GENERIC NEWTON POLYGON OF THE L-FUNCTION OF n VARIABLES OF THE LAURENT POLYNOMIAL I?

Fusheng LENG()

Academy of Mathematics and Systems Science,Chinese Academy of Sciences,Beijing 100190,China

E-mail:leng@amss.ac.cn

Abstract The Hodge bound for the Newton polygon of L-functions of T-adic exponential sums associated to a Laurent polynomial is established.We improve the lower bound and study the properties of this new bound.We also study when this new bound is reached with large p arbitrarily,and hence the generic Newton polygon is determined.

Key words exponential sum;Hodge bound;combinatorial Newton polygon

Dedicated to Professor Banghe LI on the occasion of his 80th birthday

1 Introduction

withmij=Bpi?j.

Knowing the analytic continuation ofCf(s,T),we are then interested in the location of its zeros.More precisely,we would like to determine theT-adic Newton polygon NPT(?,q)of this entire functionCf(s,T).

This is expected to be a complicated problem.LetD=D(?)be the least common denominator of all weights of vectors incone(f)(see Section 2 below).The only easy case for determining NPT(?,q)isp≡1 modD,which called the Hodge lower bound ofCf(s,T)and is denoted by HPq(?)[13].

The lowest Newton polygon GNPT(?,q)=inffNPT(?,q)exists and is called the generic Newton polygon ofCf(s,T).We have the relation

There have been many discussions on when NPT(?,q)=HPq(?).Other natural questions concern what exactly the generic Newton polygon is,and when NPT(?,q)=GNPT(?,q).There are currently only a few conclusions about these questions.Forn=1,the second issue was solved by Zhu [9]and was simplified by Blache and F′erard [2].Some special cases of ? were discussed forp≡?1 modDby Liu and Niu [12].

Ren in 2017 gave another lower bound of NPT(?,q),called the improved Hodge polygon,and calculated the points with integer multiples of ? for a parallelotope case [11].

We will give a combinatorial description of the generic lower bound and study it.In Section 2,we will give the rigorous definition of the weight of the vector and prove some of its properties.In Section 3,we will give the generic lower bound of NPT(?,q)and prove that

and

naturally.Finally,in Section 4,we will show why CNPT(?,q)will be GNPT(?,q),and do some preparatory work for the subsequent proof.

The main results here are Theorems 2.9,3.11 and 4.2.

2 Convex Polyhedron and Weight of Vector

A subsetJ(f)ofI(f)is called a generating set ofI(f)if?f0=?fwithf0=f|J(f).

Lemma 2.3The minimum generating sete(f)offis unique.e(f)is called the vertex set off.

ProofA vector v inI(f)is called a vertex ifcv=1 is satisfied for anyw-representationThencu=0 for everycu/cvand thisw-representation is unique.Notice that the weight of a vertex is equal to 1.

Lettinge(f)be the set of all such vertices,we will prove thate(f)is the unique minimum generating set off.

Suppose thatJ(f)is a local minimum generating set off;that is,any real subset ofJ(f)is not a generating set off.Clearly,e(f)?J(f).

Assuming a vector v∈J(f)e(f),v has aw-representation

withcv<1 andw(v)≤1.Then

is a representation of v with

so this is aw-representation of v as well.

Corollary 2.4For any v∈cone(f),there exists aw-representation=v ofv.

ProofFollowing Lemma 2.1,we need only to prove the case of whenw(v)=1(which means that v∈?f).This is clear by Lemma 2.3. □

ProofSufficiency is trivial,and necessity is obtained from Lemma 2.5. □

Ifei(f)=ej(f)for i,j∈cone(f),we say that i～j.“～”is an equivalence relation.

ProofClearlyw(ai+bj)=aw(i)+bw(j),by Lemma 2.1 and Lemma 2.2 and Lemma 2.5,and thusei(f)=ej(f)?eai+bj(f).

isaw-representation of i.Thereforeeai+bj(f)?ei(f),and finally,we have thatei(f)=ej(f)=eai+bj(f). □

This lemma shows that every equivalence class of“～”is actually acone without edges.All ofev(f)under the containing relationships will be a partial order,and every maximal elementez(f)={v1,···,vk}generates a co-dimension 1 facial of?fasThere are many conclusions to be drawn from this,but we are here to cover only what is needed for the main theorem.

Sincee(f)(ev(f)etc.)depends only on?=?f,we rewrite this as bye(?)(ev(?)etc.).Defi ne

Following Lemma 2.7,?is split into?vand the origin,which is called the regular splitting of?,and?vis called a regular cell(or cell for short)of?(the cell containing v is also written as Cv).Notice that any cell is convex.

We rewriteev(?)ase(?v).

Fix an orderρ:e(?)→{1,···,|e(?)|}as a bijection.

ProofThis follows from Lemma 2.5 and the definition of a basic cell. □

Jeffrey Karp, an associate professor of medicine at Harvard Medical School and co-director of the Center for Regenerative Therapeutics at Brigham and Women s Hospital, is also a senior author of the paper. Lead author is Woo Kyung Cho, a postdoc in the Harvard-MIT Division of Health Sciences and Technology (HST).

Theorem 2.9An orderρof all vertices ine(?)leads to a certain basic cell splitting.

ProofNotice that all basic cells do not intersect with each other.Then the theorem follows directly from the definition of a basic cell. □

Now we can define the global weight by orderρas follows

The global weight wρhas similar properties of the weightw.Denoting thatcelluis the basic cell containing u for any u∈cone(?),we have

Proposition 2.14

The equation holds if and only ife(celli)∪e(cellj)=e(celli+j).

ProofThis follows from Proposition 2.10 and Corollary 2.12. □

Examp le 2.17A nontrivial example ofρ-dec for dimension 2 is a pentagon which is?=convex {(?1,3),(?9,?1),(0,?10),(2,?8),(2,1)}andI(f)=?∩M(?).

Letfbeρ-dec.All vectors inI(f)that appear in the-representationsof each j∈cone(?)form a subset ofI(f),called the basis off,which we denote asB(f).

3 Combinatorial Newton Polygon

Following this lemma,Mrcan be expressed as

whereσa?1:{i1,···,ir}→M(?f)is an injective map.Repeat to use this lemma,we then have

whereσ1,···,σa?1:{i1,···,ir}→M(?f)are all injective maps.

Since Pkuu=pi?j andw(u),by the definition of?f,andkuare non-negative integers,following Lemmas 2.1 and 2.2 we have that

then for any i1,···,irinM(?f)that are different from each other,and for any permutationσof {i1,···,ir}andσ1,···,σa?1,we have

Denote thatF<α={v∈M(?f):w(v)<α}andFα={v∈M(?f):w(v)≤α}forαinWf.

Definition 3.2A subsetSofM(?f)is said to be loyal ifF<α?S?Fαfor someαinWf.We also say thatαis the weight ofSand denote this byw(S)=α.IfS=Fα,we callSthe Complete loyal set.

Obviously,we have the following lemma:

Above all,we have that

This is the Hodge lower bound ofCf(s,T).

Recalling the equation(1),we take a different approach to valuation;that is,

and hence

which is the so-called improved Hodge lower bound in [2].

We also have the third estimate of ordπMr.Before further discussing this,we will introduce some definitions.

Definition 3.4Assume thatα∈WfandS?M(?f).Denotep?as the congruence modDsuch that

We set that

For 0≤α<1,define that

Set that

We now can give the lower bound.

Therefore we can define the combinatorial Newton polygon as a lower bound of the Newton polygon ofCf(s,T).

Definition 3.6TheT-adic combinatorial Newton polygon ofCf(s,T)is defined by

The definition of the combinatorial Newton polygon does not depend on a specifi cf,but on the polytope ?f.

Lemma 3.7For any positive integerkand any 0≤α<1,

ProofIt is easy to get the proof from the definition ofgx(α). □

This lemma shows(x,grk?1+x(α))on the same line with slope?1,0 or 1.Notice thatδp,α(αk)for all of 0≤α<1 cannot equal?1 and 1 at the same time,since {p?αk}<{αk}and {p?αk}>{αk}are contradictory,and thus we have

Proposition 3.8Let C=convex {(x,Np(x)):rk?1≤x≤rk}for a fixed positive integerkand denote thatmp(αk)=Np(rk)?Np(rk?1).Then

The case {p?αk}>{αk}is similar.

The case {p?αk}={αk}is simple,sinceδp,α(αk)=0 for any 0≤α<1. □

This proposition shows that the combinatorial Newton polygon has at most one vertex betweenrk?1andrk.We denote this midpoint byηk=(xk,yk).

Since Ψx? Ψx?1=?w(pαk)」??w(αk)」is a constant forrk?1

Theorem 3.9

ProofWe omit the proof. □

Obviously CNPT(?,p)is higher than the improved Hodge polygon defined by Ren in [2].We now show that CNPT(?,p)is also higher than the Hodge polygon ofCf(s,T).

Lemma 3.10LetSbe a loyal set with|S|=rand 0≤α<1 that satisfiesgr(α)=Np(r).Then there is a permutationσofSsuch that

ProofFor the definition ofNp(r),if 0≤α0<1 satisfies thatα0<αand that

thengr(α0)>gr(α)is a contradiction.Hence,we have that

for every 0≤α0<α.This means that there is an injective map

satisfying that {w(pv)}≤{w(σ1(v))}≤α.

Similarly,we have that

for everyα<α0<1,and hence we have an injective map

satisfying thatα<{w(pσ2(v))}≤{w(v)}.

Set a permutationσofSsuch that

Then the lemma holds. □

Theorem 3.11

ProofAll we need to prove is that Ψr+Np(r)≥(p?1) Λr.

Since

by Lemma 3.7,forgr(α)=Np(r)and a loyal setSwith|S|=r,there is a permutationσofSsatisfying the conditions of the lemma,which means that

Finally we have that Ψr+Np(r)≥(p?1) Λr.The equation holds only if

(i.e.,Np(r)=0),which means that LS({h})=LS({p?h})for every 0≤h<1.Therefore the equation holds if and only ifNp(r)=0. □

Definition 3.12We say that?is monotonic if|L(α)|≤|L(α′|for anyα≤α′∈Wf.We say that?is regular if,for any complete loyal setS,there is someαsatisfying thatgC(α)=Np(|C|)for any basic cell C ofS.

For any finite subsetS?M(?f),denote thatw(S)=max(w(i))i∈S.

Lemma 3.13If?is monotonic,then

Corollary 3.14If?is monotonic,then it is regular.

ProofThe proof is omitted. □

Combining this with Lemma 3.7,we have

Lemma 3.15If?is monotonic,thenNp(x)=gr({w(Sk?1)})forrk?1≤r

ProofThe proof is omitted. □

Theorem 3.16Iffisρ-dec,then every basic cell ofρis monotonic.

ProofLet C be a basic cell ofρand let s0be the wρ-minimal nonzero vector in C.ThenL(α)+s0?L(α+)for anyα∈Wf,which is what we need. □

4 Split Theorem

then we have that

ProofSince NPT(?,q)≥HPq(?),the proof follows from Proposition 4.1. □

Recalling Definition 2.15,let

Lemma 4.3Suppose thatfisρ-dec andp>D.Then

The equation holds only ife(cellσ(i))?e(celli),sincepi and i are in the same basic cell,which means thatσ(i)is in the convex ofcelli.Thereforeσ(i)∈celli,by Theorem 2.9;i.e.,σacts separately on each basic cell.

On the other hand,whenp>D,pi?σ(i)∈celliifσ(i)∈celli.By Proposition 2.14 again,we have thatσ(celli)=cellifor any i∈Sk.

Finally we have finished the proof by Lemma 2.18. □

For a fixed basic cell C,let i1,j1∈Sk∩cone(C)satisfy that

We define thatσ0(i1)=j1.Next,let i2∈Sk∩cone(C){i1}and j2∈Sk∩cone(C){j1}satisfy that

We defineσ0(i2)=j2,and so on.

Lemma 4.4σ0∈Σk.

which leads to a contradiction.Thusα≤{w(σ0(i))}≤{w(Sk)}andw?(pi?σ0(i))=?w(pi)」??w(σ0(i))」.

The condition of Lemma 4.4 holds in many cases,but not all cases.This lemma shows that CNPT(?,p)seems to be a nearly-best lower bound of the Newton polygon ofCf(s,T).