlgebra, TangentCone
i1 : R=QQ[y,x];
i2 : I=ideal(y^3-y^2,x^3-x^2);
i3 : time IC=integralClosure(I);
-- used 0.243466 seconds
i4 : toString IC
o4 = ideal(x^3-x^2,
y^3-y^2,
y^2*x^2-y^2*x-y*x^2+y*x)
\end{verbatim}
It should be clear from these that $y^2x^2$ is not "between" $y^3$ and $x^3$,
but that $yx$ is between $y^2$ and $x^2$.
\section{ Local monomial orderings}
From this it should be clear that the triling entries are more important to an understanding of the integral closure above than the leading entries.
\begin{verbatim}
ring r=0,(y,x),ds;
> ideal i=y3-y2,x3-x2;
> list nor=normalI(i);nor;
[1]:
_[1]=-y2+y3
_[2]=-x2+x3
_[3]=-yx+y2x+yx2-y2x2
> ideal s=std(nor[1]);s;
s[1]=y2
s[2]=yx
s[3]=x2
\end{verbatim}
While we might have expected what {\tt normalI} gave us,
perhaps we (meaning at least the royal I) were not ready for
what standard (Gr\"obner) bases look like relative to local monomial orderings.
Here $x^2-x^3=x^2u_1$ and $y^2-y^3=y^2u_2$
with $u_1:=1-x$ and $u_2:=1-y$ polynomial units.
So the extra generator found is really of the form $yxu_1u_2$.
The lesson here is that interreduction is much trickier
with a local monomial ordering
in that $x^2-x^3$ could be reduced to $x^2-x^s$ for any $s>2$.
So saying that $x^2\in I$ above means only that $x^2u\in I$
for some polynomial unit $u$.
{\sc Macaulay2} is not really set up to compute integral closures
for local monomial orderings,
so instead the following gives a local answer based on the global output.
\begin{verbatim}
5 : S=QQ[y,x,MonomialOrder=>{Weights=>{-1,-1}},Global=>false];
i6 : phi=map(S,R,matrix{{y,x}});
i7 : ic=(flatten entries gens IC)/phi;
i8 : toString gens gb ideal ic
o8 = matrix {{x^2-x^3,
y*x-y^2*x-y*x^2+y^2*x^2,
y^2-y^3}}
\end{verbatim}
\section{Qth-power approach}
My {\sc Macaulay2} code based on the Qth-power algorithm for integral closures
of rings, uses a local monomial ordering, but is restricted currently to
positive characteristic, and probably runs well only for small $Q>0$.
That said,
\begin{verbatim}
i3 : R=ZZ/13[x,y,MonomialOrder=>{
Weights=>{-1,-1},
Weights=>{0,-1}},
Global=>false];
i4 : I={x^2-x^3,y^2-y^3};
i5 : wt={{-1,-1},{0,-1}};
i6 : time IC=idealClosure(R,{0},I,wt);
2 3 2 3
{x - x , y - y }
{1}
{x, y}
2 2
{x , x*y, y }
2 2
{x , x*y, y }
{1}
{x, y}
2 2
{x , x*y, y }
-- used 0.2507 seconds
i7 : toString IC
o7 = {x^2, x*y, y^2}
\end{verbatim}
does produce the local answer I'm advocating theoretically.
\end{document}