Central characters, Bernstein-Sato polynomials
and associated stratification
Viktor Levandovskyy, Jorge Martı́n-Morales, Daniel Andres
Lehrstuhl D für Mathematik, RWTH Aachen
6.9 , Summer School at Rolduc 2013
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Introduction and notations
Basic notations
K stands for a field of characteristic 0 (for this talk, let K = C)
K[x1 , . . . , xn ] the ring of polynomials in n variables.
Dn := K[x1 , . . . , xn ]h∂1 , . . . , ∂n i the ring of K-linear partial
differential operators on K[x1 , . . . , xn ], the n-th Weyl algebra:
∂i xi = xi ∂i + 1,
∂i xj = xj ∂i ,
∂j ∂i = ∂i ∂j ,
xj xi = xi xj
Dn is a Noetherian integral domain with PBW basis
{x1α1 · · · xnαn · ∂1β1 · · · ∂nβn | α, β ∈ Nn }
Dn [s] := Dn ⊗K K[s].
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Mathematician’s Heart
Consider f = (x 2 + 49 y 2 + z 2 − 1)3 − x 2 z 3 −
9 2 3
80 y z
∈ K[x, y, z].
f is reduced polynomial, defining a hyperfurface in K3 .
The singular locus of f
∂f ∂f ∂f
. . . is described via the ideal hf , ∂x
, ∂y , ∂z i.
The corresponding variety is Sing(f ) = V1 ∪ V2 , where
V1 = V (hx 2 + 9/4y 2 − 1, z),
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V2 = V (hx, y, z 2 − 1i)
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Mathematician’s Heart
Consider f = (x 2 + 49 y 2 + z 2 − 1)3 − x 2 z 3 −
9 2 3
80 y z
∈ K[x, y, z].
f is reduced polynomial, defining a hyperfurface in K3 .
The singular locus of f
∂f ∂f ∂f
. . . is described via the ideal hf , ∂x
, ∂y , ∂z i.
The corresponding variety is Sing(f ) = V1 ∪ V2 , where
V1 = V (hx 2 + 9/4y 2 − 1, z),
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V2 = V (hx, y, z 2 − 1i)
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Mathematician’s Heart
Consider f = (x 2 + 49 y 2 + z 2 − 1)3 − x 2 z 3 −
9 2 3
80 y z
∈ K[x, y, z].
f is reduced polynomial, defining a hyperfurface in K3 .
The singular locus of f
∂f ∂f ∂f
. . . is described via the ideal hf , ∂x
, ∂y , ∂z i.
The corresponding variety is Sing(f ) = V1 ∪ V2 , where
V1 = V (hx 2 + 9/4y 2 − 1, z),
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V2 = V (hx, y, z 2 − 1i)
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Mathematician’s Heart
Consider f = (x 2 + 49 y 2 + z 2 − 1)3 − x 2 z 3 −
9 2 3
80 y z
∈ K[x, y, z].
f is reduced polynomial, defining a hyperfurface in K3 .
The singular locus of f
∂f ∂f ∂f
. . . is described via the ideal hf , ∂x
, ∂y , ∂z i.
The corresponding variety is Sing(f ) = V1 ∪ V2 , where
V1 = V (hx 2 + 9/4y 2 − 1, z),
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V2 = V (hx, y, z 2 − 1i)
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Mathematician’s Heart
Sing(f ) = V (hx 2 + 9/4y 2 − 1, z) ∪ V (hx, y, z 2 − 1i).
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Stratification for mathematician’s heart
There exists a stratification of C3 into constructible
(
) sets such that bf ,p (s)
(
constant on each stratum.
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) is
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Stratification for mathematician’s heart
There exists a stratification of C3 into constructible (i. e. union of locally
closed) sets such that bf ,p (s) (the local Bernstein-Sato polynomial of f
at the point p ∈ K3 ) is constant on each stratum.
For the heart hypersurface the data for the stratification look as follows:


1
p ∈ C3 \ V (f ),





p ∈ V (f ) \ (V1 ∪ V2 ),
s + 1
2
,
bf ,p (s) = (s + 1) (s + 4/3)(s + 2/3)
p ∈ V1 \ V3 ,


2
(s + 1) (s + 4/3)(s + 5/3)(s + 2/3) p ∈ V3 ,




(s + 1)(s + 4/3)(s + 5/3)
p ∈ V2 .
where V3 = V (h19x 2 + 1, 171y 2 − 80, zi) ⊂ V1 is the variety of 4
different imaginary points.
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Stratification for mathematician’s heart
There exists a stratification of C3 into constructible (i. e. union of locally
closed) sets such that bf ,p (s) (the local Bernstein-Sato polynomial of f
at the point p ∈ K3 ) is constant on each stratum.
For the heart hypersurface the data for the stratification look as follows:


1
p ∈ C3 \ V (f ),





p ∈ V (f ) \ (V1 ∪ V2 ),
s + 1
2
,
bf ,p (s) = (s + 1) (s + 4/3)(s + 2/3)
p ∈ V1 \ V3 ,


2
(s + 1) (s + 4/3)(s + 5/3)(s + 2/3) p ∈ V3 ,




(s + 1)(s + 4/3)(s + 5/3)
p ∈ V2 .
where V3 = V (h19x 2 + 1, 171y 2 − 80, zi) ⊂ V1 is the variety of 4
different imaginary points.
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Stratification for mathematician’s heart
There exists a stratification of C3 into constructible (i. e. union of locally
closed) sets such that bf ,p (s) (the local Bernstein-Sato polynomial of f
at the point p ∈ K3 ) is constant on each stratum.
For the heart hypersurface the data for the stratification look as follows:


1
p ∈ C3 \ V (f ),





p ∈ V (f ) \ (V1 ∪ V2 ),
s + 1
2
,
bf ,p (s) = (s + 1) (s + 4/3)(s + 2/3)
p ∈ V1 \ V3 ,


2
(s + 1) (s + 4/3)(s + 5/3)(s + 2/3) p ∈ V3 ,




(s + 1)(s + 4/3)(s + 5/3)
p ∈ V2 .
where V3 = V (h19x 2 + 1, 171y 2 − 80, zi) ⊂ V1 is the variety of 4
different imaginary points.
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The Dn [s]-module K x, s, 1f • f s
Let f ∈ K[x] = K[x1 , . . . , xn ] be a non-zero polynomial.
We denote by K[x, s, 1f ] • f s the free K[x, s, 1f ]-module of rank one
generated by the formal symbol f s .
K[x, s, 1f ] • f s has a natural structure of left Dn [s]-module.
∂g
∂f 1
1 s
s
·f .
∂i (g(s, x) · f ) =
+ sg(s, x)
· f s ∈ K x, s,
∂xi
∂xi f
f
. . . and xi , s act naturally by multiplication.
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The global Bernstein-Sato polynomial
Theorem (J. Bernstein, 1971/72)
For a non-constant polynomial f ∈ C[x1 , . . . , xn ] there exists a linear
partial differential operator P(s) with polynomial coefficients in
C[x1 , . . . , xn , s] and a univariate polynomial b(s) ∈ C[s], such that
1 s
·f .
P(s) • f s+1 = b(s) · f s ∈ C x, s,
f
The set of all possible polynomials b(s) satisfying the above equation
is an ideal of C[s]. The monic generator of this ideal is denoted by
bf (s) and called the Bernstein-Sato polynomial of f .
AnnD[s] (fs ) = {Q(s) ∈ D[s] | Q(s) • f s = 0} ⊂ D[s]
is a left ideal of D[s], the annihilator ideal of f s .
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The global Bernstein-Sato polynomial
Theorem (J. Bernstein, 1971/72)
For a non-constant polynomial f ∈ C[x1 , . . . , xn ] there exists a linear
partial differential operator P(s) with polynomial coefficients in
C[x1 , . . . , xn , s] and a univariate polynomial b(s) ∈ C[s], such that
1 s
·f .
P(s) • f s+1 = b(s) · f s ∈ C x, s,
f
The set of all possible polynomials b(s) satisfying the above equation
is an ideal of C[s]. The monic generator of this ideal is denoted by
bf (s) and called the Bernstein-Sato polynomial of f .
AnnD[s] (fs ) = {Q(s) ∈ D[s] | Q(s) • f s = 0} ⊂ D[s]
is a left ideal of D[s], the annihilator ideal of f s .
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The global Bernstein-Sato polynomial
Theorem (J. Bernstein, 1971/72)
For a non-constant polynomial f ∈ C[x1 , . . . , xn ] there exists a linear
partial differential operator P(s) with polynomial coefficients in
C[x1 , . . . , xn , s] and a univariate polynomial b(s) ∈ C[s], such that
1 s
·f .
P(s) • f s+1 = b(s) · f s ∈ C x, s,
f
The set of all possible polynomials b(s) satisfying the above equation
is an ideal of C[s]. The monic generator of this ideal is denoted by
bf (s) and called the Bernstein-Sato polynomial of f .
AnnD[s] (fs ) = {Q(s) ∈ D[s] | Q(s) • f s = 0} ⊂ D[s]
is a left ideal of D[s], the annihilator ideal of f s .
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“Geometric” localization
Let p ∈ Kn and mp = hx1 − p1 , . . . , xn − pn i. Then
S = K[x1 , . . . , xn ] \ mp is multiplicatively closed Ore set in Dn [s].
Then (by Bernstein indeed) there exist a linear partial diff. op. Pp (s)
with coefficients in S −1 K[x1 , . . . , xn ] ⊗K K[s] ∼
= K[x1 , . . . , xn ]mp [s] and a
univariate monic polynomial bf ,p (s) ∈ K[s], such that
Pp (s) • f s+1 = bf ,p (s) · f s
holds.
bf ,p (s) is called the local Berstein-Sato polynomial (at p).
For any p ∈ Kn , one has bf ,p (s) | bf (s).
Theorem (Mebkhout-Narvaez, Briançon-Maisonobe)
bf (s) = LCMp∈Kn bf ,p (s) = LCMp∈Sing(f ) bf ,p (s).
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“Geometric” localization
Let p ∈ Kn and mp = hx1 − p1 , . . . , xn − pn i. Then
S = K[x1 , . . . , xn ] \ mp is multiplicatively closed Ore set in Dn [s].
Then (by Bernstein indeed) there exist a linear partial diff. op. Pp (s)
with coefficients in S −1 K[x1 , . . . , xn ] ⊗K K[s] ∼
= K[x1 , . . . , xn ]mp [s] and a
univariate monic polynomial bf ,p (s) ∈ K[s], such that
Pp (s) • f s+1 = bf ,p (s) · f s
holds.
bf ,p (s) is called the local Berstein-Sato polynomial (at p).
For any p ∈ Kn , one has bf ,p (s) | bf (s).
Theorem (Mebkhout-Narvaez, Briançon-Maisonobe)
bf (s) = LCMp∈Kn bf ,p (s) = LCMp∈Sing(f ) bf ,p (s).
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“Geometric” localization
Let p ∈ Kn and mp = hx1 − p1 , . . . , xn − pn i. Then
S = K[x1 , . . . , xn ] \ mp is multiplicatively closed Ore set in Dn [s].
Then (by Bernstein indeed) there exist a linear partial diff. op. Pp (s)
with coefficients in S −1 K[x1 , . . . , xn ] ⊗K K[s] ∼
= K[x1 , . . . , xn ]mp [s] and a
univariate monic polynomial bf ,p (s) ∈ K[s], such that
Pp (s) • f s+1 = bf ,p (s) · f s
holds.
bf ,p (s) is called the local Berstein-Sato polynomial (at p).
For any p ∈ Kn , one has bf ,p (s) | bf (s).
Theorem (Mebkhout-Narvaez, Briançon-Maisonobe)
bf (s) = LCMp∈Kn bf ,p (s) = LCMp∈Sing(f ) bf ,p (s).
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“Geometric” localization
Let p ∈ Kn and mp = hx1 − p1 , . . . , xn − pn i. Then
S = K[x1 , . . . , xn ] \ mp is multiplicatively closed Ore set in Dn [s].
Then (by Bernstein indeed) there exist a linear partial diff. op. Pp (s)
with coefficients in S −1 K[x1 , . . . , xn ] ⊗K K[s] ∼
= K[x1 , . . . , xn ]mp [s] and a
univariate monic polynomial bf ,p (s) ∈ K[s], such that
Pp (s) • f s+1 = bf ,p (s) · f s
holds.
bf ,p (s) is called the local Berstein-Sato polynomial (at p).
For any p ∈ Kn , one has bf ,p (s) | bf (s).
Theorem (Mebkhout-Narvaez, Briançon-Maisonobe)
bf (s) = LCMp∈Kn bf ,p (s) = LCMp∈Sing(f ) bf ,p (s).
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“Geometric” localization
Let p ∈ Kn and mp = hx1 − p1 , . . . , xn − pn i. Then
S = K[x1 , . . . , xn ] \ mp is multiplicatively closed Ore set in Dn [s].
Then (by Bernstein indeed) there exist a linear partial diff. op. Pp (s)
with coefficients in S −1 K[x1 , . . . , xn ] ⊗K K[s] ∼
= K[x1 , . . . , xn ]mp [s] and a
univariate monic polynomial bf ,p (s) ∈ K[s], such that
Pp (s) • f s+1 = bf ,p (s) · f s
holds.
bf ,p (s) is called the local Berstein-Sato polynomial (at p).
For any p ∈ Kn , one has bf ,p (s) | bf (s).
Theorem (Mebkhout-Narvaez, Briançon-Maisonobe)
bf (s) = LCMp∈Kn bf ,p (s) = LCMp∈Sing(f ) bf ,p (s).
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bf ,p (s): Sensitivity wrt Singularities
Lemma (classical)
For f ∈ K∗ = K \ {0}, bf (s) = 1.
For f ∈ K[x1 , . . . , xn ] \ K, (s + 1) | bf (s).
For f ∈ K[x1 , . . . , xn ] \ K and p 6∈ V (f ), bf ,p (s) = 1.
For f ∈ K[x1 , . . . , xn ] \ K and p smooth on V (f ), bf ,p (s) = s + 1.
Moreover, the roots of global (and hence local) Bernstein-Sato
polynomial are negative rational numbers! (Kashiwara)
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bf ,p (s): Sensitivity wrt Singularities
Lemma (classical)
For f ∈ K∗ = K \ {0}, bf (s) = 1.
For f ∈ K[x1 , . . . , xn ] \ K, (s + 1) | bf (s).
For f ∈ K[x1 , . . . , xn ] \ K and p 6∈ V (f ), bf ,p (s) = 1.
For f ∈ K[x1 , . . . , xn ] \ K and p smooth on V (f ), bf ,p (s) = s + 1.
Moreover, the roots of global (and hence local) Bernstein-Sato
polynomial are negative rational numbers! (Kashiwara)
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bf ,p (s): Sensitivity wrt Singularities
Lemma (classical)
For f ∈ K∗ = K \ {0}, bf (s) = 1.
For f ∈ K[x1 , . . . , xn ] \ K, (s + 1) | bf (s).
For f ∈ K[x1 , . . . , xn ] \ K and p 6∈ V (f ), bf ,p (s) = 1.
For f ∈ K[x1 , . . . , xn ] \ K and p smooth on V (f ), bf ,p (s) = s + 1.
Moreover, the roots of global (and hence local) Bernstein-Sato
polynomial are negative rational numbers! (Kashiwara)
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Algorithmics
Good news
All the objects from Bernstein’s functional equation are computable:
AnnD[s] f s , the annihilator ideal of f s
the Bernstein-Sato polynomial bf (s) ∈ K[s]
a Bernstein operator P(s) ∈ D[s] (not unique)
Not so good news
Computations rely on non-commutative Gröbner bases and thus are
hard in general.
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Algorithmics
Good news
All the objects from Bernstein’s functional equation are computable:
AnnD[s] f s , the annihilator ideal of f s
the Bernstein-Sato polynomial bf (s) ∈ K[s]
a Bernstein operator P(s) ∈ D[s] (not unique)
Not so good news
Computations rely on non-commutative Gröbner bases and thus are
hard in general.
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Bernstein Data for the heart
f = (x 2 + 94 y 2 + z 2 − 1)3 − x 2 z 3 −
9 2 3
80 y z
∈ K[x, y, z].
Then AnnD[s] fs has 13 generators with leading terms
4617 · y 3 ∂x , 513 · x 2 y ∂x , . . . , 102400 · x 2 z 5 ∂y , 37428480 · y 4 z 5 ∂y ;
bf (s) = (s + 1)2 · (s + 23 ) · (s + 34 ) · (s + 35 )
and a reduced P(S) has 1261 terms, here some leading part of them
(
1 2 3
1
7084781
1
xy z −
xy 2 z 2 )∂x3 ∂z2 + (
yz 3 −
yz 2 )∂x2 ∂y ∂z2
24
5760
177292800
4104
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Bernstein Data for the heart
f = (x 2 + 94 y 2 + z 2 − 1)3 − x 2 z 3 −
9 2 3
80 y z
∈ K[x, y, z].
Then AnnD[s] fs has 13 generators with leading terms
4617 · y 3 ∂x , 513 · x 2 y ∂x , . . . , 102400 · x 2 z 5 ∂y , 37428480 · y 4 z 5 ∂y ;
bf (s) = (s + 1)2 · (s + 23 ) · (s + 34 ) · (s + 35 )
and a reduced P(S) has 1261 terms, here some leading part of them
(
1 2 3
1
7084781
1
xy z −
xy 2 z 2 )∂x3 ∂z2 + (
yz 3 −
yz 2 )∂x2 ∂y ∂z2
24
5760
177292800
4104
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Bernstein Data for the heart
f = (x 2 + 94 y 2 + z 2 − 1)3 − x 2 z 3 −
9 2 3
80 y z
∈ K[x, y, z].
Then AnnD[s] fs has 13 generators with leading terms
4617 · y 3 ∂x , 513 · x 2 y ∂x , . . . , 102400 · x 2 z 5 ∂y , 37428480 · y 4 z 5 ∂y ;
bf (s) = (s + 1)2 · (s + 23 ) · (s + 34 ) · (s + 35 )
and a reduced P(S) has 1261 terms, here some leading part of them
(
1 2 3
1
7084781
1
xy z −
xy 2 z 2 )∂x3 ∂z2 + (
yz 3 −
yz 2 )∂x2 ∂y ∂z2
24
5760
177292800
4104
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Algorithmics in D-module Theory
Existing systems
(experimental) KAN / SM 1 by N. Takayama et al.
DMODULES . M 2
in M ACAULAY 2 by A. Leykin and H. Tsai
special algorithms by M. Schulze in S INGULAR to compute local
Bernstein-Sato polynomial in an isolated singular point
D - MODULE suite in S INGULAR :P LURAL by V. L., J. Martı́n-Morales,
D. Andres et al.; includes thorough implementation of many
important functions.
several packages in R ISA /A SIR by M. Noro et al.
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Towards the Stratification
From Berstein’s FE: hbf (s)i = (AnnD[s] f s + D[s]hf i) ∩ K[s].
Indeed, this holds true also in the local case: ∀p ∈ Kn
hbf ,p (s)i = (AnnDp [s] f s + Dp [s]hf i) ∩ K[s].
Notation:
From now on, by (D, b) we denote (D[s], bf (s)) for the global
respectively (Dp [s], bf ,p (s)) for the local situation.
Lemma
Let q(s) ∈ K[s] be a polynomial, A a K-algebra and L a left ideal in
A[s] := A ⊗K K[s] satisfying L ∩ K[s] 6= 0. The following equalities hold:
1
L + A[s]hq(s)i ∩ K[s] = L ∩ K[s] + K[s]hq(s)i,
2
L : q(s) ∩ K[s] = L ∩ K[s] : q(s),
3
L : q(s)∞ ∩ K[s] = L ∩ K[s] : q(s)∞ .
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Towards the Stratification
From Berstein’s FE: hbf (s)i = (AnnD[s] f s + D[s]hf i) ∩ K[s].
Indeed, this holds true also in the local case: ∀p ∈ Kn
hbf ,p (s)i = (AnnDp [s] f s + Dp [s]hf i) ∩ K[s].
Notation:
From now on, by (D, b) we denote (D[s], bf (s)) for the global
respectively (Dp [s], bf ,p (s)) for the local situation.
Lemma
Let q(s) ∈ K[s] be a polynomial, A a K-algebra and L a left ideal in
A[s] := A ⊗K K[s] satisfying L ∩ K[s] 6= 0. The following equalities hold:
1
L + A[s]hq(s)i ∩ K[s] = L ∩ K[s] + K[s]hq(s)i,
2
L : q(s) ∩ K[s] = L ∩ K[s] : q(s),
3
L : q(s)∞ ∩ K[s] = L ∩ K[s] : q(s)∞ .
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Towards the Stratification
From Berstein’s FE: hbf (s)i = (AnnD[s] f s + D[s]hf i) ∩ K[s].
Indeed, this holds true also in the local case: ∀p ∈ Kn
hbf ,p (s)i = (AnnDp [s] f s + Dp [s]hf i) ∩ K[s].
Notation:
From now on, by (D, b) we denote (D[s], bf (s)) for the global
respectively (Dp [s], bf ,p (s)) for the local situation.
Lemma
Let q(s) ∈ K[s] be a polynomial, A a K-algebra and L a left ideal in
A[s] := A ⊗K K[s] satisfying L ∩ K[s] 6= 0. The following equalities hold:
1
L + A[s]hq(s)i ∩ K[s] = L ∩ K[s] + K[s]hq(s)i,
2
L : q(s) ∩ K[s] = L ∩ K[s] : q(s),
3
L : q(s)∞ ∩ K[s] = L ∩ K[s] : q(s)∞ .
VL+JMM+DA (LDfM, RWTH Aachen)
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Multiplicities of Roots
Lemma
Let K = K̄. Consider f ∈ R = K[x1 , . . . , xn ] and p ∈ V (f ). Let,
moreover, mα be the multiplicity of α as a root of b(−s). Consider the
ideals L = AnnD[s] (f s ) + D[s]f and Ji = L + D[s](s + α)i+1 ⊆ D[s],
i = 0, . . . , n. The following conditions are equivalent:
1
mα > i,
2
Ji ∩ K[s] = h(s + α)i+1 i,
3
4
5
(s + α)i ∈
/ Ji ,
L : (s + α)i + D[s]hs + αi =
6 D[s],
i
L : (s + α) |s=−α 6= D.
Moreover mα = m ≤ n if and only if
D[s] ) J0 ) · · · ) Jm−1 = Jm = · · · = Jn .
VL+JMM+DA (LDfM, RWTH Aachen)
CCStrat
6.9.13
15 / 31
Multiplicities of Roots
Lemma
Let K = K̄. Consider f ∈ R = K[x1 , . . . , xn ] and p ∈ V (f ). Let,
moreover, mα be the multiplicity of α as a root of b(−s). Consider the
ideals L = AnnD[s] (f s ) + D[s]f and Ji = L + D[s](s + α)i+1 ⊆ D[s],
i = 0, . . . , n. The following conditions are equivalent:
1
mα > i,
2
Ji ∩ K[s] = h(s + α)i+1 i,
3
4
5
(s + α)i ∈
/ Ji ,
L : (s + α)i + D[s]hs + αi =
6 D[s],
i
L : (s + α) |s=−α 6= D.
Moreover mα = m ≤ n if and only if
D[s] ) J0 ) · · · ) Jm−1 = Jm = · · · = Jn .
VL+JMM+DA (LDfM, RWTH Aachen)
CCStrat
6.9.13
15 / 31
Multiplicities of Roots
Lemma
Let K = K̄. Consider f ∈ R = K[x1 , . . . , xn ] and p ∈ V (f ). Let,
moreover, mα be the multiplicity of α as a root of b(−s). Consider the
ideals L = AnnD[s] (f s ) + D[s]f and Ji = L + D[s](s + α)i+1 ⊆ D[s],
i = 0, . . . , n. The following conditions are equivalent:
1
mα > i,
2
Ji ∩ K[s] = h(s + α)i+1 i,
3
4
5
(s + α)i ∈
/ Ji ,
L : (s + α)i + D[s]hs + αi =
6 D[s],
i
L : (s + α) |s=−α 6= D.
Moreover mα = m ≤ n if and only if
D[s] ) J0 ) · · · ) Jm−1 = Jm = · · · = Jn .
VL+JMM+DA (LDfM, RWTH Aachen)
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6.9.13
15 / 31
Central character decomposition
Let A be an associative Noetherian K-algebra
and Z ⊂ A be a finitely generated commutative subalgebra of A.
In this talk, assume Z = Z (A) is the center of A.
We identify Z ∗ = HomK−alg (Z , K) with the set of maximal ideals of the
commutative algebra Z .
Let M be a finitely generated A-module and χ ∈ Z ∗ .
VL+JMM+DA (LDfM, RWTH Aachen)
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6.9.13
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Generalized Weight Decomposition
The generalized χ-weight subspace of M wrt Z is
n
o
M χ = v ∈ M | ∃n(v ) ∈ N, ∀z ∈ Z , (z − χ(z))n(v ) v = 0 .
SZ M = {χ ∈ Z ∗ |M χ 6= 0} is the support of M wrt Z .
M possesses a generalized weight decomposition if
M
M=
M χ.
χ∈Z ∗
VL+JMM+DA (LDfM, RWTH Aachen)
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6.9.13
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Generalized Weight Decomposition
The generalized χ-weight subspace of M wrt Z is
n
o
M χ = v ∈ M | ∃n(v ) ∈ N, ∀z ∈ Z , (z − χ(z))n(v ) v = 0 .
SZ M = {χ ∈ Z ∗ |M χ 6= 0} is the support of M wrt Z .
M possesses a generalized weight decomposition if
M
M=
M χ.
χ∈Z ∗
VL+JMM+DA (LDfM, RWTH Aachen)
CCStrat
6.9.13
17 / 31
Generalized Weight Decomposition
The generalized χ-weight subspace of M wrt Z is
n
o
M χ = v ∈ M | ∃n(v ) ∈ N, ∀z ∈ Z , (z − χ(z))n(v ) v = 0 .
SZ M = {χ ∈ Z ∗ |M χ 6= 0} is the support of M wrt Z .
M possesses a generalized weight decomposition if
M
M=
M χ.
χ∈Z ∗
VL+JMM+DA (LDfM, RWTH Aachen)
CCStrat
6.9.13
17 / 31
Generalized Weight Decomposition
Let Z = Z (A) be the center of A and r ∈ N.
Lemma
Let M ∼
= Ar /PM for a left submodule PM ⊂ Ar . We define
preAnn(M) =
r
\
AnnM
A ej .
j=1
Then
1
Z ∩ preAnn(M) = Z ∩ AnnA M,
2
the Zariski closure of SZ M equals V (preAnn(M) ∩ Z ).
VL+JMM+DA (LDfM, RWTH Aachen)
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6.9.13
18 / 31
Generalized Weight Decomposition
Let Z = Z (A) be the center of A and r ∈ N.
Lemma
Let M ∼
= Ar /PM for a left submodule PM ⊂ Ar . We define
preAnn(M) =
r
\
AnnM
A ej .
j=1
Then
1
Z ∩ preAnn(M) = Z ∩ AnnA M,
2
the Zariski closure of SZ M equals V (preAnn(M) ∩ Z ).
VL+JMM+DA (LDfM, RWTH Aachen)
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6.9.13
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Central Character Decomposition Theorem
Theorem (L., 2005)
Let M ∼
PM ⊂ Ar .
= Ar /PM for a left submodule
L
χ
If | SZ M |< ∞, then M = χ∈Z ∗ M and
Mχ ∼
= Ar /(PM : Jχ∞ ), where Jχ =
\
ker ψ.
ψ∈SZ M
ψ6=χ
In the D-module setup for a hypersurface f , one has
ker χ = K[s]hs − χ(s)i, where bf (χ(s)) = 0.
Recall, that χ(s) ∈ Q<0 as a root of Bernstein-Sato polynomial.
VL+JMM+DA (LDfM, RWTH Aachen)
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6.9.13
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Central Character Decomposition Theorem
Theorem (L., 2005)
Let M ∼
PM ⊂ Ar .
= Ar /PM for a left submodule
L
χ
If | SZ M |< ∞, then M = χ∈Z ∗ M and
Mχ ∼
= Ar /(PM : Jχ∞ ), where Jχ =
\
ker ψ.
ψ∈SZ M
ψ6=χ
In the D-module setup for a hypersurface f , one has
ker χ = K[s]hs − χ(s)i, where bf (χ(s)) = 0.
Recall, that χ(s) ∈ Q<0 as a root of Bernstein-Sato polynomial.
VL+JMM+DA (LDfM, RWTH Aachen)
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6.9.13
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Theorem
Let K be a field of characteristic zero, f ∈ R \ K and p ∈ V (f ) be fixed.
Consider the D[s]-module M = D[s]/(AnnD[s] f s + D[s]f ). Then
1
The annihilator of the module M, AnnD[s] (M) = D[s]b(s).
2
M possesses finite generalized weight decomposition
M
M = D[s]/(AnnD[s] f s + D[s]f ) =
M χ,
χ∈C(M)
where C(M) = SK[s] (M) ⊂ Q is the set of roots of b(s). Moreover,
for χ ∈ C(M) one has ker χ = K[s](s − χ(s)).
3
The computation of the finite generalized weight decomposition
above is algorithmic.
4
GK. dim M = n, GK. dim D[s]/D[s]b(s) = 2n and thus the module
M is generalized holonomic.
VL+JMM+DA (LDfM, RWTH Aachen)
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6.9.13
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CenCharDec and Stratification by Roots
Theorem (L.–Martı́n-Morales, 2012)
Let α be a root of bf (−s) with the multiplicity mα . For 0 ≤ i ≤ mα − 1
consider the ideals
L := AnnD[s] (f s ) + D[s]hf i,
Lα,i := L : (s + α)i + D[s]hs + αi.
Then one has
mα (p) > i ⇐⇒ p ∈ V (Lα,i ∩ C[x]).
Let Vα,i = V (Iα,i ∩ C[x]) be an affine variety. Then
∅ =: Vα,mα ⊂ Vα,mα −1 ⊂ · · · ⊂ Vα,0 ⊂ Vα,−1 := Cn ,
and mα (p) = i if and only if p ∈ Vα,i−1 \ Vα,i . We call this sequence
the stratification of Cn associated with the root α.
VL+JMM+DA (LDfM, RWTH Aachen)
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CenCharDec and Stratification by Roots
Theorem (L.–Martı́n-Morales, 2012)
Let α be a root of bf (−s) with the multiplicity mα . For 0 ≤ i ≤ mα − 1
consider the ideals
L := AnnD[s] (f s ) + D[s]hf i,
Lα,i := L : (s + α)i + D[s]hs + αi.
Then one has
mα (p) > i ⇐⇒ p ∈ V (Lα,i ∩ C[x]).
Let Vα,i = V (Iα,i ∩ C[x]) be an affine variety. Then
∅ =: Vα,mα ⊂ Vα,mα −1 ⊂ · · · ⊂ Vα,0 ⊂ Vα,−1 := Cn ,
and mα (p) = i if and only if p ∈ Vα,i−1 \ Vα,i . We call this sequence
the stratification of Cn associated with the root α.
VL+JMM+DA (LDfM, RWTH Aachen)
CCStrat
6.9.13
21 / 31
CenCharDec and Stratification by Roots
Theorem (L.–Martı́n-Morales, 2012)
Let α be a root of bf (−s) with the multiplicity mα . For 0 ≤ i ≤ mα − 1
consider the ideals
L := AnnD[s] (f s ) + D[s]hf i,
Lα,i := L : (s + α)i + D[s]hs + αi.
Then one has
mα (p) > i ⇐⇒ p ∈ V (Lα,i ∩ C[x]).
Let Vα,i = V (Iα,i ∩ C[x]) be an affine variety. Then
∅ =: Vα,mα ⊂ Vα,mα −1 ⊂ · · · ⊂ Vα,0 ⊂ Vα,−1 := Cn ,
and mα (p) = i if and only if p ∈ Vα,i−1 \ Vα,i . We call this sequence
the stratification of Cn associated with the root α.
VL+JMM+DA (LDfM, RWTH Aachen)
CCStrat
6.9.13
21 / 31
CenCharDec and Stratification by Roots
Theorem (L.–Martı́n-Morales, 2012)
Let α be a root of bf (−s) with the multiplicity mα . For 0 ≤ i ≤ mα − 1
consider the ideals
L := AnnD[s] (f s ) + D[s]hf i,
Lα,i := L : (s + α)i + D[s]hs + αi.
Then one has
mα (p) > i ⇐⇒ p ∈ V (Lα,i ∩ C[x]).
Let Vα,i = V (Iα,i ∩ C[x]) be an affine variety. Then
∅ =: Vα,mα ⊂ Vα,mα −1 ⊂ · · · ⊂ Vα,0 ⊂ Vα,−1 := Cn ,
and mα (p) = i if and only if p ∈ Vα,i−1 \ Vα,i . We call this sequence
the stratification of Cn associated with the root α.
VL+JMM+DA (LDfM, RWTH Aachen)
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6.9.13
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Example (Remember the heart? )
f = (x 2 + 9/4y 2 + z 2 − 1)3 − x 2 z 3 − 9/80y 2 z 3 ∈ C[x, y , z].
The Bernstein-Sato polynomial is
bf (s) = (s + 1)2 (s + 4/3)(s + 5/3)(s + 2/3).
Sing(f ) = V1 ∪ V2 , where V1 = V (hx 2 + 9/4y 2 − 1, z),
V2 = V (hx, y , z 2 − 1i) and V3 = V (h19x 2 + 1, 171y 2 − 80, zi) ⊂ V1 .
The stratifications, associated with each root of bf (s) are given by
α
α
α
α
=
=
=
=
−1,
−4/3,
−5/3,
−2/3,
VL+JMM+DA (LDfM, RWTH Aachen)
∅
∅
∅
∅
⊂ V1 ⊂ V (f )
⊂ V1 ∪ V2
⊂ V2 ∪ V3
⊂
V1
CCStrat
⊂
⊂
⊂
⊂
C3 ;
C3 ;
C3 ;
C3 .
6.9.13
22 / 31
Example (Remember the heart? )
f = (x 2 + 9/4y 2 + z 2 − 1)3 − x 2 z 3 − 9/80y 2 z 3 ∈ C[x, y , z].
The Bernstein-Sato polynomial is
bf (s) = (s + 1)2 (s + 4/3)(s + 5/3)(s + 2/3).
Sing(f ) = V1 ∪ V2 , where V1 = V (hx 2 + 9/4y 2 − 1, z),
V2 = V (hx, y, z 2 − 1i) and V3 = V (h19x 2 + 1, 171y 2 − 80, zi) ⊂ V1 .
The stratifications, associated with each root of bf (s) are given by
α
α
α
α
=
=
=
=
−1,
−4/3,
−5/3,
−2/3,
VL+JMM+DA (LDfM, RWTH Aachen)
∅
∅
∅
∅
⊂ V1 ⊂ V (f )
⊂ V1 ∪ V2
⊂ V2 ∪ V3
⊂
V1
CCStrat
⊂
⊂
⊂
⊂
C3 ;
C3 ;
C3 ;
C3 .
6.9.13
22 / 31
Example (Remember the heart? )
f = (x 2 + 9/4y 2 + z 2 − 1)3 − x 2 z 3 − 9/80y 2 z 3 ∈ C[x, y , z].
The Bernstein-Sato polynomial is
bf (s) = (s + 1)2 (s + 4/3)(s + 5/3)(s + 2/3).
Sing(f ) = V1 ∪ V2 , where V1 = V (hx 2 + 9/4y 2 − 1, z),
V2 = V (hx, y, z 2 − 1i) and V3 = V (h19x 2 + 1, 171y 2 − 80, zi) ⊂ V1 .
The stratifications, associated with each root of bf (s) are given by
α
α
α
α
=
=
=
=
−1,
−4/3,
−5/3,
−2/3,
VL+JMM+DA (LDfM, RWTH Aachen)
∅
∅
∅
∅
⊂ V1 ⊂ V (f )
⊂ V1 ∪ V2
⊂ V2 ∪ V3
⊂
V1
CCStrat
⊂
⊂
⊂
⊂
C3 ;
C3 ;
C3 ;
C3 .
6.9.13
22 / 31
Example (Remember the heart? )
f = (x 2 + 9/4y 2 + z 2 − 1)3 − x 2 z 3 − 9/80y 2 z 3 ∈ C[x, y , z].
The Bernstein-Sato polynomial is
bf (s) = (s + 1)2 (s + 4/3)(s + 5/3)(s + 2/3).
Sing(f ) = V1 ∪ V2 , where V1 = V (hx 2 + 9/4y 2 − 1, z),
V2 = V (hx, y, z 2 − 1i) and V3 = V (h19x 2 + 1, 171y 2 − 80, zi) ⊂ V1 .
The stratifications, associated with each root of bf (s) are given by
α
α
α
α
=
=
=
=
−1,
−4/3,
−5/3,
−2/3,
VL+JMM+DA (LDfM, RWTH Aachen)
∅
∅
∅
∅
⊂ V1 ⊂ V (f )
⊂ V1 ∪ V2
⊂ V2 ∪ V3
⊂
V1
CCStrat
⊂
⊂
⊂
⊂
C3 ;
C3 ;
C3 ;
C3 .
6.9.13
22 / 31
Recall, that there is a decomposition
M = ⊕M χ .
Moreover, define Mχ,k = m ∈ M | ∀z ∈ Z (z − χ(z))k m = 0 , which
is a submodule of M and of M χ . Hence
M ⊃ M χ ⊃ . . . ⊃ Mχ,2 ⊃ Mχ,1 ⊃ Mχ,0 = 0.
Let α be a root of b(s) of multiplicity m ≥ 2 and χ = χ(α). Then
Ann Mχ,1 = D[s](s − α) ⊃ . . . ⊃ Ann Mχ,m = Ann M χ = D[s](s − α)m .
Hence M χ = Mχ,m ) . . . ) Mχ,1 . Moreover, Mχ,i is a holonomic
D-module and a generalized holonomic D[s]-module.
Inclusion reverting correspondence between Mχ,i and Vα,i−1
Q
To a constructible set C ⊂ Cn , on which bf ,p = (s − αi )`i , one can
naturally
associate a P
submodule M ⊃ NC := ⊕Mχ(αi ),`i . Then
P
n
Ci NCi = M, where
i Ci = C .
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Recall, that there is a decomposition
M = ⊕M χ .
Moreover, define Mχ,k = m ∈ M | ∀z ∈ Z (z − χ(z))k m = 0 , which
is a submodule of M and of M χ . Hence
M ⊃ M χ ⊃ . . . ⊃ Mχ,2 ⊃ Mχ,1 ⊃ Mχ,0 = 0.
Let α be a root of b(s) of multiplicity m ≥ 2 and χ = χ(α). Then
Ann Mχ,1 = D[s](s − α) ⊃ . . . ⊃ Ann Mχ,m = Ann M χ = D[s](s − α)m .
Hence M χ = Mχ,m ) . . . ) Mχ,1 . Moreover, Mχ,i is a holonomic
D-module and a generalized holonomic D[s]-module.
Inclusion reverting correspondence between Mχ,i and Vα,i−1
Q
To a constructible set C ⊂ Cn , on which bf ,p = (s − αi )`i , one can
naturally
associate a P
submodule M ⊃ NC := ⊕Mχ(αi ),`i . Then
P
n
Ci NCi = M, where
i Ci = C .
VL+JMM+DA (LDfM, RWTH Aachen)
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Example (the heart is beating)
We see that SZ M(f ) = {−1, −4/3, −5/3, −2/3}. Moreover,
M (s+1) =
AnnD[s]
fs
D[s]
D[s]
= M(s+1,1) .
2
AnnD[s] f s + hf , s + 1i
+ hf , (s + 1) i
A stratification of C3 into constructible sets with bf ,p (s) constant on
each stratum follows:


1
p ∈ C3 \ V (f ),





p ∈ V (f ) \ (V1 ∪ V2 ),
s + 1
2
bf ,p (s) = (s + 1) (s + 4/3)(s + 2/3)
p ∈ V1 \ V3 ,


2
(s + 1) (s + 4/3)(s + 5/3)(s + 2/3) p ∈ V3 ,




(s + 1)(s + 4/3)(s + 5/3)
p ∈ V2 .
To each stratum we associate a global holonomic D[s]-module. For
instance, M(s+1,2) + M (s+4/3) + M (s+2/3) corresponds to V1 \ V3 .
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Example (the heart is beating)
We see that SZ M(f ) = {−1, −4/3, −5/3, −2/3}. Moreover,
M (s+1) =
AnnD[s]
fs
D[s]
D[s]
= M(s+1,1) .
2
AnnD[s] f s + hf , s + 1i
+ hf , (s + 1) i
A stratification of C3 into constructible sets with bf ,p (s) constant on
each stratum follows:


1
p ∈ C3 \ V (f ),





p ∈ V (f ) \ (V1 ∪ V2 ),
s + 1
2
bf ,p (s) = (s + 1) (s + 4/3)(s + 2/3)
p ∈ V1 \ V3 ,


2
(s + 1) (s + 4/3)(s + 5/3)(s + 2/3) p ∈ V3 ,




(s + 1)(s + 4/3)(s + 5/3)
p ∈ V2 .
To each stratum we associate a global holonomic D[s]-module. For
instance, M(s+1,2) + M (s+4/3) + M (s+2/3) corresponds to V1 \ V3 .
VL+JMM+DA (LDfM, RWTH Aachen)
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Example (the heart is beating)
We see that SZ M(f ) = {−1, −4/3, −5/3, −2/3}. Moreover,
M (s+1) =
AnnD[s]
fs
D[s]
D[s]
= M(s+1,1) .
2
AnnD[s] f s + hf , s + 1i
+ hf , (s + 1) i
A stratification of C3 into constructible sets with bf ,p (s) constant on
each stratum follows:


1
p ∈ C3 \ V (f ),





p ∈ V (f ) \ (V1 ∪ V2 ),
s + 1
2
bf ,p (s) = (s + 1) (s + 4/3)(s + 2/3)
p ∈ V1 \ V3 ,


2
(s + 1) (s + 4/3)(s + 5/3)(s + 2/3) p ∈ V3 ,




(s + 1)(s + 4/3)(s + 5/3)
p ∈ V2 .
To each stratum we associate a global holonomic D[s]-module. For
instance, M(s+1,2) + M (s+4/3) + M (s+2/3) corresponds to V1 \ V3 .
VL+JMM+DA (LDfM, RWTH Aachen)
CCStrat
6.9.13
24 / 31
Comparison with local multiplicity
Ongoing work
Understanding the connection between the local Bernstein-Sato
polynomial of f and local multiplicity µp at p ∈ Kn .
There exists a stratification Sµ of Kn , such that the local multiplicity is
constant at each stratum. We observe on examples, that the
presented stratification Sb (according to local Bernstein-Sato
polynomial) contains the Sµ , that is, it is finer.
Example (Once more the heart)
For the presented example, Sµ |Sing(f ) = V1 ] V2 ,
while Sb |Sing(f ) = (V1 \ V3 ) ] V3 ] V2 .
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6.9.13
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Comparison with local multiplicity
Ongoing work
Understanding the connection between the local Bernstein-Sato
polynomial of f and local multiplicity µp at p ∈ Kn .
There exists a stratification Sµ of Kn , such that the local multiplicity is
constant at each stratum. We observe on examples, that the
presented stratification Sb (according to local Bernstein-Sato
polynomial) contains the Sµ , that is, it is finer.
Example (Once more the heart)
For the presented example, Sµ |Sing(f ) = V1 ] V2 ,
while Sb |Sing(f ) = (V1 \ V3 ) ] V3 ] V2 .
VL+JMM+DA (LDfM, RWTH Aachen)
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6.9.13
25 / 31
Singularities evolve like Hanoi towers?
VL+JMM+DA (LDfM, RWTH Aachen)
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Some Literature
1
V. Levandovskyy, On preimages of ideals in certain
non-commutative algebras, in “Computational Commutative and
Non-Commutative Algebraic Geometry” (eds. Pfister, G. and
Cojocaru, S. and Ufnarovski, V.), IOS Press, 2005
2
N. Budur, M. Mustaţǎ and M. Saito Bernstein-Sato polynomials of
arbitrary varieties, Compositio Math. 142, 779-797, 2006
3
D. Andres, V. Levandovskyy and J. Martı́n-Morales: Principal
Intersection and Bernstein-Sato Polynomial of an Affine Variety .
In Proc. ISSAC 09, ACM Press, 231-238, 2009
4
V. Levandovskyy and J. Martı́n-Morales :Algorithms for Checking
Rational Roots of b-functions and their Applications, Journal of
Algebra 352, 1, 408–429, 2012
VL+JMM+DA (LDfM, RWTH Aachen)
CCStrat
6.9.13
27 / 31
Thank you!
http://www.singular.uni-kl.de/
VL+JMM+DA (LDfM, RWTH Aachen)
CCStrat
6.9.13
28 / 31
Second Example: Solitude
f = x 2 yz + y 3 z − x 2 z 2 + xy 2 + y 3 ∈ K[x, y, z].
VL+JMM+DA (LDfM, RWTH Aachen)
CCStrat
6.9.13
29 / 31
Second Example (Solitude): Stratification by roots
Example
Let f = x 2 yz + y 3 z − x 2 z 2 + xy 2 + y 3 and M = V (f ) ⊂ C3 .
The global Bernstein-Sato polynomial of f is
bf (s) = (s + 7/6)2 (s + 5/6)2 (s + 1)2 (s + 4/3)(s + 5/3)(s + 3/2).
Sing(f ) = V (hx, y i) ∪ V (hy, zi). Define
V = V (hx 2 + 4x, xz, y, z 2 + zi) = {0, q1 = (0, 0, −1), q2 = (−4, 0, 0)}.
Then the stratification associated with each root of bf (s) is given by
α
α
α
α
=
=
=
=
−7/6, −5/6,
−1,
−4/3, −5/3,
−3/2,
VL+JMM+DA (LDfM, RWTH Aachen)
∅
∅
∅
∅
⊂ V (x, y, z) ⊂ V (x, y)
⊂
V (y, z) ⊂ V (f )
⊂
V (x, y , z) = {0}
⊂
V = {0, q1 , q2 }
CCStrat
⊂
⊂
⊂
⊂
C3 ;
C3 ;
C3 ;
C3 .
6.9.13
30 / 31
Second Example (Solitude): Stratification by roots
Example
Let f = x 2 yz + y 3 z − x 2 z 2 + xy 2 + y 3 and M = V (f ) ⊂ C3 .
The global Bernstein-Sato polynomial of f is
bf (s) = (s + 7/6)2 (s + 5/6)2 (s + 1)2 (s + 4/3)(s + 5/3)(s + 3/2).
Sing(f ) = V (hx, y i) ∪ V (hy, zi). Define
V = V (hx 2 + 4x, xz, y, z 2 + zi) = {0, q1 = (0, 0, −1), q2 = (−4, 0, 0)}.
Then the stratification associated with each root of bf (s) is given by
α
α
α
α
=
=
=
=
−7/6, −5/6,
−1,
−4/3, −5/3,
−3/2,
VL+JMM+DA (LDfM, RWTH Aachen)
∅
∅
∅
∅
⊂ V (x, y, z) ⊂ V (x, y)
⊂
V (y, z) ⊂ V (f )
⊂
V (x, y, z) = {0}
⊂
V = {0, q1 , q2 }
CCStrat
⊂
⊂
⊂
⊂
C3 ;
C3 ;
C3 ;
C3 .
6.9.13
30 / 31
Second Example (Solitude): Stratification by roots
Example
Let f = x 2 yz + y 3 z − x 2 z 2 + xy 2 + y 3 and M = V (f ) ⊂ C3 .
The global Bernstein-Sato polynomial of f is
bf (s) = (s + 7/6)2 (s + 5/6)2 (s + 1)2 (s + 4/3)(s + 5/3)(s + 3/2).
Sing(f ) = V (hx, y i) ∪ V (hy, zi). Define
V = V (hx 2 + 4x, xz, y, z 2 + zi) = {0, q1 = (0, 0, −1), q2 = (−4, 0, 0)}.
Then the stratification associated with each root of bf (s) is given by
α
α
α
α
=
=
=
=
−7/6, −5/6,
−1,
−4/3, −5/3,
−3/2,
VL+JMM+DA (LDfM, RWTH Aachen)
∅
∅
∅
∅
⊂ V (x, y, z) ⊂ V (x, y)
⊂
V (y, z) ⊂ V (f )
⊂
V (x, y, z) = {0}
⊂
V = {0, q1 , q2 }
CCStrat
⊂
⊂
⊂
⊂
C3 ;
C3 ;
C3 ;
C3 .
6.9.13
30 / 31
Second Example (Solitude): Stratification by roots
Example
Let f = x 2 yz + y 3 z − x 2 z 2 + xy 2 + y 3 and M = V (f ) ⊂ C3 .
The global Bernstein-Sato polynomial of f is
bf (s) = (s + 7/6)2 (s + 5/6)2 (s + 1)2 (s + 4/3)(s + 5/3)(s + 3/2).
Sing(f ) = V (hx, y i) ∪ V (hy, zi). Define
V = V (hx 2 + 4x, xz, y, z 2 + zi) = {0, q1 = (0, 0, −1), q2 = (−4, 0, 0)}.
Then the stratification associated with each root of bf (s) is given by
α
α
α
α
=
=
=
=
−7/6, −5/6,
−1,
−4/3, −5/3,
−3/2,
VL+JMM+DA (LDfM, RWTH Aachen)
∅
∅
∅
∅
⊂ V (x, y, z) ⊂ V (x, y)
⊂
V (y, z) ⊂ V (f )
⊂
V (x, y, z) = {0}
⊂
V = {0, q1 , q2 }
CCStrat
⊂
⊂
⊂
⊂
C3 ;
C3 ;
C3 ;
C3 .
6.9.13
30 / 31
Solitude: Stratification by Local Bernstein-Sato
Example
A stratification associated with local b-functions and the corresponding
univariate polynomials follows.


1





s+1





(s + 1)(s + 7/6)(s + 5/6)
bf ,p (s) = (s + 1)2



(s + 1)(s + 7/6)(s + 5/6)(s + 3/2)





(s + 1)2 (s + 3/2)




bf (s)
VL+JMM+DA (LDfM, RWTH Aachen)
CCStrat
p
p
p
p
p
p
p
∈ C3 \ V (f ),
∈ V (f ) \ Sing(f ),
∈ V (x, y) \ {0, q1 },
∈ V (y , z) \ {0, q2 },
= q1 ,
= q2 ,
= 0.
6.9.13
31 / 31