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-- Copyright 1999-2002 by Anton Leykin and Harrison Tsai
-----------------------------------------------------------------------
-- bFunction (I, w) -> bf
-- I = holonomic ideal in a Weyl algebra with no parameters
-- w = weight
-- bf = b-function of I with respect to w (polynomial
-- in K[$s], where K is the coefficient ring)
--
-- (method: algorithms 5.1.5 and 5.1.6 in Saito-Strumfels-Takayama)
-----------------------------------------------------------------------
bFunction = method(Options => {Strategy => IntRing})
-- makes polynomial f monic (internal)
makeMonic := f -> ( if coefficientRing ring f === QQ
then (1 / (leadCoefficient f)) * f
else (1 // (leadCoefficient f)) * f
);
-- lifts the coeffs of a polynomial to QQ,
-- returns error if QQ is not a subring of a coefficient ring
-- (internal)
makeQQ := f -> (
local R;
s := symbol s;
if ring f === QQ or f === 0 then (
R = QQ[s];
if not member(QQ, R.baseRings) then
error "QQ is not a base ring of R";
if f == 0 then 0_R
else sum(listForm f, u -> lift(u#1, QQ) * s^(sum u#0))
)
else (
R = (coefficientRing ring f)[s];
if f == 0 then 0_R
else sum(listForm f, u -> u#1 * s^(sum u#0))
)
);
-- trivial intersection strategy (internal)
bfIntRing := method()
bfIntRing(Ideal, List) := (I, w) -> (
local tInfo;
-- prep work
if not (ring I).?IntRing then
createIntRing (ring I);
createDpairs (ring I);
W := ring I;
IR := W.IntRing;
dpV := W.dpairVars;
dpI := W.dpairInds;
-- sanity check
if (#(dpI#2) != 0) then
error "expected no central variables in Weyl algebra";
if (#w != ((numgens W) // 2)) then
error "expected weight vector of length " | ((numgens W) // 2);
w = apply(numgens W, i -> (
p := position(dpI#1, u -> u == i);
if p =!= null then w#p
else (
p = position(dpI#0, u -> u == i);
-w#p
)
));
-- compute in_(-w,w) (I)
inI := inw(I, w);
n := #(dpI#0);
eulerOp := sum(n, i -> w_(dpI#1#i)*(dpV#0#i)*(dpV#1#i));
elimIdeal := W.RtoIR inI + ideal (W.RtoIR eulerOp - IR_(numgens(IR) - 1));
pInfo(2, "computing elimIdealGB... ");
tInfo = toString first timing(
elimIdealGB := gens gb elimIdeal;
);
pInfo(3, " elimIdealGB = " | toString elimIdealGB);
pInfo(2, " time = " | tInfo);
-- take the generator of J and cook up the b-function
bGen := selectInSubring(1,elimIdealGB);
bfcn := (
if numgens source bGen == 0 then 0
else makeMonic (mingens ideal bGen)_(0,0)
);
makeQQ bfcn
);
-- TryGeneric or NonGeneric strategy (internal)
bfGenericOrNonGeneric := method(Options => {Strategy => TryGeneric})
bfGenericOrNonGeneric(Ideal, List) := o -> (I, w) -> (
local tInfo;
-- prep work
if not (ring I).?ThetaRing then
createThetaRing (ring I);
if not (ring I).ThetaRing.?IntRing then
createIntRing (ring I).ThetaRing;
createDpairs (ring I);
W := ring I;
T := W.ThetaRing;
TI := T.IntRing;
dpV := W.dpairVars;
dpI := W.dpairInds;
-- sanity check
if (#(dpI#2) != 0) then
error "expected no central variables in Weyl algebra";
if (#w != ((numgens W) // 2)) then
error ("expected weight vector of length " | ((numgens W) // 2));
w = apply(numgens W, i -> (
p := position(dpI#1, u -> u == i);
if p =!= null then w#p
else (
p = position(dpI#0, u -> u == i);
-w#p
)
));
-- compute in_(-w,w) (I)
inI := inw(I, w);
n := #(dpI#0);
eulerOp := sum(n, i -> w_(dpI#1#i)*(dpV#0#i)*(dpV#1#i));
-- two different strategies can be used
local intIdeal;
if (o.Strategy == TryGeneric) and
(-- "isGeneric" check
pInfo(2, "'isGeneric' check... ");
tInfo = toString first timing(
isIt := all(flatten entries gens inI, W.isGeneric)
);
pInfo(2, " time = " | tInfo);
isIt
)
then(
-- GENERIC
pInfo(1, "b-function: Using GENERIC strategy... ");
intIdeal = inI;
)
else(
-- NON-GENERIC
pInfo(1, "b-function: Using NON-GENERIC strategy... ");
pInfo(2, "Calculating intIdeal... ");
tInfo = toString first timing(
dpI' := {select(dpI#0, i -> w#i != 0), select(dpI#1, i -> w#i != 0)};
dpI'' := {select(dpI#0, i -> w#i == 0), select(dpI#1, i -> w#i == 0)};
-- want: eliminate all u_i, v_i as well as x_i, dx_i of weight 0
u := symbol u;
v := symbol v;
UV := (coefficientRing W)(monoid [ (dpI'#0) / (i -> u_i), (dpI'#1) / (i -> v_i),
(dpI''#0) / (i -> W_i), (dpI''#1) / (i -> W_i),
(dpI'#0) / (i -> W_i), (dpI'#1) / (i -> W_i),
WeylAlgebra => W.monoid.Options.WeylAlgebra,
MonomialOrder => Eliminate (2 * #dpI#0) ]);
WtoUV := map(UV, W, matrix { apply(numgens W, i -> (
if member(i, dpI'#0) then
(UV_(u_i) * substitute(W_i, UV))
else if member(i, dpI'#1) then (substitute(W_i, UV) * UV_(v_i))
else substitute(W_i, UV)
))
});
intGB := gens gb ((WtoUV inI)
+ ideal apply(#dpI'#0, i -> (UV_(u_(dpI'#0#i)) * UV_(v_(dpI'#1#i)) - 1)
));
intIdeal = ideal substitute(selectInSubring(1, intGB), W);
);
pInfo(2, " time = " | tInfo);
);
pInfo(3, " intIdeal = " | toString (intIdeal));
-- compute J = intIdeal \cap K[\theta]
pInfo(2, "computing elimIdealGB... ");
intIdealGens := first entries gens intIdeal;
tInfo = toString first timing(
elimIdeal := (T.RtoIR (if #intIdealGens==0 then ideal T
else ideal ( intIdealGens/ W.WtoT )))
+ ideal(TI_(numgens TI - 1) - T.RtoIR W.WtoT eulerOp);
elimIdealGB := gens gb elimIdeal;
);
pInfo(3, " elimIdealGB = " | toString elimIdealGB);
pInfo(2, " time = " | tInfo);
-- take the generator of J and cook up the b-function
bGen := selectInSubring(1,elimIdealGB);
bfcn := (
if numgens source bGen == 0 then 0_TI
else makeMonic (mingens ideal bGen)_(0,0)
);
makeQQ bfcn
);
bFunction(Ideal, List) := RingElement => o -> (I, w) -> (
result := (
if o.Strategy == IntRing then bfIntRing(I, w)
else if o.Strategy == TryGeneric or o.Strategy == NonGeneric
then bfGenericOrNonGeneric(I, w, o)
else error "wrong Strategy option"
);
result
);
-- factors a b-function
factorBFunction = method()
factorBFunction(RingElement) := Product => f -> (
R := ring f;
-- sanity check
if numgens R != 1 then
error "polynomial ring of one variable expected";
if coefficientRing R =!= QQ then
error "expected polynomial over QQ";
l := listForm f;
d := product(l, u -> denominator(u#1));
l = l / (u -> (u#0, lift(u#1*d, ZZ)));
R' := ZZ(monoid [R_0]);
f = sum (l, u -> u#1*R'_(u#0));
f = factor f;
f = select(f, u-> first degree u#0 > 0);
result := apply(f, u->(
if first degree u#0 != 1 then error "internal error: incorrect b-function";
coeff := listForm u#0 / (v->v#1);
Power(R_0 + (if #coeff> 1 then (coeff#1/coeff#0) else 0), u#1)
));
if #result==0 then 1_R' else result
);-- end factorBFunction
bFunctionRoots = method()
bFunctionRoots RingElement := List => f -> (
if f==1 then {} else apply(toList factorBFunction f,
u -> - leadCoefficient substitute(u#0, {(ring u#0)_0 => 0_(ring u#0)}) )
);
getIntRoots = method()
getIntRoots RingElement := List => f -> (
roots := bFunctionRoots f;
roots = select(roots, u -> denominator u == 1);
apply(roots, u -> numerator u)
);-- end getIntRoots
TEST ///
Dtrace 1
pInfo(1, "testing globalBFunction...")
for str in {IntRing, TryGeneric, NonGeneric, GeneralBernsteinSato} do (
print str;
x = symbol x; dx = symbol dx;
R = QQ[x, dx, WeylAlgebra => {x=>dx}];
n = 10;
f = x^n;
b = globalBFunction(f, Strategy => str);
assert ((n^n * b) == (
use ring b;
s := (ring b)_0;
product(n, i -> n * (s + 1) - i)
))
);
clearAll()
pInfo(1, "testing generalB...")
for str in {InitialIdeal, StarIdeal} do (
pInfo(1, "Strategy=>" | toString str);
R = QQ[x_1..x_4];
F = {x_3*x_1^2 + x_4*x_2^3};
b = {1_R,x_1,x_2} / (g->toString factorBFunction generalB (F,g,Strategy=>str));
assert(toString b == "{(s+1)*(s+2)*(s+3/2)*(s+4/3)*(s+5/3)*(s+5/6)*(s+7/6), (s+1)*(s+2)*(s+5/2)*(s+4/3)*(s+5/3)*(s+11/6)*(s+13/6), (s+1)*(s+2)*(s+3/2)*(s+5/3)*(s+7/3)*(s+7/6)*(s+11/6)}")
);
assert(toString factorBFunction generalB (F,1_R,Exponent=>2) == "(s+1)*(s+2)^2*(s+3)*(s+3/2)*(s+5/2)*(s+4/3)*(s+5/3)*(s+7/3)*(s+8/3)*(s+5/6)*(s+7/6)*(s+11/6)*(s+13/6)")
///
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