1.Introduction Jen-ChiehHsiaoandChing-JuiLai Thestrongmonodromyconjectureformonomialidealsontoricvarieties

The strong monodromy conjecture for monomial ideals on toric varieties

Jen-Chieh Hsiao and Ching-Jui Lai

Department of Mathematics, National Cheng Kung University, Tainan, Taiwan

Dedicated to Professor Gennady Lyubeznik on the occasion of his 60th birthday.

ABSTRACT

We compute Denef and Loeser’s motivic zeta function associated to a monomial ideal on an affine toric variety, generalizing a result of Howald, Mustat¸
a, and Yuen. We also investigate the relation between the poles of the motivic zeta function and the roots of their corresponding Bernstein–Sato polynomial defined by the first author and Matusevich.

ARTICLE HISTORY Received 8 November 2017 Revised 8 February 2018 Communicated by U.Walther

KEYWORDS

Bernstein–Sato polynomials;

motivic zeta functions;

strong monodromy conjecture; toric varieties 2010 MATHEMATICS SUBJECT

CLASSIFICATION 14M2514B0514E1814F10

1. Introduction

The strong monodromy conjecture of Igusa states that the real part of a pole of the p-adic zeta function associated to a hypersurface defined by a polynomial f is a root of the Bernstein–Sato polynomial of f [12]. A fascinating consequence of this conjecture is a link between the poles of Igusa zeta functions and the monodromy eigenvalues of Milnor fibers of f. As a generalization of Igusa zeta function, Denef and Loeser introduce their motivic zeta function using motivic integra- tion [6, 7]. We will focus on the special type of motivic zeta function Z(X, I ;s) associated to an ideal sheaf I on an algebraic variety X, whose definition will be recalled in Section 2. For more information about this subject, we refer to the surveys [5, 17, 14].

On the other hand, Budur, Mustat¸
a, and Saito define a generalization of Bernstein–Sato poly- nomial b^X_IðsÞ for an ideal I on a smooth affine variety X [2]. Their definition is adopted in [11]

to extend the notion of Bernstein–Sato polynomial b^UI^rðsÞ for an ideal I on an affine (normal) toric variety Ur. It is then curious for us to see whether the analogous strong Monodromy Conjecture holds in the toric setting. In [10], this conjecture under the p-adic setting is verified for the case where I is a monomial ideal on an affine spaceAⁿ. The key ingredient in their proof is an explicit formula of the p-adic zeta function associated to the pair (Aⁿ, I). It turns out that the real parts of candidate poles of the p-adic zeta function associated to (Aⁿ, I) come from the rays of the normal fanDIof I. So one can test the conjecture using the combinatorial description in [3] of the roots of b^A_IⁿðsÞ in terms of the Newton polyhedron of I.

CONTACTJen-Chieh Hsiao jhsiao@mail.ncku.edu.tw National Cheng Kung University, No 1 University Road East District, Tainan 701, Taiwan.

ß 2019 Taylor & Francis Group, LLC

https://doi.org/10.1080/00927872.2018.1444172

The purpose of this paper is to test the conjecture for the motivic zeta function Z(Ur, I;s) and the Bernstein–Sato polynomial b^UI^rðsÞ assocaited to a monomial ideal I on an affine toric variety Ur. To do this, we need an explicit formula for Z(Ur,I;s). This relies on a change of variables formula of motivic zeta functions obtained in [7] applying to a toric log resolution h : X(D)!Urof (Ur, I) and the description of the contact locus of a monomial ideal on a toric variety in [13] or Section 3. As a result, the motivic zeta function Z(Ur, I;s) has a similar description as in the case studied in [10], except that a pole of Z(Ur, I;s) may come from a ray of the fanD that is not a ray of the normal fan DI. In fact, we show in Example 6.2 and 6.3 that such a ray does possibly produce a pole of Z(Ur, I;s) that is not a root of the b-function b^U_I^rðsÞ. However, under a mild assumption we can still prove in Theorem 6.4 that the poles obtained from the rays of the normal fanDI are roots of b^U_I^rðsÞ. See Section 2.4 for the definition of a pole of the motivic zeta function Z(X, I ; s) as well as the analogous strong Monodromy Conjecture in the motivic setting (Conjecture 2.1).

Our explicit computation of the toric motivic zeta functions is one of the first few examples where the ambient space is singular. We mention the work of Veys on the motivic zeta function of an effective divisor E on a Q-Gorenstein variety X [15] with the condition that the singular locus of X is contained in the support of E. See also [16] for a study of relevant zeta functions on varieties with Kawamata log terminal singularities.

2. Motivic Igusa zeta functions of Denef and Loeser We state some definitions and results from [7].

2.1.

Let k be a field of characteristic 0 and let M be the Grothendieck ring of algebraic varieties over k. SetL :

¼ ½A¹k 2 M and denote ^M the completion of M[L^1] with respect to the filtration fF^mM½L^1g_m2Z, where F^mM½L^1 is the subgroup of M[L^1] generated by f[S]L^ijidimS  mg.

For an algebraic variety X of dimension d over k, denote L(X) and Lm(X) the arc space and the m-th jet scheme of X respectively. Let pm: LðXÞ ! LmðXÞ be the canonical morphism corre- sponding to truncation of arcs. For a semi-algebraic subset A of L(X), Denef and Loeser define the motivic measure lðAÞ 2 cM of A. We will not recall the definition of a semi-algebraic of L(X) nor the construction of the motivic measure l. Instead, we only mention some properties that will be used later.

1. If a semi-algebraic set A of L(X) is contained in L(S) for some closed subvariety S of X with dim S<dim X, then l(A) ¼ 0.

2. If AP i, i2N, are semi-algebraic and mutually disjoint, then A :¼ ⋃;i2NAi is semi-algebraic and

i2NlðAiÞ converges to l(A) in cM.

3. If BLm(X) is constructible then p^1_m ðBÞ is semi-algebraic. Such semi-algebraic set p^1_m ðBÞ will be called a cylinder or a constructible set in L(X). Moreover, if X is smooth and A ¼ p^1_m ðBÞ is constructible, then

lðAÞ ¼½pmðAÞ

L^ðmþ1Þd2 cM which does not depend on m.

2.2.

Let I be a coherent ideal sheaf on X, and consider the function ordtI:L(X)!N[f1g given by c7! mingordtgð~cÞ, where the minimum is taken over all g in the stalk Ip0ðcÞof I at p0(c) and~c 2

LðXÞðkcÞ ¼ Xðkc½½tÞ is the corresponding rational point of c2L(X) over the residue field kc of c on L(X). The motivic Igusa zeta function associated the ideal I on X is defined as

ZðX; I; sÞ ¼Ð

LðXÞL^ordtIs dl:¼

X

n0

lðContⁿðIÞÞ  L^ns 2 cM½½L^s (2.1)

where the fiber ContⁿðIÞ :¼ fc 2 LðXÞ j ordtIðcÞ ¼ ng ¼ ðordtIÞ^1ðnÞ of the function ord_tI is the n-th contact locus of I .

2.3.

For a log resolution h:Y!X of the pair (X,I ), let L(h):L(Y)!L(X) be the induced map on the corresponding arc spaces. There is a change of variables formula

ð

LðXÞL^ordtIsdl ¼ ð

LðYÞL^ðordtI8LðhÞsþordtJachÞdl :¼

X

i;j2N

lðContⁱðI OYÞ\Cont^jðJac_hÞÞL^ðisþjÞ: (2.2)

Here, the Jacobian ideal sheaf Jachof h is by definition the ideal sheaf ofOYsuch that the image of the morphism h^X^dX! X^dY induced by pulling-back d-forms can be written as Jac_hX^dY. Note thatX^dX is the dth wedge product of the sheaf of K€ahler differentials XX on X. For more details, except for [7], the reader is also referred to [8] in which another proof of this formula is given.

2.4.

Now, let I be an ideal on an affine variety X. Assume X is either smooth or a normal toric var- iety. Then we have the notion of b-function b^X_IðsÞ for the pair (X,I) [2, 11]. The strong Monodromy Conjecture in the motivic setting can be formulated as the following.

Conjecture 2.1. There exists a finite subset S ofZ>0 Z>0 such that ZðX; I; sÞ 2 cM L^s; 1

1 L^asb

 

ða;bÞ2S

and such that ^b_a is a root of b^X_IðsÞ.

Under this conjecture, we can express the motivic zeta function Z(X,I ;s) as a rational function

NðL^sÞ

DðL^sÞ for NðL^sÞ; DðL^sÞ 2M½L^^s. A rational number ^b_a will be called a pole of the motivic zeta function Z(X,I ;s) if 1L^asbdivides D(L^s) but not divides N(L^s).

3. Arc spaces of toric varieties

We recall some results in [13] concerning the arc spaces of toric varieties. Throughout the paper, we will use freely the results in toric geometry as described in [9].

3.1.

Let M be the free abelian groupZ^dand N its dual Hom_Z(M,Z) together with the canonical pair- ing h,i:NM!Z. We denote M ZR and N ZR by MR and N_R, respectively. The pair h,i natur- ally extends to a pairing N_R M_R! R which we also denote by h,i. For a linear subspace WN_R, the induced pairing ðN_R=WÞ  W^?! R is also denoted by h,i. Let q : NR! N_R=W be the projection. We have hv,ui ¼ hq(v), ui for v2NR and u2W^?.

Let X ¼ X(D) be the toric variety over a field k corresponding to a d-dimensional fan D in NR

and denote by T: ¼ Spec k[M] the dense open torus in X. For a cone s 2D, denote XðsÞ ¼ orb s the orbit closure corresponding to s. The toric variety X(s) corresponds to the the fan D^s in NR=Rs that consists of the image r in NR=Rs of the cones r 2 D, s<r. The affine toric subvari- ety Ur  XðsÞ is Spec k½s^?\ r^Ú\ M. Denote by Nsthe image of N in N_R=Rs.

3.2.

For a field extension K k, let g be the generic point of Spec K[[t]]. The arc space L(X) naturally decomposes as L(X) ¼ ts2DL(X)(s) where

LðXÞðsÞ ¼ fc 2 LðXÞ j cðgÞ 2 orb sg

whose closure in L(X) is LðXÞðsÞ ¼ LðXðsÞÞ. This decomposition can be further refined to L(T)- orbits of L(X) under the action of L(T). In fact, for each s2D, there is a one-to-one correspond- ence

fLðTÞ  c j c 2 LðXÞðsÞg $ jDsj \ Ns

that sends the L(T)-orbit of c 2 L(X)(s) to a lattice point v 2 jDsj \ Ns satisfying, for u2M, ordtc^ðx^uÞ ¼ hv; ui. Under this correspondence, we write L(T)v for the L(T)-orbit L(T)c. Note that the orbits L(T)vare cylinders in L(X). The motivic measure

lðLðTÞ_vÞ ¼ 0; for 0 “ s and v 2 jDsj \ Ns; (3.1) since L(T)vL(X(s)) and since X(s) is a closed subvariety of X with dim X(s)<dim X.

3.3.

For a torus invariant ideal sheaf I on X, the n-th contact locus Contⁿ(I ) is a disjoint union of L(T)-orbits. In fact, it follows from (3.1) above and from Theorem 4.1 and Lemma 5.9 in [13]

that we have the following proposition.

Proposition 3.1 ([13]). An orbit L(T)c ¼ L(T)v, v 2 jDsj \ Ns, is contained in Contⁿ(I ) if and only if, for any cone r 2 D^s containing v, we have LðTÞ_v LðUrÞ and

n ¼ min

x^u2IðU_rÞordtc^ðx^uÞ ¼ min

x^u2IðU_rÞhv; ui:

In particular, the motivic measure lðContⁿðIÞÞ ¼P

vlðLðTÞ_vÞ where the sum runs through all v2jDj\N satisfying n ¼ minx^u2IðUrÞhv; ui for all r2D containing v.

4. Bernstein–Sato polynomials on toric varieties 4.1.

Let R be the cone Rⁿ_0 in Rⁿ and U_R¼ Aⁿk be the corresponding affine space over a field k of characteristic 0. Denote by D the n-th Weyl algebra. In [2], Budur, Mustat¸
a, and Saito define a generalization of Bernstein–Sato polynomials b^UJ^RðsÞ for an ideal J ¼ hf1; :::; fri on U_Ras follows.

Let s1; :::; sr be indeterminates over k, consider k½x1; :::; xn

Y

i¼1

f_i^1; s1; :::; sr

" #

Y

i¼1

f_i^si:

This is a D[sij]-module, where sij¼ sit^1_i tj, and the action of the operator ti is given by tiðsjÞ ¼ sjþ dij (the Kronecker delta). The Bernstein–Sato polynomial or b-function associated to J ¼

hf1; :::; fri is defined to be the monic polynomial b^U_J^RðsÞ of the lowest degree in s ¼Pr

i¼1si satis- fying a relation of the form

b^U_J^RðsÞ

Y

i¼1

f_i^si ¼

X

k¼1

Pktk

Y

i¼1

f_i^si; (4.1)

where P1; :::; Pr2 D½sijj i; j 2 f1; :::; rg. It is shown in [2] that b^UJ^RðsÞ is a nonzero polynomial and is independent of the generating set of J.

4.2.

When J is a monomial ideal on an affine space U_R, the roots of b^U_J^RðsÞ can be completely described in terms of the Newton polyhedron PJof the ideal J [3].

Denote C :¼ R^Ú\ Zⁿ and CI:¼ fu 2 C j x^u2 Ig. Let Q be a face of P_J. Denote by VQ the vector subspace of R^d generated by Q. Let MQ the subset of Zⁿ such that MQ1 is the semi- subgroup ofZⁿgenerated by fa  b j a 2CJ and b 2CJ\ Qg, where 1 ¼ (1, … ,1)2Zⁿ. Let M_Q⁰ :

¼ MQþ b0 be the shift of MQ by any lattice point b02 CJ\ Q, which does not depend on the choice of b0.

Theorem 4.1 ([3]). The set of roots of b^UJ^RðsÞ is the union of the sets RQ:¼ fLQðaÞ j a 2 ðMQn M⁰_QÞ \ VQg

for faces Q of PJthat is not contained in any coordinate hyperplane, where LQ: Rⁿ! R is a lin- ear functional satisfying LQ(a) ¼ 1 for all a2Q.

4.3.

Now, let Ur be the affine toric variety corresponding a d-dimensional cone r in R^dand let I be an ideal on Ur. Apply the same functional equation (4.1) using the differential operators on Ur, one can define the b-function b^U_I^rðsÞ for the pair (Ur,I). In [11], the b-function b^U_I^rðsÞ is related to the b-function b^U_J^RðsÞ of certain ideal J on an affine space UR. We recall the statement as follows.

Let fv1; :::; vng be the set of primitive generators of the rays of r, each of which corresponds to a facet of r^Ú. Consider the linear map F: R^d! Rⁿ that sends u2R^d to ðhv1; ui; :::; hvn; uiÞ 2 Rⁿ. The map F induces a ring homomorphism k½r^Ú\ Z^d ! k½R^Ú\ Zⁿ by sending f :¼P

jkjx^uj to Fðf Þ:¼P

jkjy^FðujÞ, where x^a:¼Q

ix^a_iⁱ and y^b:¼Q

jy^b_j^j for a 2 r^Ú\ Z^d and b 2R^Ú\ Zⁿ. For an ideal I ¼ hf1; :::; fri of k½r^Ú\ Z^d, consider the ideal J ¼ hFðf1Þ; :::; FðfrÞi in k½R^Ú\ Zⁿ.

Theorem 4.2 ([11]). b^UI^rðsÞ ¼ b^UJ^RðsÞ

We mention that Theorem 4.2 is used in [11] to relate the b-function b^U_I^rðsÞ with multiplier ideals of the pair (Ur,I).

5. The case of monomial ideals on affine spaces

Let X ¼ Us be the d-dimensional affine space over a field k of characteristic 0 where sN_R is a pointed smooth cone. Let J be a nonzero proper monomial ideal on X.

5.1.

By Proposition 3.1, the motivic zeta function in (2.1) can be express as ZðUs; J; sÞ ¼

X

n0

lðContⁿðJÞÞ  L^ns¼

X

v2s\N

lðLðTÞ_vÞ  L^JðvÞs

whereJðvÞ :¼ minx^u2Jhv; ui.

5.2.

Let e1; :::; ed be the primitive generators on the rays of s^Ú. Then x^e1; :::; x^ed generate the polyno- mial algebra k[s^Ú\M]. For m  max1idhv; eii, consider the constructible set Cm LmðXÞ con- sisting of m-jets c with ordtc^ðx^eiÞ ¼ hv; eii. Then p^1_m ðCmÞ ¼ LðTÞ_vis a cylinder and

½pmðLðTÞ_vÞ ¼ ½Cm ¼ ðL1Þ^d L^md L^hv;

P

ieii

2 M:

Putting es¼P

iei, the motivic measure of L(T)v can be expressed as lðLðTÞ_vÞ ¼ ½Cm

L^mþ1d¼ ð1  L^1Þ^d L^hv;esi: (5.1) Therefore, the motivic zeta function

ZðUs; J; sÞ ¼

X

v2s\N

ð1  L^1Þ^d ðL^1Þ^{JðvÞsþhv;esi}: (5.2)

5.3.

LetDJbe the normal fan of the Newton polyhedron PJof J. The fanDJconsists of cones r_Q:¼ fv 2 NRjhv; ui  hv; u⁰i for u 2 Q and u⁰2 PJg

corresponding to faces Q of PJ. Note that if w is a vertex of PJsuch that v2s\N is contained in r_w, thenJ(v) ¼ hv,wi.

The following theorem collects the main results in [10]. Although it was originally stated for the coneR ¼ Rⁿ0, we rewrite it for the smooth cone s by using the natural bijection s ffiR.

Theorem 5.1 ([10]). The motivic zeta function Z(U^s,J;s) is a rational function ofL^s. Each pole of Z(Us,J;s) can be written as the form ^hv;e_JðvÞ^sifor a primitive generator v of some ray of the normal fan DJ of J. Moreover, any such number ^hv;e_JðvÞ^si is a root of the Bernstein–Sato polynomial b^UJ^sðsÞ.

In particular, Conjecture 2.1 holds for monomial ideals on affine spaces.

6. The case of monomial ideals on affine toric varieties

Let X ¼ Urbe an affine toric variety of dimension d that corresponds to some cone r in N_R. Let I be a nonzero proper monomial ideal on X. Consider a toric log resolution h:Y ¼ X(D)!X of the pair (X,I). The toric map h factors through the normalized blowing-up X(DI) of X along I. By the change of variables formula (2.2), the motivic zeta function (2.1) can be written as

ZðUr; I; sÞ :¼

X

i;j2N

lðContⁱðI  O^YÞ \ Cont^jðJachÞÞ  L^ðisþjÞ: (6.1)

Let v 2r \ N and let s 2D be a d-dimensional cone containing v. The following observations are helpful in computing the motivic zeta function Z(Ur,I;s).

1. For a L(T)-orbit LðTÞ_v ContⁱðI  OYÞ \ Cont^jðJac_hÞ, it follows from Proposition 3.1 that i ¼ min

x^u2IOYðUsÞhv; ui and j ¼ min

x^u2JachðUsÞhv; ui:

2. By (5.2) the motivic measure lðLðTÞ_vÞ ¼ ð1L^1Þ^d L^hv;esi; where es is the sum of primi- tive generators on the rays of s^Ú.

3. Denote by Vert(PI) the set of vertices of the Newton polyhedron PIof I, then we have

IðvÞ :¼ min

u2PIhv; ui ¼ min

u2VertðPIÞhv; ui ¼ min

x^u2IOYðUsÞhv;ui:

4. Define ⁰ðvÞ :¼ hv; esi. Then ⁰ðvÞ ¼ min_xu2X^dYðUsÞhv; ui. Here, we identify the canonical sheaf X^dY of the smooth toric variety Y with the torus invariant ideal sheaf whose local sections on Uscorrespond to lattice points in the relative interior of s^Ú[9, Section 4.3].

5. Since the sheafX^dX of K€ahler differential d-forms is M-graded [4, Section 3], the image of the toric map h^X^dXðUsÞ ! X^dYðUsÞ is also M-graded. This implies that the Jacobian ideal JachðUsÞ is a monomial ideal in the polynomial ring O(Us). So we can define

⁰⁰ðvÞ :¼ min

x^u2JachðUsÞhv; ui

6. Notice that the two valuations ⁰ and ⁰⁰ extend to the function field of Us, so they do not depend on the choice of s. So we put

DðvÞ :¼ ⁰ðvÞþ⁰⁰ðvÞ ¼ min

x^u2X^dYJachðUsÞhv; ui:

Again, we abuse the notationX^dY to denote the ideal sheaf whose local sections on Uscorres- pond to lattice points in the relative interior of s^Ú. Note that the function D is linear on each cone of the fanD.

With the above observations, we conclude that the motivic zeta function (6.1) of the pair (Ur, I) can be computed as

ZðUr; I; sÞ ¼

X

v2r\N

lðLðTÞ_vÞ  L^{½IðvÞsþ00ðvÞ}

¼

X

v2r\N

ð1L^1Þ^d L^0ðvÞ L^{½IðvÞsþ00ðvÞ}

¼

X

v2r\N

ð1L^1Þ^d ðL^1Þ^{IðvÞsþDðvÞ}

(6.2)

Note that the change of variables formula is valid for any log resolution, so the zeta function Z(Ur,I;s) does not depend on the fan D. Notice also that the expression (6.2) of Z(U^r,I;s) does generalize the expression in (5.2). Indeed, if r is a pointed smooth cone, then by taking h to be the identity map

DðvÞ ¼ min

x^u2X^dX

hv; ui ¼ min

u2intðr^ÚÞ\Mhv; ui ¼ hv; eri:

Since the zeta functions (6.2) and (5.2) have similar expressions, we have the following gener- alization of the first part of Theorem 5.1 whose proof goes exactly the same as in [10] using their results of zeta functions associated to cones.

Theorem 6.1. The motivic zeta function Z(Ur,I;s) is a rational function of L^s. Each pole of Z(Ur,I;s) is of the form ^_^D_IðvÞ^ðvÞ, where v is the primitive generator on a ray of a fan D so that X(D)!Ur is a toric log resolution of (Ur,I).

Proof. Rewrite the motivic zeta function (6.2) as ZðUr; I; sÞ ¼ ð1L^1Þ^d

X

s2DI

Zs;I;DðsÞ; where

Z_s;I;DðsÞ :¼

X

v2intðsÞ\Z^d

ðL^1Þ^{IðvÞsþDðvÞ}:

Since I and D are linear on each s2DI, the statement of the theorem follows from [10,

Theorem 1.3]. w

Next, we investigate whether poles of the motivic zeta function Z(Ur,I;s) are roots of the Bernstein–Sato polynomial b^UI^rðsÞ as defined in [11].

Example 6.2. Consider the affine toric variety Ur whose cone r ¼ Coneðe2; de1 e2Þ, d2. The dual cone r^Ú ¼ Coneðe^₁; e^1þ de^₂Þ. Introduce variables x,y so that the coordinate ring of Ur is represented as k½r^Ú\ M ¼ k½x; xy; :::; xy^d: Let I be the ideal hx³y²; x⁴yi in k[r^Ú\M].

Let h : X(D)!Ur be the minimal log resolution of (Ur,I), namely D ¼ s1[ s2[ s3 where s1¼ Coneðe1; de1 e2Þ, s2¼ Coneðe1; e1þ e2Þ, and s3¼ Coneðe1þ e2; e2Þ. We identify the coordinate ring of the local charts Usi of X(D) as

k½s^Ú₁\ M ¼ k½y^1; xy^d; k½s^Ú2 \ M ¼ k½xy^1; y; and k½s^Ú3 \ M ¼ k½x; x^1y: The Jacobian ideals of the map h on the local charts are

JachðUs1Þ ¼ hxy^di; JachðUs2Þ ¼ hxyi; and JachðUs3Þ ¼ hx²i:

Put w1¼ (2,2d1) and w2¼ ð2; 1Þ ¼ w3. Then we have

DðvÞ ¼ hv; wii for i ¼ 1; 2; 3:

Put v1 ¼ (d,1), v2 ¼ (1,0), v3 ¼ (1,1), and v4 ¼ (0,1). These are primitive generators on the rays ofD. Note that v2 comes from the ray ofD that is not a ray of the normal fan DI. We have

Dðv1Þ ¼ 1 ¼ Dðv4Þ, Dðv2Þ ¼ 2 and Dðv3Þ ¼ 3. On the other hand, we have Iðv1Þ ¼ 3d  2,

Iðv2Þ ¼ 3, Iðv3Þ ¼ 5, and Iðv4Þ ¼ 1, so the candidate poles of Z(Ur,I;s) are

1 3d  2; 2

3 ; 3

5 ; and 1:

For d ¼ 2, the b-function for the pair (Ur,I) is b^U_I^rðsÞ ¼ ðs þ 1Þ²

 s þ1

2

2 s þ1

4



s þ3 4



s þ2 5



s þ3 5



s þ4 5



s þ6 5





 s þ 7

10



s þ 9 10



s þ11 10



s þ13 10

 :

One sees that^1₄ ;^3₅ ; and  1 are roots of b^UI^rðsÞ, but^2₃ is not. We point out that b-function algorithms have been developed and implemented, see [1].

In fact, Theorem 6.4 shows that a candidate pole coming from a ray of the normal fan DIis always a root of b^U_I^rðsÞ. Moreover, it follows from [10, Lemma 3.1] that^2₃ does appear as a pole of Z(Ur,I;s) (not just a candidate). Indeed, one can verify this by explicitly computing Z(Ur,I;s).

Set T ¼ L^s and Bi:¼ ðTÞ^{IðviÞsþDðviÞ}. Then by [10, Lemma 3.1], we have

ZðUr; I; sÞ ¼

X

1i3

1 1  Bi

 1

1  Biþ1



X

i¼2;3

1 1  Bi

¼ð1B1B3Þð1B

Q

1i4ð1  BiÞ

¼ð1T^9sþ4Þð1T^sþ1ÞþT^sþ1ð1T^3sþ2Þð1T^4sþ1Þ ð1  T^4sþ1Þð1  T^3sþ2Þð1  T^5sþ3Þð1  T^sþ1Þ :

Example 6.3. Even for principal ideals, the situation is not as expected. Consider r ¼ Coneð3e1 2e2; e2Þ and I ¼ hx³y²i a principal ideal on Ur whose coordinate ring is represented as k½x; xy; x²y³. Let h : X(D)!Ur be the minimal log resolution of (Ur,I), namely D ¼ s1[ s2[ s3

where s1¼ Coneð3e1 2e2; 2e1 e2Þ, s2¼ Coneð2e1 e2; e1Þ, and s3¼ Coneðe1; e2Þ. We identify the coordinate ring of the local charts Usi of X(D) as

k½s^Ú₁ \ M ¼ k½x²y³; x^1y^2; k½s^Ú2 \ M ¼ k½xy²; y^1; and k½s^Ú3 \ M ¼ k½x; y:

The Jacobian ideals of the map h on the local charts are

Jac_hðUs1Þ ¼ hx²y³i; JachðUs2Þ ¼ hx; x²y³i; and JachðUs3Þ ¼ hxi:

One verifies that the minimal log resolution h produces three candidate poles of Z(Ur,I;s),

1

5 ;^12 ;^23. Only^2₃ is not a root of b^U_I^rðsÞ ¼ ðsþ1Þ²ðs þ¹₂Þðs þ¹₅Þðs þ²₅Þðs þ³₅Þðs þ⁴₅Þ.

Theorem 6.4. Let h:X(D)!U^r be a toric log resolution of (Ur,I) such that I OXðDÞ  Jach XXðDÞ.

Here, we identify XX(D) with the torus invariant ideal ofOX(D) whose local sections on each chart Us, s2D, correspond to the lattice points in the interior of s^Ú. If v is a primitive generator on a ray ofDIsuch thatI(v)6¼0, then ^_^D_IðvÞ^ðvÞis a root of the Bernstein–Sato polynomial b^UI^rðsÞ.

Proof. We use freely the notation in Section 4. Let v be a primitive generator on a ray of DIsuch thatI(v) 6¼ 0. Let Q be the facet of the Newton polyhedron PIthat corresponds to v. Note that Q is not contained in any faces of r. Note also that the assumption I OXðDÞ Jac_h XXðDÞ implies thatIðvÞ  DðvÞ: Let s be a d-dimensional cone of D that contains v and let w 2 s^Ú\M be a lat- tice point that corresponds to a monomial in the ideal Jach XXðDÞðUsÞ so that D ¼ hv,wi. We haveI(v)hv,wi1.

Let ~v 2R\Qⁿ be such that F(~v) ¼ v where F^: ðRⁿÞ^! ðR^dÞ^ is the dual of the map F.

Then

hv; ui ¼ hF^ð~vÞ; ui ¼ h~v; FðuÞi; for u 2 Z^d: This implies thatIðvÞ ¼ minx^u2Ih~v; FðuÞi ¼ Jð~vÞ and that

 ^DðvÞ

IðvÞ ¼ hv; wi

Jð~vÞ¼ h~v; FðwÞi

Jð~vÞ :

Set LðaÞ:¼^h~v;ai_Jð~vÞ. Then L(a) takes the value 1 on a face ~Q containing F(Q). By Theorems 4.1 and 4.2, it suffices to show that FðwÞ 2 ðM_~Qn M⁰_~QÞ \ V_~Q. Since Q is a facet of PI, it is certainly true that w 2 VQ and hence FðwÞ 2 FðVQÞ  V_~Q. Moreover, pick any b 2CJ\ ~Q. Since w lies in the relative interior of r, we have F(w)12intR and a: ¼ F(w)1þb2CJ. Hence FðwÞ ¼1 þ a  b 2 M~Q.

To show that FðwÞ 62 M⁰_~Q, it suffices to observe that the linear functional L takes values that are strictly greater than 1 on M⁰_~Q, but LðFðwÞÞ ¼^hv;wi_IðvÞ 1. w

Acknowledgements

The first author thanks Laura Matusevich for carefully reading the first draft of this paper. He is also grateful to Uli Walther for his comment on the expression of the poles of the motivic zeta funcions.

Funding

J.C.H. is partially supported by MOST grant 105-2115-M-006-015-MY2. C. J. L. is partially supported by MOST grant 106-2115-M-006 -019.

References

[1] Berkesch, C., Leykin, A. (2010). Algorithms for Bernstein-Sato polynomials and multiplier ideals. In: ISSAC 2010—Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, ACM, New York, pp. 99–106. MR 2920542

[2] Budur, N., Mustat¸a, M., Saito, M. (2006). Bernstein-Sato polynomials of arbitrary varieties. Compos. Math.

142(3):779–797. MR 2231202 (2007c:32036)

[3] Budur, N., Mustat¸a, M., Saito, M. (2006). Combinatorial description of the roots of the Bernstein-Sato poly- nomials for monomial ideals. Comm. Algebra 34(11):4103–4117. MR 2267574 (2007h:32041)

[4] Corti~nas, G., Haesemeyer, C., Walker, M. E., Weibel, C. (2009). The K-theory of toric varieties. Trans. Am.

Math. Soc. 361(6):3325–3341. MR 2485429

[5] Denef, J. (1991). Report on Igusa’s local zeta function. Asterisque 1990/91(201–203):359–386 (1992).

Seminaire Bourbaki, Exp. No. 741. MR 1157848 (93g:11119)

[6] Denef, J., Loeser, F. (1998). Motivic Igusa zeta functions. J. Algebraic Geom. 7(3):505–537. MR 1618144 (99j:14021)

[7] Denef, J., Loeser, F. (1999). Germs of arcs on singular algebraic varieties and motivic integration. Invent.

Math. 135(1):201–232. MR 1664700

[8] Ein, L., Mustat¸
a, M. (2009). Jet schemes and singularities. Summer Institute in Algebraic Geometry at the University of Washington, Seattle from July 25 to August 12, 2005. In: Part 2, Proc. Sympos. Pure Math., Vol. 80. Providence, RI: Am. Math. Soc., pp. 505–546. MR 2483946

[9] Fulton, W. (1993). Introduction to Toric Varieties. Annals of Mathematics Studies, Vol. 131. Princeton, NJ:

Princeton University Press (The William H. Roever Lectures in Geometry). MR 1234037

[10] Howald, J., Mustat¸
a, M., Yuen, C. (2007). On Igusa zeta functions of monomial ideals. Proc. Am. Math.

Soc. 135(11):3425–3433. MR 2336554

[11] Hsiao, J.-C., Matusevich, L. F. (2016). Bernstein–Sato Polynomials on Normal Toric Varieties, ArXiv e- prints.

[12] Igusa, J.-I. (2000). An Introduction to the Theory of Local Zeta Functions. AMS/IP Studies in Advanced Mathematics, Vol. 14. Providence, RI/Cambridge, MA: American Mathematical Society/International Press.

MR 1743467

[13] Ishii, S. (2004). The arc space of a toric variety. J. Algebra 278(2):666–683. MR 2071659

[14] Nicaise, J. (2010). An Introduction to p-Adic and Motivic Zeta Functions and the Monodromy Conjecture.

Algebraic and Analytic Aspects of Zeta Functions and L-Functions, MSJ Mem., Vol. 21. Tokyo: Math. Soc.

Japan, pp. 141–166. MR 2647606

[15] Veys, W. (2001). Zeta functions and “Kontsevich invariants” on singular varieties. Canad. J. Math. 53(4):

834–865. MR 1848509 (2002e:14025)

[16] Veys, W. (2003). Stringy zeta functions for Q-Gorenstein varieties. Duke Math. J. 120(3):469–514. MR 2030094 (2004m:14075)

[17] Veys, W. (2006). Arc Spaces, Motivic Integration and Stringy Invariants. Singularity Theory and Its Applications, Adv. Stud. Pure Math., Vol. 43. Tokyo: Math. Soc. Japan, pp. 529–572. MR 2325153