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-- Copyright 1999-2002 by Anton Leykin and Harrison Tsai
------------------------------------------------------------------------
-- This function computes the differential operators of order k or less
-- on an affine variety defined by an ideal I in a polynomial ring
------------------------------------------------------------------------
diffOps = method()
diffOps (RingElement, ZZ) := (f, k) -> (diffOps(ideal f, k))
diffOps (Ideal, ZZ) := (I, k) -> (
R := ring I;
W := makeWeylAlgebra( R, SetVariables=>false);
createDpairs W;
-- make coeffs of derivations
F := gens I;
-- make Dk
Dbasis := matrix {flatten apply( toList(0..k-1), i -> (
(entries symmetricPower(k-i, matrix{W.dpairVars#1}))#0 ) )};
xbasis := matrix {flatten apply( toList(0..k-1), i -> (
(entries symmetricPower(k-i, vars R))#0 ) )};
smallxbasis := matrix {flatten apply( toList(0..k-2), i -> (
(entries symmetricPower(k-1-i, vars R))#0 ) )};
pInfo(1, "Making the matrix of partial derivatives of elements in I...");
temp := transpose diff(transpose xbasis, F |
flatten ((transpose smallxbasis)*F) );
addon := directSum toList(rank target temp:matrix{{F}});
full := temp | addon;
pInfo(1, "Computing syzygies on an " | rank source full | " by " |
rank target full | " matrix...");
syzy := (syz full)^{0..rank source temp - 1};
pInfo(1, "Reducing matrix of " | rank source syzy | " by " |
rank target syzy | " syzygies with respect to I...");
syzy = gens gb (syzy % I);
hashTable {PolyGens => syzy, BasisElts => Dbasis}
)
----------------------------------------------------------------
-- Puts Rbasis of diffOps into the Weyl algebra representation
----------------------------------------------------------------
putWeylAlgebra = method()
putWeylAlgebra (HashTable) := (h) -> (
(h.BasisElts) * substitute(h.PolyGens, ring h.BasisElts)
)
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