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# eulerRestriction(CentralArrangement,List,ZZ) -- form the Euler restriction of a central multiarrangement

## Synopsis

• Function: eulerRestriction
• Usage:
eulerRestriction(A, m, i)
• Inputs:
• Outputs:
• , the Euler restriction of (A,m)

## Description

The Euler restriction of a multiarrangement (introduced by Abe, Terao, and Wakefield in The Euler multiplicity and addition–deletion theorems for multiarrangements, J. Lond. Math. Soc. (2) 77 (2008), no. 2, 335348.) generalizes restriction to multiarrangements in such a way that addition-deletion theorems hold. The underlying simple arrangement of the Euler restriction is simply the usual restriction; however, the multiplicities are generally smaller than the naive ones.

If all of the multiplicities are $1$, the same is true of the Euler restriction:

 i1 : R = QQ[x,y,z] o1 = R o1 : PolynomialRing i2 : A = arrangement {x,y,z,x-y,x-z} o2 = {x, y, z, x - y, x - z} o2 : Hyperplane Arrangement  i3 : (A'',m'') = eulerRestriction(A,{1,1,1,1,1},1) o3 = ({z, x, x - z}, {1, 1, 1}) o3 : Sequence i4 : restriction(A,1) o4 = {x, z, x, x - z} o4 : Hyperplane Arrangement  i5 : trim oo -- same underlying simple arrangement, different multiplicities o5 = {z, x, x - z} o5 : Hyperplane Arrangement 

If $({\mathcal A},m)$ is a free multiarrangement and so is $({\mathcal A},m')$, where $m'$ is obtained from $m$ by lowering a single multiplicity by one, the Euler restriction is free as well, and the modules of logarithmic derivations form a short exact sequence. See the paper of Abe, Terao and Wakefield for details.

 i6 : m = {2,2,2,2,1}; m' = {2,2,2,1,1}; i8 : (A'',m'') = eulerRestriction(A,m,3) o8 = ({z, y, y - z}, {2, 3, 1}) o8 : Sequence i9 : prune image der(A,m) 3 o9 = R o9 : R-module, free, degrees {3:3} i10 : prune image der(A,m') 3 o10 = R o10 : R-module, free, degrees {2..3, 3} i11 : prune image der(A'',m'') 2 o11 = (QQ[y..z]) o11 : QQ[y..z]-module, free, degrees {2:3}

It may be the case that the Euler restriction is free, while the naive restriction is not:

 i12 : A = arrangement "bracelet"; i13 : (B,m) = eulerRestriction(A,{1,1,1,1,1,1,1,1,1},0) o13 = ({x + x , x , x , x + x + x , x + x , x }, {1, 1, 1, 1, 1, 1}) 3 4 3 4 2 3 4 2 4 2 o13 : Sequence i14 : C = restriction(A,0) o14 = {x , x , x , x + x , x + x , x + x , x + x , x + x + x } 2 3 4 2 4 3 4 2 4 3 4 2 3 4 o14 : Hyperplane Arrangement  i15 : assert(isFreeModule prune image der B) -- one is free i16 : assert(not isFreeModule prune image der C) -- the other is not