Generic Newton polygon for exponential sums in two variables with triangular base
aa r X i v : . [ m a t h . N T ] J a n GENERIC NEWTON POLYGON FOR EXPONENTIAL SUMS IN TWOVARIABLES WITH TRIANGULAR BASE
RUFEI REN
Abstract.
Let p be a prime number. Every two-variable polynomial f p x , x q over a finite field ofcharacteristic p defines an Artin–Schreier–Witt tower of surfaces whose Galois group is isomorphicto Z p . Our goal of this paper is to study the Newton polygon of the L -functions associated to afinite character of Z p and a generic polynomial whose convex hull is a fixed triangle ∆. We denotethis polygon by GNP p ∆ q . We prove a lower bound of GNP p ∆ q , which we call the improved Hodgepolygon IHP p ∆ q , and we conjecture that GNP p ∆ q and IHP p ∆ q are the same. We show that ifGNP p ∆ q and IHP p ∆ q coincide at a certain point, then they coincide at infinitely many points.When ∆ is an isosceles right triangle with vertices p , q , p , d q and p d, q such that d is notdivisible by p and that the residue of p modulo d is small relative to d , we prove that GNP p ∆ q andIHP p ∆ q coincide at infinitely many points. As a corollary, we deduce that the slopes of GNP p ∆ q roughly form an arithmetic progression with increasing multiplicities. Contents
1. Introduction 12. Dwork trace formula 43. Improved Hodge polygon for a triangle ∆ 94. The case when ∆ is an isosceles right triangle I. 225. The case when ∆ is an isosceles right triangle II. 30References 461.
Introduction
We shall state our main results and their motivation after recalling the notion of L -functions for Witt coverings. Let p be a prime number. Let f p x , x q : “ ÿ P P Z ě a P x P x x P y be a two-variable polynomial in F p r x , x s and writeˆ f p x , x q : “ ÿ P P Z ě ˆ a P x P x x P y for its Teichm¨uller lift, where ˆ a P denotes the Teichm¨uller lift of a P . We use F p p f q to denotethe extension of F p generated by all coefficients of f and set n p f q : “ r F p p f q : F p s . Theconvex hull of the set of points p , q Y P | a P ‰ ( is called the polytope of f and denotedby ∆ f . Date : November 8, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Artin–Schreier–Witt towers, T -adic exponential sums, Slopes of Newton polygon, T -adicNewton polygon for Artin–Schreier–Witt towers, Eigencurves. Let p G m q be the two-dimensional torus over F p n p f q . The main subject of our study isthe L -function associated to finite characters χ : Z p Ñ C ˆ p of conductor p m χ given by L ˚ f p χ, s q : “ ź x P|p G m q | ´ χ ` Tr Q pn p f q deg p x q { Q p p ˆ f p ˆ x qq ˘ s deg p x q , where |p G m q | is the set of closed points of p G m q and ˆ x is the Teichm¨uller lift of a closedpoint x in p G m q . The characteristic power series C ˚ f p χ, s q is a product of reciprocals of L -functions:(1.1) C ˚ f p χ, s q “ ź j “ L ˚ f p χ, p jn p f q s q ´p j ` q . We can alternatively express L ˚ f p χ, s q in terms of C ˚ f p χ, s q as L ˚ f p χ, s q “ ´ C ˚ f p χ, s q C ˚ f p χ, p n p f q s q C ˚ f p χ, p n p f q s q ¯ ´ . Therefore, C ˚ f p χ, s q and L ˚ f p χ, s q determine each other. Definition 1.1.
From [LWei], we know that L ˚ f p χ, s q ´ : “ p p mχ ´ q Area p ∆ f q ÿ i “ v i s i is a polynomial of degree 2 p p m χ ´ q Area p ∆ f q in Z p r ζ p mχ sr s s , where ζ p mχ is a primitive p m χ -th root of unity. We call the lower convex hull of the set of points ` i, p m χ ´ p p ´ q v p n p f q p v i q ˘ the normalized Newton polygon of L ˚ f p χ, s q ´ , which is denoted by NP p f, χ q L ´ . Here, v p n p f q p´q is the p -adic valuation normalized so that v p n p f q p p n p f q q “ . Similarly, we writeNP p f, χ q C for the normalized Newton polygon of C ˚ f p χ, s q .In [DWX], Davis, Wan and Xiao studied the p -adic Newton slopes of L ˚ f p χ, s q when f p x q is a one-variable polynomial whose degree d is coprime to p . They concluded that, foreach character χ : Z p Ñ C ˆ p of relatively large conductor, NP p f, χ q L ´ depends only on itsconductor. We briefly introduce their proof as follows.They proved a lower bound of NP p f, χ q C when χ is the so-called universal characterand an upper bound by the Poincar´e duality of roots of L ˚ f p χ , s q for a particular character χ of conductor p . The lower bound is called the Hodge polygon in their paper. Then theyverified that the upper bound coincides with the lower bound at x “ kd for any non-negativeinteger k . Since the Newton polygon of C ˚ f p χ, s q is confined between these two bounds, italso passes through their intersections. See more details in [DWX].We also mention here that the aforementioned proof strongly inspired the proof ofspectral halo conjecture by Liu, Wan, and Xiao in [LWX]; we refer to [RWXY] for thediscussion on the analogy of the two proofs. Motivated by the attempt of extending spectralhalo type results beyond the case of modular forms, it is natural to ask whether one cangeneralize the main results of [DWX] to more general cases of exponential sums and Artin–Schreier–Witt towers. For example, in a joint work with Wan, Xiao, and Yu, we examinedthe case when the Galois group of the Artin–Schreier–Witt tower is canonically isomorphicto Z p ℓ .In this paper, we mainly deal with the generic Newton polygon of L -functions for two-variable polynomials. We want to apply the methods in [DWX] to this case. Therefore,it is crucial for us to give a lower bound and an upper bound for C ˚ f p χ, s q . However, the ENERIC NEWTON POLYGON FOR EXPONENTIAL SUMS IN TWO VARIABLES WITH TRIANGULAR BASE3
Hodge polygon provided by Liu and Wan in [LWan] is no longer optimal, and is in generalstrictly lower than the upper bound we obtain by Poincare duality. Our main contributionin this paper is to find an improved lower bound for NP p f, χ q C , which we call the improvedHodge bound IHP p ∆ q . We conjecture that our improved Hodge polygon is optimal, and isequal to the generic Newton polygon, that is the lowest Newton polygon for all polynomials f with the same convex hull.When ∆ f is an isosceles right triangle with vertices p , q , p d, q and p , d q , we willgive an equivalent condition to verify the coincidence of improved Hodge polygon with theNewton polygon (at infinitely many points), and we will show that this condition is met fora generic polynomial with convex hull ∆ f .We now turn to stating our main results more rigorously. Notation 1.2.
For a two-dimensional convex polytope ∆ which contains p , q , we denoteits cone by Cone p ∆ q : “ ! P P R ˇˇ kP P ∆ for some k ą ) , and put M p ∆ q : “ Cone p ∆ q X Z to be the set of lattice points in Cone p ∆ q .Moreover, we write T k p ∆ q (resp. T k p ∆ q ) for the set consisting of all points in M p ∆ q with weight w (See Definition 2.12) strictly less than k (resp. less than or equal to k ), anddenote its cardinality by x k p ∆ q (resp. x k p ∆ q ). Notation 1.3.
For integers a and b , we denote by a % b the residue of a modulo b . Definition 1.4.
The generic Newton polygon of ∆ is defined byGNP p ∆ q : “ inf χ : Z p { p mχ Z p Ñ C ˆ p ∆ f “ ∆ ´ NP p f, χ q L ´ ¯ , where χ : Z p Ñ C ˆ p runs over all finite characters, and f runs over all polynomials in F p r x , x s such that ∆ f “ ∆. The following are our main results. Theorem 1.5.
Let ∆ be a right isosceles triangle with vertices p , q , p , d q , p d, q , where d is a positive integer not divisible by p . Let p be the residue of p modulo d . Suppose d ě p p ` p q . Then the generic Newton polygon GNP p ∆ q passes through points p x k p ∆ q` i, h k p ∆ q ` ki q for any k ě and ď i ď kd ` , where x k p ∆ q “ p kd ` q kd and h k p ∆ q “ p p ´ qp k ´ q k p k ` q d ` k ÿ P P T p ∆ q t pw p P q u . The points p x k , h k p ∆ qq are vertices for the improved Hodge polygon IHP p ∆ q (see Def-inition 2.18 and Proposition 3.16). So the essential content of the proof is to show thatthe generic Newton polygon GNP p ∆ q also passes through these points. The proof of Theo-rem 1.5 consists of two parts: first we show that, for a fixed polynomial f with convex hull∆ f , if IHP p ∆ q coincides with the corresponding Newton polygon NP p f, χ q C at x , thenthese two polygons agree at all points x “ x k p ∆ q ` i for k ě ď i ď kd `
1. Thisis proved in Theorem 3.1, which in fact holds with less constraints on ∆. Next, we provethat, for a generic polynomial f , NP p f, χ q C agrees with IHP p ∆ q at x “ x p ∆ q . For this,we look at the leading term of r v x p ∆ q for the universal polynomial f univ with convex hull ∆and show that this term is non-zero when d ě p p ` p q . This is proved in Theorem 4.2,which in fact holds under a weaker condition on p .From [LWei, Theorem 1.4], for a finite character χ of conductor p m χ , we know that L ˚ f p χ, s q ´ has degree of p p m χ ´ q d . RUFEI REN
Theorem 1.6.
Under the hypotheses of Theorem 1.5, if we put p α , . . . , α p p mχ ´ q d q to bethe sequence of p n p f q -adic Newton slopes of L ˚ f p χ, s q ´ (in non-decreasing order), then forfirst p p mχ ´ q d ` p p mχ ´ q d -th slopes we have $’’&’’% α x i ` , . . . , α x i ` P p ip m ´ , i ` p mχ ´ q for i “ , , . . . , p m χ ´ ´ ,α x i ` , . . . , α x i “ ip mχ ´ for i “ , , , . . . , p m χ ´ ´ ,α x pmχ ´ ` , . . . , α x pmχ ´ ´ “ . In fact, points p x k p ∆ q , h p T k qq are vertices of the improved Hodge polygon (see Defini-tion 2.18 and Proposition 3.16).We do not know if Theorem 1.5 still holds for polytopes which are not right isoscelestriangle. However, for an arbitrary multi-variable polynomial f in F p r x s , we are still ableto get an improved Hodge polygon for NP p f, χ q . Especially, when f is a two-variablepolynomial, it is expected that the slopes of the improved Hodge polygon form certaingeneralized arithmetic progression. We plan to address this in a forthcoming paper.”The Newton polygon for exponential sums was explicitly computed in the “ordinary”case by Adolphson–Sperber [AS], Berndt–Evans [BE], and Wan [W] in many special cases,and in general (namely the T -adic setup) by Liu–Wan [LWan]. For the ∆ we considered inTheorem 1.5, the ordinary condition amounts to requiring p ” p mod d q . Blache, Ferard,and Zhu in [BFZ] proved a lower bound for the Newton polygon of one-variable Laurentpolynomial over F q of degree p d , d q , which is called a Hodge-Stickelberger polygon. Theyalso showed that when p approaches to infinite, the Newton polygon coincides the Hodge-Stickelberger polygon.Going beyond the ordinary case, there has been many researches on understandingthe generic Newton polygon of L f p χ, s q when f is a polynomial of a single variable. Thefirst results are due to Zhu [Z1] and Scholten–Zhu [SZ], when p is large enough. In [BF],Blache and Ferard worked on the generic Newton polygon associated to characters of largeconductors. In [OY], Ouyang and Yang studied the one-variable polynomial f p x q “ x d ` a x .A similar result can be found in [OZ], where Ouyang and Zhang studied the family ofpolynomials of the form f p x q “ x d ` a d ´ x d ´ .Our Theorem 1.5 maybe considered as the first step beyond the ordinary case when thebase polynomial is multivariable. A similar result is obtained by Zhu in [Z2] independentlywhich shows that GNP p ∆ f q and IHP p ∆ f q coincide for characters of Z p of conductor p . Acknowledgments.
The author would like to thank his advisor Liang Xiao for the extra-ordinary support in this paper and also thank Douglass Haessig, Hui June Zhu, and DaqingWan for helpful discussion. 2.
Dwork trace formula
Let p be an odd prime and let f p x , x q : “ ř P P Z ě a P x P x x P y be a two-variable polyno-mial in F p r x , x s . Denote F p p f q to be the finite field generated by the coefficients of f , whichwe call the coefficient field of f . The convex hull of the set of points tp , qu Y P | a P ‰ ( is called the polytope of f and denoted by ∆ f .Our discussion will focus on a fixed f until Proposition 4.4. We put F q “ F p p f q and n “r F q : F p s . Let ˆ a P P Z q be the Teichm¨uller lift of a P . We call ˆ f p x , x q : “ ř P P Z ě ˆ a P x P x x P y the Teichm¨uller lift of f p x q .For convenience, we put v p p´q (resp. v q p´q ) be the p -adic valuation normalized so that v p p p q “ v q p q q “ ENERIC NEWTON POLYGON FOR EXPONENTIAL SUMS IN TWO VARIABLES WITH TRIANGULAR BASE5 T -adic exponential sums.Notation 2.1. We recall that the
Artin–Hasse exponential series is defined by(2.1) E p π q “ exp ` ÿ i “ π p i p i ˘ “ ź p ∤ i, i ě ` ´ π i ˘ ´ µ p i q{ i P ` π ` π Z p rr π ss . Putting E p π q “ T ` Z p J π K – Z p J T K . Definition 2.2.
For each power series in Z q J T K , say g p T q , we define its T -adic valuation as the largest k such that g P T k Z q J T K and denote it by v T p g q . Definition 2.3.
For each k ě
1, the T -adic exponential sum of f over F ˆ q k is S ˚ f p k, T q : “ ÿ p x ,x qPp F ˆ qk q p ` T q Tr Q qk { Q p p ˆ f p ˆ x , ˆ x qq P Z p rr T ss . Definition 2.4.
The T -adic L -function of f is defined by L ˚ f p T, s q “ exp ´ ÿ k “ S ˚ f p k, T q s k k ¯ and its corresponding T -adic characteristic power series is defined by C ˚ f p T, s q : “ exp ´ ÿ k “ ´p q k ´ q ´ S ˚ f p k, T q s k k ¯ (2.2) “ ÿ k “ u k p T q s k P Z p J T.s K , We put u k p T q “ u k,j T j P Z p rr T ss .Moreover, they determine each other by relations:(2.3) C ˚ f p T, s q “ ´ ź j “ L ˚ f p T, q j s q j ` ¯ ´ and(2.4) L ˚ f p T, s q “ ´ C ˚ f p T, s q C ˚ f p T, q s q C ˚ f p T, qs q ¯ ´ . It is clear that for a finite character χ : Z p Ñ C ˆ p , we have L ˚ f p χ, s q “ L ˚ f p T, s q ˇˇ T “ χ p q´ and C ˚ f p χ, s q “ C ˚ f p T, s q ˇˇ T “ χ p q´ , where L ˚ f p χ, s q and C ˚ f p χ, s q are defined in the introduction. Notation 2.5.
Recall that we put E p π q “ T `
1. We put E f p x , x q : “ ź P P Z ě E p ˆ a P πx P x x P y q“ ÿ P P Z ě e P p T q x P x x P y P Z q J T KJ x , x K . (2.5) RUFEI REN
Dwork’s trace formula.
Recall that ∆ f is the convex hull of f p x , x q and M p ∆ f q is defined in Notation 1.2 as a set consisting of all the lattice points in the Cone p ∆ f q . Let D be the smallest positive integer such that w p M p ∆ f qq Ă D Z . Definition 2.6.
We fix a D-th root T { D of T. Define B “ ! ÿ P P M p ∆ f q b P p T { D x q P x p T { D x q P y ˇˇˇ b P P Z q J T { D K , v T p b P q Ñ `8 , when w p P q Ñ 8 ) . Let ψ p denote the operator on B such that ψ p ´ ÿ P P M p ∆ f q b P x P x x P y ¯ : “ ÿ P P M p ∆ f q b p pP q x P x x P y . Recall that n “ r F q : F p s . Definition 2.7.
Define(2.6) ψ : “ σ ´ ˝ ψ p ˝ E f p x , x q : B ÝÑ B , and its n -th iterate ψ n “ ψ np ˝ n ´ ź i “ E σ i Frob f p x p i , x p i q , where σ Frob represents the arithmetic Frobenius acting on the coefficients, and for any g P B we have E f p x , x qp g q : “ E f p x , x q ¨ g .One can easily check that ψ p ˝ E f p x , x q ` x P x x P y ˘ “ ÿ Q P M p ∆ f q e pQ ´ P p T q x Q x x Q y , where e pQ ´ P p T q is defined in (2.5). Theorem 2.8 (Dwork Trace Formula) . For every integer k ą , we have p q k ´ q ´ S ˚ f p k, π q “ Tr B { Z q rr π ss ` ψ nk ˘ . Proof.
This was proved by [LWei, Lemma 4.7]. (cid:3)
One can see [W] for a a thorough treatment of the universal Dwork trace formula.
Proposition 2.9 (Analytic trace formula) . The theorem above has an equivalent multi-plicative form: C ˚ f p T, s q “ det ` I ´ sψ n | B { Z q rr π ss ˘ . (2.7) Proof.
Also see [LWei, Theorem 4.8]. (cid:3)
Definition 2.10.
The normalized Newton polygon of C ˚ f p T, s q , denoted by NP p f, T q C , isthe lower convex hull of the set of points ! ´ i, v T p u i q n ¯ ) . Notation 2.11.
In this paper, we fix ∆ to be a triangle with vertices at p , q , P : “ p a , b q and P : “ p a , b q . Definition 2.12.
For each lattice point P in Z , assume that Q is the intersection of thelines OP and P P . Then we call w p P q : “ ÝÝÑ OP ÝÝÑ OQ the weight of P . ENERIC NEWTON POLYGON FOR EXPONENTIAL SUMS IN TWO VARIABLES WITH TRIANGULAR BASE7
The weight function w is linear, i.e. Any two points P and Q in Z ě satisfy(2.8) w p P ` Q q “ w p P q ` w p Q q . Equality (2.8) does not always hold for a general polytope.We shall frequently work with multisets, i.e. sets of possibly repeating elements. Theyare often marked by a superscript star to be distinguished from regular sets, e.g. S ‹ . Thedisjoint union of two multiset S ‹ and S is denoted by S ‹ Z S as a multiset. Definition 2.13.
Let S be a subset of M p ∆ q . Then we write S ‹ m (resp. S ‹8 ) for the unionof m (resp. countably infinite) copies of S as a multiset. Notation 2.14.
For any sets S ‹ and S ‹ in M p ∆ q ‹8 of the same cardinality, we denote byIso p S ‹ , S ‹ q the set of all bijections (as multisets) from S ‹ to S ‹ . When S ‹ “ S ‹ “ S ‹ , wedenote Iso p S ‹ q : “ Iso p S ‹ , S ‹ q . Definition 2.15.
For a bijection τ in Iso p S ‹ , S ‹ q , we define(2.9) h p S ‹ , S ‹ , τ q : “ ÿ P P S ‹ P w p pτ p P q ´ P q T . For any submultiset S ‹ of S ‹ , we write τ | S ‹ for the restriction of τ to S ‹ . Moreover,the minimum of h p S ‹ , S ‹ , τ q is denoted by(2.10) h p S ‹ , S ‹ q : “ min τ P Iso p S ‹ , S ‹ q p h p S ‹ , S ‹ , τ qq , where τ varies among all bijections from S ‹ to S ‹ . Definition 2.16.
We call a bijection from S ‹ to S ‹ minimal , if it reaches the minimum in(2.10). When S ‹ “ S ‹ , we call it a minimal permutation of S ‹ and abbreviate h p S ‹ , S ‹ , ‚q (resp. h p S ‹ , S ‹ q ) to h p S ‹ , ‚q (resp. h p S ‹ q ). Remark 2.17. If S ‹ i for i “ , M p ∆ q , we suppress the star from the notation. Definition 2.18.
The improved Hodge polygon of ∆, denoted by IHP p ∆ q , is the lowerconvex hull of the set of points ! ˆ ℓ, min S ‹ P M ℓ p n q h p S ‹ q n ˙ ) , where M ℓ p n q represents for the setconsisting of all multi-subsets of M p ∆ q ‹ n of cardinality nℓ , note M ℓ p q “ M ℓ .we shall prove in Proposition 3.16 later thatmin S ‹ P M ℓ p n q h p S ‹ q “ n ¨ min S P M ℓ h p S q , and hence the IHP p ∆ q is independent of n . In particular, IHP p ∆ q is the convex hull of theset of points ˆ ℓ, min S P M ℓ h p S q ˙ . Notation 2.19.
We denote by „ m m ¨ ¨ ¨ m ℓ ´ n n ¨ ¨ ¨ n ℓ ´ M the ℓ ˆ ℓ -submatrix formed by elements of a matrix M whose row indices belong to t m , m , . . . , m ℓ ´ u and whose column indices belong to t n , n , . . . , n ℓ ´ u .Put ∆ f “ ∆. Recall that we define T in Notation 1.2. Lemma 2.20.
We have e O p T q “ and v T p e Q p T qq ě P w p Q q T for all Q P M p ∆ q . RUFEI REN
Proof. (1) It follows from the definition of e O p T q in (2.5).(2) Let S p f q : “ t P P T | a P ‰ u “ t Q , Q , . . . , Q t u , where t a P u is the set of coefficients of f p x , x q and t is the cardinality of S p f q .Expanding each E p ˆ a Q i πx p Q i q x x p Q i q y q to be a power series in variables x and x , we get E f p x , x q “ t ź i “ E p ˆ a Q i πx p Q i q x x p Q i q y q “ ÿ ~j P Z t ě c ~j t ź i “ p ˆ a Q i πx p Q i q x x p Q i q y q j i , where t ˆ a P u is the set of coefficients of ˆ f p x , x q and c ~j belongs to Z q .It is not hard to get that e Q p T q “ ÿ! ~j ˇˇˇ t ř i “ j i Q i “ Q ) c ~j t ź i “ p ˆ a Q i π q j i “ ÿ! ~j ˇˇˇ t ř i “ j i Q i “ Q ) ´ c ~j t ź i “ p ˆ a Q i q j i π t ř i “ j i ¯ . Since w p Q i q ď Q i P S p f q , then for each ~j such that t ř i “ j i Q i “ Q , we have v T ´ c ~j t ź i “ p ˆ a Q i q j i π t ř i “ j i ¯ “ t ÿ i “ j i ě t ÿ i “ j i w p Q i q “ w p Q q , where T “ E p π q ´
1. Therefore, we immediately get that v T p e Q p T qq ě w p Q q . Since v T p e Q p T qq is an integer, we have v T p e Q p T qq ě P w p Q q T . (cid:3) Notation 2.21.
We label points in M p ∆ q such that M p ∆ q “ t P , P , . . . u . Proposition 2.22.
The normalized Newton polygon NP p f, T q C lies above IHP p ∆ f q .Proof. We write N for the standard matrix of ψ p ˝ E f corresponding to the basis t x p P q x x p P q y , x p P q x x p P q y , ¨ ¨ ¨ u of the Banach space B . By [RWXY, Corollary 3.9], we know that the standard matrix of ψ n corresponding to the same basis is equal to σ n ´ p N q ˝ σ n ´ p N q ˝ ¨ ¨ ¨ ˝ N. Then by [RWXY,Proposition 4.6], for every ℓ P N we have u ℓ p T q “ ÿ t P m , ,P m , ,...,P m ,ℓ ´ uP M ℓ t P m , ,P m , ,...,P m ,ℓ ´ uP M ℓ ... t P mn ´ , ,P mn ´ , ,...,P mn ´ ,ℓ ´ uP M ℓ det ˆ n ´ ź j “ „ m j ` , m j ` , ¨ ¨ ¨ m j ` ,ℓ ´ m j, m j, ¨ ¨ ¨ m j,ℓ ´ σ j Frob p N q ˙ , (2.11)where m n,i : “ m ,i for each 0 ď i ď ℓ ´ ENERIC NEWTON POLYGON FOR EXPONENTIAL SUMS IN TWO VARIABLES WITH TRIANGULAR BASE9
Then for S j “ t P m j, , P m j, , . . . , P m j,ℓ ´ u , we have v T ´ det ` n ´ ź j “ „ m j ` , m j ` , ¨ ¨ ¨ m j ` ,ℓ ´ m j, m j, ¨ ¨ ¨ m j,ℓ ´ σ j Frob p N q ˘¯ “ v T ´ n ´ ź j “ ÿ τ j “ Iso p S j ` , S j q sgn p τ j q ź P P S j ` σ j Frob p e pτ j p P q´ P q ¯ ě n ´ ÿ j “ h p S j ` , S j qě h p n ´ ě j “ S ‹ j q , (2.12)where S n “ S . Therefore, it is easily seen that (cid:3) (2.13) v T p u ℓ p T qq ě min S ‹ P M ℓ p n q h p S ‹ q . Improved Hodge polygon for a triangle ∆Recall that ∆ is a triangle with vertices p , q , P : “ p a , b q and P : “ p a , b q and aswe defined in Notation 1.2, x k “ x k p ∆ q (resp. x k “ x k p ∆ q ) is the number of lattice pointsin M p ∆ q whose weight is strictly less than (resp. less than or equal to) k . For the rest ofthis paper, we restrict p to be a prime satisfying p ∤ a b ´ a b and p ą p b a ´ b a q gcd p a ´ a , b ´ b q ` . The goal of this section is to show that if NP p f, χ q C (See Definition 1.1) and IHP p ∆ q coincideat a certain point, then they will coincide at infinitely many points. More precisely, we havethe following. Theorem 3.1.
Let f p x , x q be a two-variable polynomial with convex hull ∆ . Supposethat there exists a nontrivial finite character χ : Z p Ñ C ˆ p and an integer k ą such that NP p f, χ q C coincides with IHP p ∆ q at x “ x k . Then for any finite character χ and positiveinteger k , IHP p ∆ q and NP p f, χ q C coincide at x k ` i k for all ď i k ď x k ´ x k .Moreover, the leading coefficients u x k ,nh p T k q p resp . u x k ,nh p T k q q of the x k -th p resp . x k -th q terms of the characteristic power series (see (2.2) for precise definition) are Z p -units. The proof of this theorem will occupy the rest of this section.
Notation 3.2.
Without loss of generality, we can assume that a b ´ a b ą
0. We call theparallelogram with vertices O, P , P and P ` P (excluding the upper and right sides) the fundamental parallelogram of ∆, and denote it by (cid:3) ∆ , i.e. the shadow region in Figure 1.We put (cid:3) Int∆ to be the set of lattice points in (cid:3) ∆ , which contains a b ´ a b points.Let Λ be the lattice generated by P and P . For each point P in Cone p ∆ q , we write P %for its residue in (cid:3) ∆ modulo Λ. Lemma 3.3.
The map η : (cid:3) Int∆ Ñ (cid:3) Int∆ P ÞÑ p pP q % is a permutation. xy P : “ p a , b q P : “ p a , b q Figure 1.
The fundamental parallelogram.
Proof. since p and a b ´ a b are coprime, there exist integers p and n such that pp ´ “p a b ´ a b q n . For a point P “ p P x , P y q , we have p pp ´ q P “ n p b P x ´ a P y q P ` n p´ b P x ` a P y q P P Λ . It implies that composite (cid:3)
Int∆ P ÞÑ pP % ÝÝÝÝÝÑ (cid:3)
Int∆ P ÞÑ p P % ÝÝÝÝÝÝÑ (cid:3)
Int∆ is the identity map. (cid:3)
The key to proving Theorem 3.1 is to gain precise control of the improved Hodgepolygon in Proposition 2.22. In [W], Wan made use of the following coarser estimate of thisHodge polygon, for each multiset S ‹ of M p ∆ q ‹8 we have h p S ‹ q ě h p S ‹ q : “ p p ´ q ÿ P P S ‹ w p P q . It is however important for our method to understand the difference between h p S ‹ q and h p S ‹ q (or more generally h p S ‹ , τ q .For each r P R , we put R p r q : “ r r s ´ r ; and for a permutation τ of S ‹ , we set(3.1) U ‹ p S ‹ , τ q : “ ! R p w p pτ p P q ´ P qq | P P S ‹ ) ‹ , (3.2) U ‹ p S ‹ , τ q ď a : “ ! r P U ‹ p S ‹ , τ q | r ď a ) ‹ and U ‹ p S ‹ , τ q ă a : “ ! r P U ‹ p S ‹ , τ q | r ă a ) ‹ to measure the distance of these weights to the next integer values.Write(3.3) h p S ‹ , τ q : “ ÿ r P U ‹ p S ‹ ,τ q r and h p S ‹ q “ min τ P Iso p S ‹ q ! h p S ‹ , τ q ) where Iso p S ‹ q is set consisting all permutations of S ‹ (see Notation 2.14). ENERIC NEWTON POLYGON FOR EXPONENTIAL SUMS IN TWO VARIABLES WITH TRIANGULAR BASE11
Lemma 3.4.
We have (3.4) h p S ‹ , τ q “ h p S ‹ q ` h p S ‹ , τ q and h p S ‹ q “ h p S ‹ q ` h p S ‹ q . Proof.
By the definition of h p S ‹ , τ q in (2.9), we have h p S ‹ , τ q “ ÿ P P S ‹ P w ` pτ p P q ´ P ˘T “ ÿ P P S ‹ w ` pτ p P q ´ P ˘ ` ÿ P P S ‹ !P w ` pτ p P q ´ P ˘T ´ w ` pτ p P q ´ P ˘) “ ÿ P P S ‹ pw ` τ p P q ˘ ´ ÿ P P S ‹ w p P q ` ÿ P P S ‹ R ` w ` pτ p P q ´ P ˘˘ “ p p ´ q ÿ P P S ‹ w p P q ` h p S ‹ , τ q“ h p S ‹ q ` h p S ‹ , τ q . Taking the minimal over all τ P Iso p S ‹ q implies h p S ‹ q “ h p S ‹ q ` h p S ‹ q . (cid:3) Lemma 3.5.
For any two permutations τ , τ of S ‹ , if U ‹ p S ‹ , τ q “ U ‹ p S ‹ , τ q , then h p S ‹ , τ q “ h p S ‹ , τ q and h p S ‹ , τ q “ h p S ‹ , τ q . Proof.
This lemma follows from the definition of h and Lemma 3.4. (cid:3) Lemma 3.6.
Suppose S ‹ , S ‹ Ă M p ∆ q ‹8 satisfy (3.5) ! w p P q | P P S ‹ ) ‹ “ ! w p P q | P P S ‹ ) ‹ as multisets. Then we have h p S ‹ q “ h p S ‹ q and h p S ‹ q “ h p S ‹ q . (3.6) Proof.
Let ξ : S ‹ Ñ S ‹ be a bijection such that the induced map of weights from ! w p P q | P P S ‹ ) ‹ to ! w p P q | P P S ‹ ) ‹ realizes the equality (3.5). Then ξ induces a bijection from Iso p S ‹ q to Iso p S ‹ q . Moreover, we have h p S ‹ , τ q “ h p S ‹ , ξ ´ τ ξ q . We immediately get the first equality in (3.6). Combining it with Lemma 3.4, we caneasily check the second equality. (cid:3)
We prove Theorem 3.1 in two steps.
Step I.
Recall that a permutation τ of S ‹ that achieves the minimum of h p S ‹ q or equiv-alently the minimum of h p S ‹ q is called minimal. The first core result in this section isProposition 3.10. It shows for a given S ‹ Ă M p ∆ q ‹8 how to construct an explicit minimalpermutation τ of S ‹ .First, we construct a minimal permutation inductively for a general subset S ‹ of M p ∆ q ‹8 as follows. Construction 3.7.
We choose a pair of points p P , Q q in S ‹ ˆ S ‹ such that R p w p pQ ´ P qq reaches the minimum among all pairs p P, Q q in S ‹ ˆ S ‹ . Define τ p P q : “ Q .Then we take out P from the first S ‹ and Q from the second S ‹ . We pick another pairof points p P , Q q from S ‹ zt P u ˆ S ‹ zt Q u such that R p w p pQ ´ P qq reaches the minimumamong all pairs p P, Q q in S ‹ zt P u ˆ S ‹ zt Q u , and define τ p P q : “ Q . Similarly, we picka “minimal” pair of points p P , Q q from S ‹ zt P , P u ˆ S ‹ zt Q , Q u . Define τ p P q : “ Q .Iterating this process defines τ . Lemma 3.8.
Let τ be a minimal permutation constructed in Construction 3.7, and let τ be an arbitrary permutation of S ‹ . Suppose (3.7) U ‹ p S ‹ , τ q ă r “ U ‹ p S ‹ , τ q ă r for some rational number r . Then by taking finite times of the following operations:(1) swapping the images of two points of the same weight; and(2) swapping the preimages of two points of the same weight,we obtain a permultation τ from τ such that τ p P q “ τ p P q for all P P S ‹ satisfying R ` w p pτ p P q ´ P q ˘ ď r and U ‹ p S ‹ , τ q “ U ‹ p S ‹ , τ q .In particular, U ‹ p S ‹ , τ q defined in (3.1) is independent of the choice made in Construc-tion 3.7.Proof. Assuming (3.7) holds for every r ď r . Then we induce a permutation r τ from τ bytaking finite times of operations (1) and (2) such that r τ p P q “ τ p P q for all P P S ‹ satisfying R ` w p p r τ p P q ´ P q ˘ ă r and U ‹ p S ‹ , r τ q “ U ‹ p S ‹ , τ q . It is not hard to check that it is enoughfor us to show this lemma works for r τ . In other words, we can assume that(3.8) τ p P q “ τ p P q for each P P S ‹ satisfying R ` w p pτ p P q ´ P q ˘ ă r . Suppose that τ p P q “ τ p P q for all P P S ‹ satisfying R ` w p pτ p P q ´ P q ˘ ď r . Then weare done. Otherwise there exists a point P in S ‹ such that R ` w p pτ p P q ´ P q “ r , but τ p P q ‰ τ p P q . By Construction 3.7, it is not hard to see that at least one of the followingcases will happen:(a) R ` w p pτ p P q ´ P ˘ “ r .(b) There exists a point P in S ‹ such that w p P q “ w p P q and τ p P q “ τ p P q .When case (a) happens, we put Q “ τ p P q and define τ : S ‹ Ñ S ‹ to be the samepermutation as τ except we swap the preimages of Q and τ p P q .Otherwise, we define τ to be the same permutation as τ except we swap the images of P and P , where P is defined in (b).By (3.8), we know that either R ` w p pQ ´ τ ´ p Q qq ˘ ě r or R ` w p pτ p P q ´ P q ˘ ě r , which implies that τ also satisfies (3.8) and ! P P U ‹ p S ‹ , τ q “ r ˇˇ τ p P q “ τ p P q ) ě ! P P U ‹ p S ‹ , τ q “ r ˇˇ τ p P q “ τ p P q ) ` . If τ does not satisfy the wanted property, then we run the same argument with τ inplace of τ to obtain another permutation τ . Iterating this process eventually gives usa permutation τ of S ‹ . It is easy to check that τ p P q “ τ p P q for all P P S ‹ satisfying R ` w p pτ p P q ´ P q ˘ ď r . Since both operations (1), (2) do not change the set U ‹ p S ‹ , τ q , wehave U ‹ p S ‹ , τ q “ U ‹ p S ‹ , τ q . We now prove the last statement of the lemma. Let τ and τ be two permuta-tions constructed in Construction 3.7 by different choices of pairs of points. Suppose that U ‹ p S ‹ , τ q ‰ U ‹ p S ‹ , τ q . Then there is a rational number r , such that U ‹ p S ‹ , τ q ă r “ U ‹ p S ‹ , τ q ă r and U ‹ p S ‹ , τ q ď r ‰ U ‹ p S ‹ , τ q ď r . Without loss of generality, we assume that U ‹ p S ‹ , τ q ď r ( U ‹ p S ‹ , τ q ď r . From the argument in this lemma above, we know that there exists a permutation τ of S ‹ such that τ p P q “ τ p P q for all P P S ‹ satisfying R ` w p pτ p P q ´ P q ˘ ď r and U ‹ p S ‹ , τ q “ U ‹ p S ‹ , τ q . ENERIC NEWTON POLYGON FOR EXPONENTIAL SUMS IN TWO VARIABLES WITH TRIANGULAR BASE13
Therefore, we have U ‹ p S ‹ , τ q ď r “ U ‹ p S ‹ , τ q ď r “ U ‹ p S ‹ , τ q ď r , a contradiction. (cid:3) Corollary 3.9.
For τ and τ as in the last lemma, we have h p S ‹ , τ q “ h p S ‹ , τ q .Proof. Since U ‹ p S ‹ , τ q “ U ‹ p S ‹ , τ q , then it follows directly from the definition of h in(3.3). (cid:3) Proposition 3.10.
The permutation τ in Construction 3.7 is a minimal permutation of S ‹ , i.e. h p S ‹ , τ q “ h p S ‹ q . Proof.
By Lemma 3.4, it is enough to prove that τ minimizes h p S ‹ , ‚q among all permuta-tions of S ‹ .Assume that τ is a minimal permutation of S ‹ . If U ‹ p S ‹ , τ q “ U ‹ p S ‹ , τ q , we are doneby Lemma 3.5. Otherwise we shall construct below a finite sequence of permutations p τ “ τ, τ , . . . , τ m q satisfying(1) for each 0 ď i ď m , we have h p S ‹ , τ i q “ h p S ‹ q . (2) U ‹ p S ‹ , τ m q “ U ‹ p S ‹ , τ q . Combining Property (2) with Lemma 3.5, we would deduce h p S ‹ , τ m q “ h p S ‹ , τ q , which completes the proof.Now we come to the inductive construction of the sequence p τ “ τ, τ , . . . , τ m q . Wetake induction on i . First, we put τ “ τ , where τ is a minimal permutation of S . Hence,it satisfies h p S ‹ , τ q “ h p S ‹ q . Suppose that we have defined τ i . If U ‹ p S ‹ , τ i q “ U ‹ p S ‹ , τ q , we terminate this induction.Otherwise let t i be the smallest number in the multiset U ‹ p S ‹ , τ qz U ‹ p S ‹ , τ i q . Let τ i : S ‹ Ñ S ‹ be the permutation constructed from τ i in Lemma 3.8. It is not hard to check that ‚ h p S ‹ , τ i q “ h p S ‹ , τ i q . ‚ τ i p P q “ τ p P q holds for each P P S ‹ satisfying w p P ´ τ i p P qq ď t i . ‚ U ‹ p S ‹ , τ q ď t i ) U ‹ p S ‹ , τ i q ď t i .Therefore, there exists a point P i in S ‹ with Q i “ τ p P i q , τ i p P i q “ Q i and τ p P i q “ Q i such that ‚ R p w p pQ i ´ P i qq “ t i , ‚ R p w p pQ i ´ P i qq ě t i , ‚ R p w p pQ i ´ P i qq ą t i , and ‚ R p w p pQ i ´ P i qq ą t i .We define τ i ` to be the same permutation as τ i except we swap the images of P i and P i .This process gives us a sequence with elements from Iso p S ‹ q , whose length, say m , is lessthan or equal to p S ‹ q .Then we check h p S ‹ , τ i ` q “ h p S ‹ , τ i q . From the induction we know that h p S ‹ , τ i q “ h p S ‹ q . Hence, we ha h p S ‹ , τ i ` q ě h p S ‹ , τ i q . On the other hand, from the definition of h , we have h p S ‹ , τ i ` q ´ h p S ‹ , τ i q “ h p S ‹ , τ i ` q ´ h p S ‹ , τ i q“ R ` w p pτ i ` p P i q ´ P i q ˘ ` R p w ` pτ i ` p P i q ´ P i q ˘ ´ R ` w p pτ i p P i q ´ P i q ˘ ´ R ` w p pτ i p P i q ´ P i q ˘ ă t i ` R ` w p pτ i ` p P i q ´ P i q ˘ ´ t i ´ R ` w p pτ i p P i q ´ P i q ˘ ď ´ R ` w p pτ i p P i q ` P i q ˘ ď . (3.9)From the linearity of w , we have w ` pτ i ` p P i q ´ P i ˘ ` w ` pτ i ` p P i q ´ P i ˘ “ w ` pτ i p P i q ´ P i ˘ ` w ` pτ i p P i q ´ P i ˘ . It implies that h p S ‹ , τ i ` q ´ h p S ‹ , τ i q is an integer. Combining it with (3.9), we have h p S ‹ , τ i ` q ď h p S ‹ , τ i q . Combining these inequalities, we obtain(3.10) h p S ‹ , τ i ` q “ h p S ‹ , τ i q “ h p S ‹ q . Finally, from the termination condition of this induction, we know that U ‹ p S ‹ , τ m q “ U ‹ p S ‹ , τ q . (cid:3) Remark 3.11.
Notice that this result crucially depends on the linearity of w as in (2.8).For general polytopes, we have a similar definition for the weight of a point, however, it isno longer linear. We will address this case in a future work.For any subset S of M p ∆ q , we write S ` P for the shift of S by P . Corollary 3.12.
Let S be a subset of M p ∆ q , and let Q , Q , . . . , Q k be points of M p ∆ q with integer weights. Put S ‹ “ S ‹ Z k ě i “ p (cid:3) Int∆ ` Q i q ‹ , then h p S ‹ q “ h p S ‹ q .Proof. Recall that for each point P P M p ∆ q , we denote by P % its residue in (cid:3) ∆ . Lemma 3.3defines a permutation of (cid:3) Int∆ such that η p P q “ p pP q %. We write η i : (cid:3) Int∆ ` Q i Ñ (cid:3) Int∆ ` Q i P ÞÑ η ´ p P ´ Q i q ` Q i . It is easy to check that R p pη i p P q ´ P q “ P in (cid:3) Int∆ ` Q i . Then we apply Construction 3.7 to S and get a minimal permutation τ S of S . It isnot hard to see that τ S , η , . . . , η k together construct a permutation τ of S ‹ which agreeswith Construction 3.7. Therefore, we have h p S ‹ q “ h p S , τ S q ` k ÿ i “ h p (cid:3) Int∆ ` Q i , η i q“ h p S , τ S q“ h p S q . (cid:3) Corollary 3.13.
Let S be a subset of M p ∆ q , and let Q , Q , . . . , Q k be points of M p ∆ q ofinteger weights. Put S ‹ “ k Ţ i “ p S ` Q i q . Then we have h p S ‹ q “ kh p S q . ENERIC NEWTON POLYGON FOR EXPONENTIAL SUMS IN TWO VARIABLES WITH TRIANGULAR BASE15
Proof.
Let τ be the minimal permutation of S constructed in Construction 3.7. For each1 ď i ď k , put τ i : S ` Q i Ñ S ` Q i P ÞÑ τ p P ´ Q i q ` Q i . It is not hard to see that τ , τ , . . . , τ k together induce a permutation τ of S ‹ which agreeswith Construction 3.7 and for each 1 ď i ď k we have h p S ` Q i , τ i q “ h p S , τ q “ h p S q . Therefore, h p S ‹ q “ h p S ‹ , τ q“ k ÿ i “ h p S ` Q i , τ i q“ kh p S q . (cid:3) Lemma 3.14. we have h p T k q “ kh p T q . Proof.
We can decompose T k into a disjoint union of sets as follows:(3.11) T k “ k ´ ğ i “ p T ` p k ´ ´ i q P ` i P q \ k ´ ğ i “ i ğ j “ p (cid:3) Int∆ ` i P ` j P q . Applying Corollary 3.12 and 3.13 to (3.11), we complete the proof of this lemma. (cid:3)
Lemma 3.15.
Let l ` (resp. l ` ) represent the number of lattice points on closedsegment O P (resp. O P ). For each k ą , we have p q x k “ k x ` k p k ´ q p a b ´ a b q . p q h p T k q “ p p ´ q k ´ ř i “ “ p a b ´ a b qp i ` q´ p l ` l q ‰ p i ` q` k “ h p T q`p p ´ qp k ´ q x ‰ . Proof. (1) Since there are totally a b ´ a b points in (cid:3) Int∆ , (1) follows directly from (3.11)above.(2) A tautological computation shows that h p T k q “ k “ h p T q ` p p ´ qp k ´ q x ‰ ` p p ´ q k ´ ÿ i “ “ a b ´ a b ´ p l ` l q ´ ` i p a b ´ a b q ‰ p i ` q . For each k ě
1, we know from Lemma 3.14 that h p T k q “ h p T k q ` h p T k q“ k “ h p T q ` p p ´ qp k ´ q x ‰ ` kh p T q`p p ´ q k ´ ÿ i “ “ p a b ´ a b qp i ` q ´ p l ` l q ´ ‰ p i ` q . Combining it with h p T q “ h p T q ` h p T q , we complete the proof. (cid:3) Step II.
The following proposition is the second core result of studying the improved Hodgepolygon IHP p ∆ q at x “ x k . Proposition 3.16.
We have (3.12) min S ‹ P M ℓ p n q h p S ‹ q “ n ¨ min S P M ℓ h p S q . Therefore, we give a simpler expression of
IHP p ∆ q as the lower convex hull of the set ofpoints ˆ ℓ, min S P M ℓ h p S q ˙ , which is independent of n . We will prove this proposition after two lemmas.
Lemma 3.17.
For any two distinct points
P, Q P M p ∆ q if w p P q ‰ w p Q q , then | w p P ´ Q q| ě gcd p a ´ a , b ´ b q b a ´ b a . Proof.
It is easily known that w pp , qq “ a ´ a b a ´ b a and w pp , qq “ b ´ b b a ´ b a . Since each point in Z is a linear combination of p , q and p , q , this lemma follows fromthe linearity of w (cid:3) Lemma 3.18.
Let M be a subset of M p ∆ q ‹8 and let S ℓ be the set consisting of all subsetsof M of cardinality ℓ . Choose a multiset S ‹ min P S ℓ such that ÿ P P S ‹ min w p P q “ min S ‹ P S ℓ ´ ÿ P P S ‹ w p P q ¯ . Suppose that p ą p b a ´ b a q gcd p a ´ a ,b ´ b q ` . Then if r S ‹ P S ℓ satisfy (3.13) ÿ P P r S ‹ w p P q “ ÿ P P S ‹ min w p P q p resp. ÿ P P S ‹ min w p P q ą min S ‹ P S ℓ ´ ÿ P P S ‹ w p P q ¯ q , we have (3.14) h p r S ‹ q “ h p S ‹ min q p resp. h p r S ‹ q ą h p S ‹ min qq . In other words, the minimal h p S ‹ q is achieved by exactly those S ‹ for which the sum ofweights of S ‹ is minimal.Proof. For a subset r S ‹ P S ℓ , if ! w p P q | P P r S ‹ ) ‹ “ ! w p P q | P P S ‹ min ) ‹ , then byLemma 3.6, we know h p r S ‹ q “ h p S ‹ min q . Otherwise we construct a sequence p S ‹ “ r S ‹ , S ‹ , . . . , S ‹ m q in S ℓ of length less than or equalto ℓ such that ‚ for any 0 ď i ď m ´ h p S i q ą h p S i ` q , and ‚ ! w p P q | P P S ‹ m ) ‹ “ ! w p P q | P P S ‹ min ) ‹ .The following is the construction:Assume that we have constructed S ‹ i . If ! w p P q | P P S ‹ i ) ‹ “ ! w p P q | P P S ‹ min ) ‹ , thenwe stop. Otherwise there exists a rational number t i such that ! w p P q ă t i | P P S ‹ i ) ‹ “ ! w p P q ă t i | P P S ‹ min ) ‹ ENERIC NEWTON POLYGON FOR EXPONENTIAL SUMS IN TWO VARIABLES WITH TRIANGULAR BASE17 and ! w p P q ď t i | P P S ‹ i ) ‹ ( ! w p P q ď t i | P P S ‹ min ) ‹ . Then there exist points P i P S ‹ min ´ S ‹ i and Q i P S ‹ i ´ S ‹ min such that w p P i q “ t i ă w p Q i q . Put S ‹ i ` to be the set induced from S ‹ i by simply substituting Q i with P i . Then we get asequence p S ‹ “ S ‹ , S ‹ , . . . , S ‹ m q in S ℓ of length, say m , less than or equal to ℓ , which satisfiesthe following conditions.(1) S ‹ “ r S ‹ .(2) ! w p P q | P P S ‹ m ) ‹ “ ! w p P q | P P S ‹ min ) ‹ .(3) For each 0 ď i ď m ´
1, we have S ‹ i ` ´ S ‹ i “ t P i u and S ‹ i ´ S ‹ i ` “ t Q i u .(4) The points above satisfy w p P i q ă w p Q i q .From Lemma 3.6, we know that (2) implies that h p S ‹ m q “ h p S ‹ min q . Therefore, it is enough to show that h p S ‹ i q ą h p S ‹ i ` q holds for each 0 ď i ď m ´ ď i ď m ´
1, let τ i P Iso p S ‹ i q be a minimal permutation of S ‹ i , i.e. h p S ‹ i , τ i q “ h p S ‹ i q .We denote by τ i ` a permutation of S ‹ i ` induced from τ i by simply substituting Q i with P i , i.e. τ i ` p P q “ $’&’% τ i p Q i q if P “ P i P i if τ i p P q “ Q i τ i p P q otherwise.Now we claim that h p S ‹ i , τ i q ´ h p S ‹ i ` , τ i ` q ą . We need to consider the following twocases.
Case 1:
When τ i p Q i q “ Q i , we have h p S ‹ i , τ i q ´ h p S ‹ i ` , τ i ` q“ P pw p Q i q ´ w p Q i q T ´ P pw p P i q ´ w p P i q T “ P p p ´ q w p Q i q T ´ P p p ´ q w p P i q T ě P p p ´ qp w p Q i q ´ w p P i qq T ´ ě Q p p ´ q gcd p a ´ a , b ´ b q b a ´ b a U ´ ą . Case 2:
When τ i p Q i q ‰ Q i , let Q i “ τ ´ i p Q i q . Then we have h p S ‹ i , τ i q ´ h p S ‹ i ` , τ i ` q“ P pw ` τ i p Q i q ˘ ´ w p Q i q T ´ P pw ` τ i p Q i q ˘ ´ w p P i q T ` P pw p Q i q ´ w p Q i q T ´ P pw p P i q ´ w p Q i q T ě ´ P w p Q i q ´ w p P i q T ` P pw p Q i q ´ pw p P i q T ´ ě P p p ´ q ` w p Q i q ´ w p P i q ˘T ´ ě Q p p ´ q gcd p a ´ a , b ´ b q b a ´ b a U ´ ą . Then this lemma follows from the following strict inequality h p S ‹ i q “ h p S ‹ i , τ i q ą h p S ‹ i ` , τ i ` q ě h p S ‹ i ` q . (cid:3) Proof of Proposition 3.16.
First, we fix a subset S P M ℓ such that h p S q “ min S P M ℓ ` h p S q ˘ . Let r S ‹ be an arbitrary submultiset in M ℓ p n q such that h p r S ‹ q “ min S ‹ P M ℓ p n q ` h p S ‹ q ˘ . By Lemma 3.18, we know that ÿ P P S w p P q “ min S P M ℓ ´ ÿ P P S w p P q ¯ and ÿ P P r S ‹ w p P q “ min S ‹ P M ℓ p n q ´ ÿ P P S ‹ w p P q ¯ . It is not hard to see that ÿ P Pp S q n w p P q “ ÿ P P r S ‹ w p P q . Therefore, by Lemma 3.18 again, we have h p r S ‹ q “ h pp S q n q . By Corollary 3.13, we knowthat h pp S q n q “ nh p S q . Combining these equalities above gives us (3.12). (cid:3)
Definition 3.19.
For any subset S of M p ∆ q , we writedet p S q f “ ÿ τ P Iso p S q sgn p τ q ź P P S e pτ p P q´ P . Then as a corollary of Proposition 3.16, we get the following.
Proposition 3.20.
We have v T ´ n ´ ź j “ σ j Frob p det p T k q f q ´ u x k ,nh p T k q T nh p T k q ¯ ě nh p T k q ` , where u x k ,nh p T k q is defined in (2.2) .Proof. By Lemma 3.18, we know that T ‹ kn is the only element in M x k p n q which makes(2.13) an equality. Therefore, we have v T ´ u x k ,nh p T k q T nh p T k q ´ n ´ ź j “ ` ÿ τ j “ Iso p T k q sgn p τ j q ź P P T k σ j Frob p e pτ j p P q´ P q ˘¯ ě nh p T k q ` . Then this proposition follows directly from checking the definition of det p T k q f . (cid:3) Notation 3.21.
For each character χ : Z p Ñ C ˆ p of conductor p , from [LWei, Theorem 1.4], L ˚ f p χ, s q ´ is a polynomial of degree a b ´ a b . We denote its q -adic Newton slopes by ´ α , α , . . . , α a b ´ a b ¯ in a non-descending order and put Σ p χ q : “ x ř j “ α j . Lemma 3.22.
For each character χ : Z p Ñ C ˆ p of conductor p , the normalized Newtonpolygon of C ˚ f p χ, s qq , i.e. NP p χ, s q C is not above points ´ x k , “ p a b ´ a b q k ´ ÿ i “ i ´ p l ` l q k ´ ÿ i “ i ` x p k ´ q k ` k Σ p χ q ‰ p p ´ q ¯ for all integers k ě . ENERIC NEWTON POLYGON FOR EXPONENTIAL SUMS IN TWO VARIABLES WITH TRIANGULAR BASE19
Proof.
First recall that n “ r F q : F p s is the degree of the coefficient field of f (see section2). It is well known that the roots of L ˚ f p χ, s q ´ are Weil numbers of weight 0, n , or 2 n .We put them into three classes according to the Weil weights:Weil weight the number of roots of L ˚ f p χ, s q ´ n l ` l n a b ´ a b ´ l ´ l ´ α i ’s are the q -adic Newton slopes of L ˚ f p χ, s q ´ , we know easily that they belong to r , q . Moreover, an algebraic number, say z , and its complex dual z are both roots of L ˚ f p χ, s q ´ or both not. Suppose that they are roots of L ˚ f p χ, s q ´ and z as Weil weight t .Then we have v p p z q` v p p z q “ t . Therefore, the sum of all q -adic Newton slopes of L ˚ f p χ, s q ´ can be computed as follows: p a b ´ a b ´ l ´ l ´ q ˆ ` p l ` l q ˆ ` ˆ . “ a b ´ a b ´ p l ` l q ´ . (3.15)On the other hand, from (1.1), i.e.(3.16) C ˚ f p χ, s q “ ź j “ L ˚ f p χ, q j s q ´p j ` q , we know that(3.17) ` k ´ ě i “ t α ` i, α ` i, . . . , α a b ´ a b ` i u ‹ ¯ i ` Z ´ t α ` k ´ , α ` k ´ , . . . , α x ` k ´ u ‹ ¯ k is included in the set of q -adic Newton slopes of C ˚ f p χ, s q as multisets and its cardinalityis equal to x k . Since elements in this set are not necessary to be the smallest x k Newtonslopes of C ˚ f p χ, s q , then the height of NP p χ, s q C at x “ x k is not above the sum p p ´ q ” k ´ ÿ i “ i a b ´ a b ÿ j “ p i ´ ` α j q ` k x ÿ j “ p k ´ ` α j q ı “p p ´ q ” p a b ´ a b q k ´ ÿ i “ i ´ p p l ` l q ` q k ´ ÿ i “ i ` x p k ´ q k ` k Σ p χ q ı , (3.18)where p ´ p χ, s q C . (cid:3) Lemma 3.23.
For each k ě , p q both p x k , h p T k qq and p x k , h p T k qq are vertices of IHP p ∆ q , and p q the segment with endpoints p x k , h p T k qq and p x k , h p T k qq is contained in IHP p ∆ q .Proof. (1) Suppose the lemma were false. Then there exists an integer k and a segment inIHP p ∆ q , say P P , such that P : “ p x k , h p T k qq is either a point strictly above P P or aninterior point on P P . From Proposition 3.16, we know that P “ ´ x k ´ i , min S P M x k ´ i p h p S qq ¯ and P “ ´ x k ` i , min S P M x k ` i p h p S qq ¯ for some positive integers i and i .Put S to be an element of M x k ´ i such that(3.19) ÿ P P S w p P q “ min S P M x k ´ i ´ ÿ P P S w p P q ¯ . By Lemma 3.18, we get(3.20) h p S q “ min S P M x k ´ i ` h p S q ˘ . It is easy to know that S is a subset of T k . We denote its complement in T k by S , whichis of cardinality i . By Lemma 3.17, we know that each point P in T k satisfies w p P q ď k ´ gcd p a ´ a , b ´ b q a b ´ a b . Combining it with h p S q ď S “ i , we have h p T k q ď h p S q ` h p S q “ h p S q ` h p S q ` h p S qď h p S q ` i p p ´ q „ k ´ gcd p a ´ a , b ´ b q a b ´ a b ` i . It simply implies that the slope of P P is less than or equal to p p ´ q „ k ´ gcd p a ´ a , b ´ b q a b ´ a b ` . On the other hand, by a similar argument, we choose an element S from M x k ` i suchthat ÿ P P S w p P q “ min S P M x k ` i ` ÿ P P S w p P q ˘ . By Lemma 3.18 again, we have(3.21) h p S q “ min S P M x k ` i ` h p S q ˘ . We also easily know that T k is included in S . Let τ be a minimal permutation of S . Weshall construct below a finite sequence of permutations of S , denoted by p τ “ τ, τ , . . . , τ m q ,satisfying(a) the length m of this sequence is less than or equal to i ,(b) h p τ i ` q ď h p τ i q `
1, and(c) τ m fixes every point in S z T k .Put τ “ τ . Assume that we have τ i already. If it fixes each point in S z T k , then weare done. Otherwise put P i to be a point in S z T k such that τ i p P i q ‰ P i . Then we define τ i ` the same permutation as τ i except we swap images of P i and τ ´ i p P i q . Iterating thisprocess gives us a sequence of permutations of S . If τ m is the last element in this sequence,we know that it fixes each point in S z T k , namely, τ m p P q “ P for each P P S z T k . Since there are at most i points in S whose images are changed by these modifications,we know that m ď i . Put Q i “ τ i p P i q and P i “ τ ´ i p P i q . We compute h p τ i q ´ h p τ i ` q“ r w p pQ i ´ P i q s ` r w p pP i ´ P i q s ´ r w p pQ i ´ P i q s ´ r w p pP i ´ P i q s ě S satisfies condi-tions (a)-(c) above. Moreover, we have h p S , τ q ě h p S , τ m q ´ i . ENERIC NEWTON POLYGON FOR EXPONENTIAL SUMS IN TWO VARIABLES WITH TRIANGULAR BASE21 As τ is minimal, we get h p S q “ h p S q ` h p S , τ qě h p S q ` h p S , τ m q ´ i “ h p S , τ m q ´ i . (3.22)Since the restriction of τ m on T k is a permutation of T k and w p P q ě k for any point P in S z T k , we have h p S , τ m q “ h ´ T k , τ m ˇˇ T k ¯ ` h ´ S z T k , τ m ˇˇ S z T k ¯ ě h p T k q ` h p S z T k qě h p T k q ` i p p ´ q k. (3.23)By (3.22) and (3.23), the slope of P P is greater than or equal to k p p ´ q ´ . Under the assumption p ą p a b ´ a b q gcd p a ´ a ,b ´ b q ` P P is strictly greater than the slope of P P , which is a contradiction.By a similar argument, we know that p x k , h p T k qq is also a vertex of IHP p ∆ q .(2) Let 0 ď i ď x k ´ x k . By Lemma 3.18, there exists T k Ă S i Ă T k such that h p S i q “ min S P M x k ` i p h p S qq . Since all points in T k z T k have integer weight k , the we have h p S i q “ h p T k q ` ik p p ´ q , which implies (2) immediately. (cid:3) Lemma 3.24.
Let χ be a nontrivial finite character. Suppose that NP p f, χ q C coincideswith IHP p ∆ q at point ` x k , h p T k q ˘ for a positive integer k . Then p q ` x k , h p T k q ˘ is also a vertex of NP p f, χ q C , and p q u x k ,nh p T k q is a Z p -unit.Proof. (1) This follows from the fact that the normalized Newton polygon NP p f, χ q C alwayslies above the improved Hodge polygon IHP p ∆ q by Proposition 2.22 and that ` x k , h p T k q ˘ is a vertex of IHP p ∆ q by Lemma 3.23.(2) Suppose that u x k ,nh p T k q is not a Z p -unit. Since we know that p x k , h p T k qq isa vertex of NP p f, χ q C , then specializing NP p f, T q C to T “ χ p q ´ p f, χ q C strictly higher than NP p f, T q C at x “ x k . By Lemma 2.22, it is also strictly higher thanIHP p ∆ q at this point, which leads to a contradiction. (cid:3) Proof of Theorem 3.1.
Let χ : Z p Ñ C ˆ p be a nontrivial character of conductor p . SinceNP p f, χ q C is not below IHP p ∆ q and the expression in (3.18) is an upper bound for IHP p ∆ q at x “ x k for each k ě
1, then we have k “ h p T q p ´ ` x p k ´ q ‰ ` k ´ ÿ i “ rp a b ´ a b qp i ` q ´ p l ` l qsp i ` qď x p k ´ q k ` k Σ p χ q ` p a b ´ a b q k ´ ÿ i “ i ´ “ p l ` l q ` ‰ k ´ ÿ i “ i. (3.24)A simplification of these inequalities above shows that they all equivalent to(3.25) h p T q ď p p ´ q Σ p χ q , an equality independent of k .Since NP p f, χ q C coincides with IHP p ∆ q at ` x k , h p T k q ˘ for a finite character χ andan integer k , by Lemma 3.24, we know that u x k ,nh p T k q is a Z p -unit. It implies thatNP p f, χ q C also coincides with IHP p ∆ q at p x k , h p T k qq . Therefore, by Lemma 3.23 (2),the slopes of segments in NP p f, χ q C after points x “ x k are all greater than or equal to p p ´ q k .One the other hand, recall that in Notation 3.21 we put t α , α , . . . , α a b ´ a b u (in anon-decreasing order) to be the set of q -adic Newton slopes for L ˚ f p χ , s q , which is containedin r , q . Therefore, each q -adic Newton slope of L ˚ f p χ , q i s q belongs to r i, i ` q . Then fromthe decomposition of C ˚ f p χ , s q in (3.16), we know that α j ` k ´ ě k for all j ě x ` x k roots of C ˚ f p χ , s q whose q -adic valuations are lessthan or equal to k , which is a contradiction to the statement in the previous paragraph.From the argument above, we see that (3.24) must be an equality, and when k “ k ,the height of NP p χ , s q C coincides with its upper bound given in (3.18). Hence, we have h p T q “ p p ´ q Σ p χ q . Notice that (3.25) is independent of k . Then the inequalities in (3.24) are equalities forall k ě
0. Combining it with Proposition 3.24, we have that u x k ,nh p T k q is a Z p -unit foreach k ě
0. Therefore, it is not hard to show that NP p f, χ q C coincides with IHP p ∆ q at p x k , h p T k qq for any integer k and nontrivial finite character χ . Then by Lemma 3.23 again,we know that α i ě x ď i ď x . Combining it with Poincar´e duality, we have α i “ x ` ď i ď x . It implies that IHP p ∆ q and NP p f, χ q C coincide at x “ x k ` i k for any0 ď i k ď x k ´ x k .By a similar argument to Lemma 3.24 (2), we know that u x k ,nh p T k q is also a Z p -unit. (cid:3) The case when ∆ is an isosceles right triangle I. In order to apply Theorem 3.1 we need that NP p f, χ q C coincides with IHP p ∆ q at x “ x k for some character χ and some integer k . This however seems to be a verydifficult question. We have the following folklore conjecture. Conjecture 4.1.
Let ∆ be a triangle with vertices at p , q , P “ p a , b q , P “ p a , b q .We assume the hypotheses (as in Theorem 3.1) on the prime p . In the moduli space of allpolynomials f p x , x q of convex hull ∆ , there exists an open dense subspace over which thecorresponding Newton polygon NP p f, χ q C agrees with IHP p ∆ q at x “ x k ` i k for all finitecharacters χ , integers k ě and ď i k ď x k ´ x k . Generically, the Newton polygon of C ˚ f p χ, s q should be as low as possible, namely,coinciding with the improved Hodge bound IHP p ∆ q .In this section, we will study a special case when ∆ is an isosceles right triangle withvertices at p , q , p d, q and p , d q , where p ∤ d . We claim that Conjecture 4.1 holds truewhen the residue of p modulo d is small enough. More precisely, we will prove the following. Theorem 4.2.
Let p be the residue of p modulo d , and let d be the residue of d modulo p . Conjecture 4.1 holds when (4.1) d ě $&% p ” ln 3 ` ` p ` q ln p ı ` p if h ě , p ´ h ” ln 3 ` ` p ` q ln p ı ` p if h ă , where h : “ log p p p ´ d q . In particular, the condition d ě p p ` p q implies (4.1); so Theorem 1.5 follows fromthis. We will complete the proof at the end of this section. ENERIC NEWTON POLYGON FOR EXPONENTIAL SUMS IN TWO VARIABLES WITH TRIANGULAR BASE23
An interpretation of Theorem 4.2.
First, we consider the “ universal polynomial ” f univ p x , x q “ ÿ P P ∆ X M p ∆ q r a P x P x x P y whose coefficients are treated as variables. Notation 4.3.
Recall the infinite matrix N defined in Proposition 2.22. Let r N be thematrix given by substituting ˆ a P in N by r a P . More rigorously, we put E f univ p x , x q : “ ÿ P P Z ě r e P p r a, T q x P x x P y P Z p r r a s J T, x , x K , and write r e P : “ r e P p r a, T q . Similar to Lemma 2.20, we have(4.2) r e P P T r w p P q s Z p r r a s J T K and r e O “ . Then, using the list p P , P , . . . q of points in M p ∆ q given in Notation 2.21, we define r N tobe the infinite matrix whose p i, j q entry is r e pP i ´ P j P Z p r r a s J T K . Similar to Definition 3.19, we putdet p T q univ “ ÿ τ P Iso p T q sgn p τ q ź P P T r e pτ p P q´ P “ ÿ i “ h p T q r v i T i P Z p r r a s J T K , where sgn p τ q is the sign of τ as a permutation.Since all results in Section 3 for the “fixed” f actually hold for on general polynomials f p x , x q P F p r x , x s , we have the following. Proposition 4.4.
The polygons
GNP p ∆ q and IHP p ∆ q coincide at ` x , h p T q ˘ if and onlyif r v h p T q is not divisible by p .Moreover, when either condition holds, Conjecture 4.1 holds for that ∆ .Proof. We first prove the “only if” part. Suppose that r v h p T q is divisible by p . For anypair of a two-variable polynomial ˚ f p x , x q P F p r x , x s with convex hull ∆ and a finitecharacter ˚ χ of conductor p m ˚ χ , we write ˚ v h p T q P Z p n p ˚ f q r ζ p m ˚ χ s as the specialization of r v h p T q at T “ ˚ χ p q ´ r a P equals to the Teichmuller lifts of the coefficients of f , where ζ p m ˚ χ is a primitive p m ˚ χ -th root of unity. Then we have v p p ˚ v h p T q q ě h p T q p m ˚ χ ´ p p ´ q ` . As in (1.1), we denote C ˚ ˚ f p ˚ χ, s q “ ź j “ L ˚ ˚ f p ˚ χ, p jn p ˚ f q s q ´p j ` q “ ÿ k “ ˚ u k s k P Z p r ζ p m ˚ χ s J s K . By Proposition 3.20, we know that v p ´ n p f q´ ź i “ σ i Frob p ˚ v h p T q q ´ ˚ u x ¯ ě n p ˚ f q h p T q ` p m ˚ χ ´ p p ´ q , where σ Frob represents the arithmetic Frobenius acting on the coefficients.
Combining the equalities above, we get that v p p ˚ u k q ě n p ˚ f q h p T q ` p m ˚ χ ´ p p ´ q . Since we choose ˚ f and ˚ χ arbitrarily, we know that GNP p ∆ q is strictly above IHP p ∆ q at x “ x , a contradiction.We prove the “if” part. Let r v h p T q be the image of r v h p T q in the quotient ring Z p r r a s{ p Z p r r a s – F p r r a s . Since r v h p T q is not divisible by p , we know that r v h p T q ‰ T “ t P P M p ∆ q | w p P q ď u and x to be its cardinality inNotation 1.2. Let f p x , x q “ ř P P T b P x P x x P y P F p r x , x s be a polynomial satisfies that ‚ f has convex hull ∆. ‚ r v h p T q | r a P “ b P ‰ χ of conductor p , NP p f , χ q L coincides withIHP p ∆ q at x “ x . Since GNP p ∆ q is not below IHP p ∆ q and the set of polynomials whichsatisfy these conditions forms a Zariski open subset in the affine space A x F p , we completethe proof. (cid:3) Now we are left to show that r v h p T q is not divisible by p . Definition 4.5.
We label the elements in T as T : “ Q , Q , . . . , Q x ( such that Q : “ p d, q and Q : “ p , d q . Each point P P M p ∆ q can be written as a linearcombination of points in T with non-negative integer coefficients, namely P “ x ÿ i “ b P,i Q i . We call the vector ~b P P Z x ě (or the linear combination) P -minimal if it satisfies x ÿ i “ b P,i “ P w p P q T . Definition 4.6. A combo , denoted by p τ,~b ‚ ,τ q , is a pair consisting of an arbitrary permu-tation τ of T together with, for each P P T , a vector ~b P,τ P Z x ě such that(4.3) x ÿ i “ b P,τ,i Q i “ pτ p P q ´ P. A combo p τ,~b ‚ ,τ q is called optimal if τ is minimal and for each P P T vector ~b P,τ is ` pτ p P q ´ P ˘ -minimal.A combo p τ,~b ‚ ,τ q is optimal if and only if ÿ P P T x ÿ i “ b P,τ,i “ h p T q . We have the following explicit expression of the leading coefficient r v h p T q . ENERIC NEWTON POLYGON FOR EXPONENTIAL SUMS IN TWO VARIABLES WITH TRIANGULAR BASE25
Lemma 4.7.
We have (4.4) r v h p T q “ ÿ p τ,~b ‚ ,τ q optimal sgn p τ q ź P P T x ź i “ p r a Q i q b P,τ,i b P,τ,i ! , where the sum runs over all optimal combos, and sgn p τ q is the sign of τ .Proof. We let ź Q P T e r a Q πx Qx x Qy “ ÿ p P Z ě r e P x P x x P y . For any optimal combo p τ,~b ‚ ,τ q , we have b P,τ,i ď p ´ P P T and 1 ď i ď x , which implies thatdet p T q univ “ ÿ τ P Iso p T q sgn p τ q ź P P T r e pτ p P q´ P ` O p T h p T q` q“ ÿ p τ,~b ‚ ,τ q ´ sgn p τ q ź P P T x ź i “ p r a Q i q b P,τ,i b P,τ,i ! ¯ T ř P P T x ř i “ b P,τ,i ` O p T h p T q` q , (4.5)where p τ,~b ‚ ,τ q runs over all combos for T and O p T h p T q` q represents for a power series in Z p r r a s J T K of T -adic valuation greater than or equal to h p T q `
1. Then this lemma followsfrom the last statement in Definition 4.6. (cid:3)
Definition 4.8.
Lemma 4.7 gives an explicit expression of r v h p T q as the sum of terms labeledby optimal combos. For a combo p τ,~b ‚ ,τ q , we callsgn p τ q ź P P T x ź i “ p r a Q i q b P,τ,i b P,τ,i !its corresponding monomial .Two combos p τ,~b ‚ ,τ q and p τ ,~b ,τ q have a same corresponding monomial (with possiblydifferent coefficients) if and only if ÿ P P T b P,τ,i “ ÿ P P T b P,τ,i for all 1 ď i ď x . Recall that our task is to prove that r v h p T q is not divisible by p . For this it is enoughto show that r v h p T q has a monomial whose coefficient is not divisible by p . To this end, werestrict our study to those monmials corresponding to some “extreme” optimal combos. Lemma 4.9.
For each combo p τ,~b ‚ ,τ q for T , we have (4.6) ÿ P P T b P,τ, ď ÿ P P T Y pP x d ] and ÿ P P T b P,τ, ď ÿ P P T Y pP y d ] , where P x and P y are the x, y -coordinates of P . Proof.
We will prove the first inequality, and the proof of the second is similarRecall that Q “ p d, q . By equality (4.3), we get(4.7) b τ ´ p P q ,τ, ď Y pP x d ] . Hence, we have that ÿ P P T b P,τ, “ ÿ P P T b τ ´ p P q ,τ, ď ÿ P P T Y pP x d ] . (cid:3) Definition 4.10.
We call a combo special if it is optimal and both inequalities in (4.6) areequalities.Notice that these two sums are the exponents of r a Q and r a Q in the correspondingmonomial; so special combos contribute to terms in r v h p T q with maximal degrees in thecoefficients r a Q and r a Q at the two vertices of ∆.Recall that for each point P in M p ∆ q , we denoted by P % its residue in (cid:3) ∆ . Notation 4.11.
We put T “ T , \ T , , where T , “ t P P T | p pP q % P T u and T , isthe complement of T , in T . In other words, we have T , “ t P P T | p pP q % R T u . Example 4.12.
When d “ p “
17, the following graph shows the distribution of T , and T , in T , where “ ˆ ” and “ ‚ ” represent points in T , and T , respectively. xy ˆˆˆˆˆˆˆ ˆˆˆˆ ˆ ˆˆˆ ˆ ˆˆ ˆ‚‚ ‚‚‚‚ ‚ ‚‚ , Figure 2.
The distributions of T , and T , when d “ p “ Lemma 4.13.
A combo p τ,~b ‚ ,τ q is special if and only if it satisfies the following two con-ditions. p q For each P P T , , we have τ ´ p P q “ p pP q % and all other b τ ´ p P q ,τ, ˚ ’s are zero except b τ ´ p P q ,τ, and b τ ´ p P q ,τ, which are equalto i P, and i P, . ENERIC NEWTON POLYGON FOR EXPONENTIAL SUMS IN TWO VARIABLES WITH TRIANGULAR BASE27 p q For each P P T , , assume that Q j P “ p pP q % ´ τ ´ p P q , then we have p pP q % ´ τ ´ p P q P T and all other b τ ´ p P q ,τ, ˚ ’s are zero except b τ ´ p P q ,τ, , b τ ´ p P q ,τ, and b τ ´ p P q ,τ,j P whichare equal to i P, , i P, and .In particular, if p τ,~b ‚ ,τ q is a special combo, then τ uniquely determines ~b ‚ ,τ .Proof. “ ðù ”. Let p τ,~b ‚ ,τ q be any combo for T which satisfies these two conditions.We easily see that for p τ,~b ‚ ,τ q to be special, it is enough to show that τ is minimal,which follows directly from h p T q ě ÿ P P T t w p P q u and r w p pτ p P q ´ P q s “ t w p P q u . “ ùñ ”. Assume that p τ,~b ‚ ,τ q fails one of these conditions. Then it is easy to check thatthe monomial corresponding to this combo either has degree greater than equal to h p T q orthe exponent of r a Q or r a Q is not maximal, a contraction to p τ,~b ‚ ,τ q being special. (cid:3) Example 4.14.
The following gives an example of a minimal permutation of T in the caseof Example 4.12. Let τ ´ p , q “ p , q , τ ´ p , q “ p , q , τ ´ p , q “ p , q ,τ ´ p , q “ p , q , τ ´ p , q “ p , q , τ ´ p , q “ p , q ,τ ´ p , q “ p , q , τ ´ p , q “ p , q , τ ´ p , q “ p , q ,τ ´ p , q “ p , q , τ ´ p , q “ p , q , τ ´ p , q “ p , q ,τ ´ p , q “ p , q , τ ´ p , q “ p , q , τ ´ p , q “ p , q ,τ ´ p , q “ p , q , τ ´ p , q “ p , q , τ ´ p , q “ p , q ,τ ´ p , q “ p , q , τ ´ p , q “ p , q , τ ´ p , q “ p , q .τ ´ p , q “ p , q , τ ´ p , q “ p , q , τ ´ p , q “ p , q ,τ ´ p , q “ p , q , τ ´ p , q “ p , q , τ ´ p , q “ p , q . From the last statement in Lemma 4.13, it determines a unique special combo. We leave itto the reader to complete its corresponding special combo.
Lemma 4.15.
There is at least one special (optimal) combo among all combos for T .Proof. In Definition 4.21, we will give a correspondence between the set of special combosand the set of special bijections (See Definition 4.21); and in (5.19), we construct an explicitspecial bijection r β . This lemma follows from an easy check that the combo correspondingto r β is special.Since the construction of r β requires nothing but p ą d ` d to be relatively largewith respect to p (the residue of p modulo d ), this is not a circular argument. (cid:3) Definition 4.16.
We write r v sp h p T q for ÿ p τ,~b ‚ ,τ q special sgn p τ q ź P P T x ź i “ p r a Q i q b P,τ,i b P,τ,i ! , where the sum runs over all special combos.By Lemma 4.15, for Theorem 4.2 to hold, it is enough to prove the following. Proposition 4.17.
There is a monomial in r v sp h p T q with coefficient not divisible by p . Its proof will be given later.By the last statement of Lemma 4.13, we are reduced to studying minimal permu-tations in special combos, which will be further reduced by the correspondence given inDefinition 4.21 soon.
Definition 4.18.
For each point P P (cid:3) ∆ , we call point p d, d q ´ P its mirror reflection anddenote it by m p P q . Notation 4.19.
Let Y be the set consisting of all lattice points strictly inside the upperright triangle of (cid:3) ∆ . We put Y : “ ! p pQ q % ˇˇ Q P T , ) to be a subset of Y . Lemma 4.20.
We have ! p pP q % ˇˇ P P T ) “ Y \ p T z m p Y qq . Proof.
Suppose that there exists a point P P T such that P “ p pQ q % and m p P q “ pQ %for two points Q , Q P T . Let p be an integer such that p p ” p mod d q . We know that p p P q % “ Q and r p m p P qs % “ Q are mirror reflections, a contradiction. (cid:3) Figure 3 shows the distribution of tp pP q % | P P T u in the case of Example 4.12, where“ ‚ ” and “ ˆ ” represent points in Y and tp pP q % | P P T uz Y respectively. xy ˆˆˆˆˆˆˆˆ ˆˆ ˆˆˆ ˆˆˆˆ ˆ ˆ ˆ‚ ‚‚ ‚‚ ‚‚‚‚˝˝˝˝ ˝˝ ˝˝ ˝ . Figure 3.
The distributions of tp pP q % | P P T u when d “ p “ Definition 4.21.
A bijection β : Y Ñ m p Y q is called special if P ´ β p P q P T for eachpoint P P Y .We define a one-to-one correspondence between special combos and special bijections β : Y Ñ m p Y q is defined as follows: ENERIC NEWTON POLYGON FOR EXPONENTIAL SUMS IN TWO VARIABLES WITH TRIANGULAR BASE29
For a special combo ` τ,~b ‚ ,τ ˘ , we assign it a special bijection β from Y to m p Y q givenby β p P q “ τ ´ pp p P q % q for each P P Y .In the opposite direction, for a special bijection β : Y Ñ m p Y q , we assign to it aspecial combo τ given by τ ´ p P q “ p pP q % if P P T , ,β pp pP q % q if P P T , . Since the composite of these map is the identity map, it is truly a one-to-one corre-spondence.
Example 4.22.
The special bijection β : Y Ñ m p Y q corresponding to the special comboin Example 4.14 is given by β p , q “ p , q , β p , q “ p , q , β p , q “ p , q ,β p , q “ p , q , β p , q “ p , q , β p , q “ p , q ,β p , q “ p , q , β p , q “ p , q , β p , q “ p , q . Notation 4.23. (1) For a special bijection β , we let τ p β q P Iso p T q denote the minimalpermutation of the corresponding special combo. In view of Lemma 4.13, τ p β q uniquely determines β .(2) The composite m ˝ β can be viewed as a permutation of Y . Then we denote bysgn p β q the sign of this permutation and also call it the sign of β . Lemma 4.24.
We have (4.8) r v sp h p T q “ ÿ β special sgn p β q ź P P T x ź i “ p r a Q i q b P,τ p β q ,i b P,τ p β q ,i ! . Proof.
First, by the one-to-one correspondence in Definition 4.21, we know that the sum of r v sp h p T q over all special combos is the same as the sum over all special β ’s. Let β be specialand let τ p β q be the corresponding minimal permutation of T . Since the restriction of τ p β q to τ p β q ´ p T , q “ T z m p Y q is symmetric about y “ x , we know that sgn p τ p β qq dependsonly on sgn p β q . More precisely, we havesgn p β q “ sgn ` τ p β q ˘ , (cid:3) which completes the proof of this lemma. Lemma 4.25. p q The contribution to r v sp h p T q in (4.8) of terms coming from P P T , issame for all special bijections, namely, for two special bijections β , β : Y Ñ m p Y q , wehave ź P P T , x ź i “ p r a Q i q b τ p β q´ p P q ,τ p β q ,i b τ p β q ´ p P q ,τ p β q ,i ! “ ź P P T , x ź i “ p r a Q i q b τ p β q´ p P q ,τ p β q ,i b τ p β q ´ p P q ,τ p β q ,i ! . p q For the contributions of terms coming from P P T , , we have that the equality ź P P T , x ź i “ p r a Q i q b τ p β q´ p P q ,τ p β q ,i b τ p β q ´ p P q ,τ p β q ,i ! “ ź P P T , x ź i “ p r a Q i q b τ p β q´ p P q ,τ p β q ,i b τ p β q ´ p P q ,τ p β q ,i ! holds if and only if we have the following equality (4.9) ! P ´ β p P q ˇˇ P P Y ) ‹ “ ! P ´ β p P q ˇˇ P P Y ) ‹ as multisets.Proof. The first statement directly follows from condition (1) for a special permutation inLemma 4.13.For any special bijection β , we have ź P P T , x ź i “ p r a Q i q b τ p β q´ p P q ,τ p β q ,i b τ p β q ´ p P q ,τ p β q ,i ! “ ź P P T , ź i “ p r a Q i q b τ p β q´ p P q ,τ p β q ,i b τ p β q ´ p P q ,τ p β q ,i ! ˆ ź P P T , r a p pP q % ´ τ p β q ´ p P q “ ź P P T , ź i “ p r a Q i q b τ p β q´ p P q ,τ p β q ,i b τ p β q ´ p P q ,τ p β q ,i ! ˆ ź P P Y r a P ´ β p P q . Since ź P P T , ź i “ p r a Q i q b τ p β q´ p P q ,τ p β q ,i b τ p β q ´ p P q ,τ p β q ,i !is same to all special permutations, we complete the proof of the second statement. (cid:3) Definition 4.26.
We call β, β : Y Ñ m p Y q related if they satisfy equality (4.9). Corollary 4.27. If β , β : Y Ñ m p Y q are two related special bijections and sgn p β q “ sgn p β q , then they contribute to a same monomial in r v h p T q . Proposition 4.28. p q There exists a special r β : Y Ñ m p Y q such that every β related to r β is even, i.e. sgn p β q “ , and the number of such β is equal to i for some integer i . p q Therefore, there exists a monomial in r v x ,h p T q such that its coefficient is in the formof i N , where N is an integer which is not divisible by p . Its proof will be completed in section 5.Theorems 4.2, 1.5 and 1.6 would follow from this proposition. proof of Theorem 4.2 assuming Proposition 4.28.
This theorem follows directly from Propo-sition 4.28, Proposition 4.4 and Theorem 3.1. (cid:3)
Proof of Theorem 1.5.
It is easy to check that d ě p p ` p q satisfies (4.1). Therefore,the only task left is to compute x k , x k and h p x k q explicitly. It follows directly from applyingLemma 3.15 to this specific ∆. (cid:3) Proof of Theorem 1.6.
Its proof follows from Theorem 1.5 and a consideration of Poincar´eduality. (cid:3) The case when ∆ is an isosceles right triangle II. Overview.
The goal of this section is to prove Proposition 4.28 by constructing ex-plicitly the special bijection r β : Y Ñ m p Y q . This is done in several steps. First, for a largesubset L of Y , we shall define a bijection r β p i.e. r β | L q : L Ñ m p L q which is “diagonal”,namely the line segment r β p P q P is parallel to the line y “ x . For the remaining points in Y , we divide them into two subsets L and L as in (5.1), where L is contained in K (see ENERIC NEWTON POLYGON FOR EXPONENTIAL SUMS IN TWO VARIABLES WITH TRIANGULAR BASE31 the blue region in Figure 4) as will be proved in Proposition 5.11, and L is contained inthe green region in Figure 4 by definition.The map r β p i.e. r β | L q will map L into the union of appropriate shifts of the subset K (see the yellow region in Figure 4). More precisely, we write L as the disjoint union L ,i \ ¨ ¨ ¨ \ L ,i r (for some non-negative integers i , . . . , i r ) and r β is the union of maps L ,i k Ñ p K ` p i k p , ´ i k p qq X m p Y q such that the line segments r β p P q P are parallelfor all points P in a fixed L ,i k . We extend r β to a map s p r β q on L \ m p Im p L qq byrequiring s p r β qp P q “ m ˝ β ´ ˝ m p P q for any point P P m p Im p β qq and hence determinethe preimages of points in m p L q under the map r β (see the pink region in Figure 4). Atlast, we write r β p i.e. r β | Y zp L Y Dom p s p r β qq q for the unique diagonal symmetric bijection from Y z ´ L Y Dom p s p r β qq ¯ to m p Y qz ´ m p L q Y Im p s p r β qq ¯ . Finally we will show that r β , s p r β q and r β altogether define the needed special bijection r β : Y Ñ m p Y q .5.2. Construction of r β .Hypothesis 5.1. Recall that we put p “ p % d . From now on we assume that p ą d ` p ă d . Notation 5.2.
Here is a list of symbols: ‚ D k : the set consisting of all lattice points on the diagonal line y “ x ` k . ‚ W k : the set consisting of all lattice points on the anti-diagonal line x ` y “ k . ‚ For an interval I Ď R , we write D I : “ š i P I X Z D i and W I : “ š i P I X Z W i . ‚ K : “ W r d ´ p , d s X D r´ p ,p q X Y (see an example in Figure 4). Definition 5.3.
Let β be an injection from a subset of Y to m p Y q .(1) We call β weakly symmetric if its domain and image are symmetric about the line y “ d ´ x ; i.e. Dom p β q “ m p Dom p β qq and Im p β q “ m p Im p β qq .(2) We call β symmetric if each point P P Dom p β q satisfies β ` m p β p P qq ˘ “ m p P q . (3) If there exists some symmetric map β with domain included in Y such thatDom p β q “ Dom p β q Y m ` Im p β q ˘ and β ˇˇ Dom p β q “ β, then we call β the symmetric closure of β and denoted it by s p β q . Lemma 5.4.
Let β be an injection from a subset of Y to m p Y q . If Dom p β q X m p Im p β qq “H , then s p β q exists.Proof. Let ‚ s p β q ˇˇ Dom p β q “ β , and ‚ for any point P P m p Im p β qq let s p β qp P q “ m ˝ β ´ ˝ m p P q . It is a trivial check that s p β q is the symmetric closure of β . (cid:3) Definition 5.5.
A vector is called diagonal if it is parallel to the line y “ x . Let V ‹ be amultiset of vectors. We call it diagonal if each ~v P V ‹ is diagonal. We write V : “ ! V ‹ ˇˇ V ‹ is a diagonal multiset ) . We define the weight on vectors on R so that w pÝÝÑ OP q “ w p P q . For each V ‹ P V and each r P R , we write p V ‹ q ě r : “ ! ~v P V ‹ ˇˇ w p ~v q ě r ) . xy K m p K q K . Figure 4.
Regions K and K when d “ p “ We define a total order “ ă ” on V as follows: Definition 5.6.
For any two sets V ‹ , V ‹ P V , we denote V ‹ ă V ‹ if one of the followingcases happens:Case 1: V ‹ ă V ‹ ;Case 2: V ‹ “ V ‹ and there exists a real number r such that p V ‹ q ě r “ p V ‹ q ě r forall r ą r but p V ‹ q ě r ă p V ‹ q ě r .To construct r β needed for Proposition 4.28, we shall construct it so that for a largestpossible subset L Ď Y , ÝÝÝÝÑ r β p P q P is diagonal for all P P L , or equivalently, the set t P ´ β p P q | P P Y u contains as many diagonal vectors (and as highest weight) as possible. Definition 5.7.
Let S be an arbitrary subset of Y , and let β : S Ñ m p Y q be an injection.We set V ‹ p β q : “ ! P ´ β p P q ˇˇ P P S ) ‹ . If V ‹ p β q is diagonal, then we also call β diagonal . Definition 5.8.
We call a pair p P, Q q in Y ˆ m p Y q eligible if it satisfies the following twoconditions:(a) ÝÝÑ QP is diagonal with weight less than or equal to 1, and(b) either w p P q ą or w p Q q ă .We write E : “ ď S Ă Y ! β : S Ñ m p Y q | p P, β p P qq is eligible for each P P S ) , ENERIC NEWTON POLYGON FOR EXPONENTIAL SUMS IN TWO VARIABLES WITH TRIANGULAR BASE33 where S runs over all subsets of Y . For simplicity of notation, we put V p E q : “ t V ‹ p β q ˇˇ β P E u . Definition 5.9.
Define r β to be an element in E such that V ‹ p r β q is a maximal elementin ` V p E q , ă ˘ .In fact, r β can be constructed in the following way:Assume that r β has been defined on some subset of Y , say Y . If there is no eligiblepair in Y zp Y q ˆ m p Y qz r β p Y q , then we call the definition of r β is completed. Otherwise,we choose an eligible pair p P , Q q in Y z Y ˆ m p Y qz r β p Y q which maximizes the weight w p P ´ Q q , and define r β p P q “ Q . Lemma 5.10.
The map r β is the unique maximal element in the totally ordered set p V p E q , ă q .Proof. Assume that we have defined β on the subset Y by the construction above and p P , Q q and p P , Q q are two eligeble pairs in Y zp Y q ˆ m p Y qz r β p Y q which maximize w p P ´ Q q “ w p P ´ Q q . Since
ÝÝÝÑ Q P and ÝÝÝÑ Q P are required to be diagonal, we know that P ‰ P and Q ‰ Q .Therefore, the definition of r β is independent of the choices of pairs. (cid:3) Write L for the domain of r β . Since we require r β to be the maximal element in V p E q ,tt is easily known that r β is symmetric. Put(5.1) L : “ ! P P Y z L ˇˇ w p P q ą ) and L : “ ! P P Y z L ˇˇ w p P q ď ) . Then we obtain a disjoint decomposition of Y as Y “ L \ L \ L . We next will(1) define a map r β on L ,(2) find its symmetric closure s p r β q ,(3) define a map r β on the complement of L Y Dom p s p r β qq in Y , and(4) put together the maps r β , s p r β q , and r β to get a bijection r β : Y Ñ m p Y q whichsatisfies the conditions in Proposition 4.28.5.3. Study of L . Recall that L defined in (5.1) is the subset where we cannot define r β diagonally and where the weight of the points is strictly bigger than .”In this subsection, we will complete the definition of r β : L Ñ m p Y q . We start witha proposition about the distribution of its domain L in Y , which plays an important rolein its construction. Proposition 5.11.
The subset L is included in K (See Notation 5.2).Proof. The proof will occupy the entire Section 5.2 and it will follow from Propositions 5.24and 5.35 below. (cid:3)
Lemma 5.12.
Recall that p “ p % d . Let P be any point in Y . If P ` p p i, j q is containedin Y for a pair of integers p i, j q , then it is also contained in Y . Proof.
Before proving the lemma, we refer to Figure3, where the bullet points in the upper-right triangle are periodic with period 3. Since the upper-right triangle in (cid:3) ∆ is convex, itis enough to show that the lemma holds for p i, j q “ p , q , p´ , q , p , q , p , ´ q , p , ´ q , and p´ , q . We will just prove the case when i “ j “ ´
1, and the rest can be handledsimilarly.Let Q be the point in T such that p pQ q % “ P . It is easy to check that P ` p p , ´ q ” pQ ` p p , ´ q” pQ ` p p , ´ q“ p p Q ` p , ´ qq p mod d q . In fact, the point Q ` p , ´ q is strictly contained in ∆ f , for otherwise P ` p p , ´ q is onthe boundary of (cid:3) ∆ f , which is a contradiction to P ` p p , ´ q P Y . Then by the definitionof Y , we know that P ` p p , ´ q belongs to Y . (cid:3) Notation 5.13.
We call the square with vertices p d ´ p , d ´ p q , p d ´ p , d ´ q , p d ´ , d ´ p q and p d ´ , d ´ q the fundamental cell , denoted by C , and write C : “ C X Y . Back to the example in Figure 3, the corresponding subset C “ tp , q , p , q , p , qu . Corollary 5.14.
We know that Y distributes periodically in Y of period p . More precisely,each point in Y is a shift of some point in C by p ip , jp q , where p i, j q is a pair of integers.Proof. It follows directly from Lemma 5.12. (cid:3)
Corollary 5.15.
Let k , k , j and j be integers satisfying k , k ě d and p |p k ´ k q . Ifboth W k X D r j ,j ` p q and W k X D r j ,j ` p q are contained in (cid:3) ∆ , then(1) ´ Y X W k X D r j ,j ` p q ¯ “ ´ Y X W k X D r j ,j ` p q ¯ . (2) If moreover we have p |p j ´ j q , we have the following equality of sets Y X W k X D r j ,j ` p q “ Y X W k X D r j ,j ` p q ` p k ´ k qp , q ` p j ´ j qp´ , q . Proof.
In fact, this corollary follows directly from previous corollary. (cid:3)
Since Y is distributed periodically of period p , by Corollary 5.14, it is enough for usto understand C . The following two lemmas show the details. Lemma 5.16.
The distribution of C in C has the following properties:(1) There is no point of Y (or C ) on the top row or the first column of C .(2) If the point p i, j q is in C and i ` j “ d ´ p , then it is also in C .(3) For each point P in C , either P or p d ´ p , d ´ p q ´ P is contained in C .Proof. The first two statements are straightforward. Therefore, we only prove Property (3).Let p i , j q and p i , j q be two points in C symmetric about y “ d ´ p ´ x . Withoutloss of generality, we assume that the weight of p i , j q is less than the weight of p i , j q .Then it is easy to check that they satisfy ‚ p d ´ p q ď i ` j ă d ´ p , ‚ i ` j “ d ´ p , and ‚ i ` j “ d ´ p . ENERIC NEWTON POLYGON FOR EXPONENTIAL SUMS IN TWO VARIABLES WITH TRIANGULAR BASE35
Suppose that both p i , j q and p i , j q are in C . Then there are two points p i , j q and p i , j q in T such that p p p i , j qq % “ p i , i q and p p p i , j qq % “ p i , i q . It is easy to show that p p i ` j q ” d ´ p p mod d q and p p i ` j q ” d ´ p p mod d q . Since p d, p q “ p ” p p mod d q , we have i ` j ” ´ p mod d q and i ` j ” ´ p mod d q . Combining these two congruence equations with d ¨ w p i , j q “ i ` j ď d ´ d ¨ w p i , j q “ i ` j ď d ´ , we get that d ¨ w p i , j q “ d ´ d ¨ w p i , j q “ d ´ . It forces p p i ` j q ” ´ p p mod d q , which is a contradiction to2 p d ´ p q ď i ` j ă d ´ p . By a similar argument, we check that at least one of p i , j q and p i , j q belongs to C ,which completes the proof. (cid:3) Notation 5.17. (1) Let d “ d p ` d and let 0 ď d ă p be the integer such that d d ” p mod p q .(2) For any two points P , P in Z ě , if they satisfy that P ´ P “ p Q for some point Q in Z ě , then we denote P ” P p mod p q . Proposition 5.18.
For any ă k ď p , we have ` p W d ´ k Y W d ´ p ´ k q X C ˘ “ p ´ if k “ p ; kd % p ´ otherwise . We need some preparations before giving the proof of this proposition after Lemma 5.23.
Lemma 5.19.
Let i , j be two positive integers. If i ` j ď p , then ` p p id , jd q ˘ % P Y . Proof.
Since p p id , jd q ” p d ´ id , d ´ jd q p mod d q , it is enough to prove1 ă w ` p d ´ id , d ´ jd q ˘ ă , which follows directly from i ` j ď d . (cid:3) Notation 5.20.
Put A : “ ! p id , jd q ˇˇ i, j ą i ` j ď p ) . By Lemmas 5.19 and 5.12, for any point P P A there exists a point P in C such that P ” p pP q % p mod p q . It automatically gives us a map from A to C , denoted by γ . Nowwe will show that γ is a bijection. Notation 5.21.
For simplicity of notation, we put P i,j : “ p id , jd q . Lemma 5.22.
The map γ is a bijection. Proof.
Any two points P i ,j and P i ,j in A satisfy γ p P i ,j q ´ γ p P i ,j q ”p pP i ,j q % ´ p pP i ,j q % “p d ´ i d , d ´ j d q ´ p d ´ i d , d ´ j d q“pp i ´ i q d , p j ´ j q d q p mod p q . Now if γ p P i ,j q “ γ p P i ,j q , we know that pp i ´ i q d , p j ´ j q d q ” O p mod p q .Since p d , p q “ , | i ´ i | ă p and | j ´ j | ă p , we have i “ i and j “ j , which implies γ is an injection. By Lemma 5.16, there are p p p ´ q points in C , which is equal to the cardinality of A . Therefore, γ is a bijection. (cid:3) Lemma 5.23.
Any two points P i ,j and P i ,j of the same weight satisfy ˇˇˇ d ¨ w ` γ p P i ,j q ´ γ p P i ,j q ˘ˇˇˇ “ or p . Proof.
We know easily that γ p P i ,j q ” p d ´ i d , d ´ j d q and γ p P i ,j q ” p d ´ i d , d ´ j d q p mod p q , which implies that d ¨ w ` γ p P i ,j q ˘ ´ p d ´ i d ` d ´ j d q and d ¨ w ` γ p P i ,j q ˘ ´ p d ´ i d ` d ´ j d q are both divisible by p .Since P i ,j and P i ,j have the same weight, we know i ` j “ i ` j . Therefore, wehave(5.2) p ˇˇ d ¨ w ` γ p P i ,j q ´ γ p P i ,j q ˘ . On the other hand, γ p P i ,j q and γ p P i ,j q both belong to C , which together with (5.2)force ˇˇ d ¨ w ` γ p P i ,j q ´ γ p P i ,j q ˘ˇˇ to be 0 or p . (cid:3) Proof of Proposition 5.18.
By Lemma 5.23, we know that ` p W d ´ k Y W d ´ p ´ k q X C ˘ “ ! p i, j q ˇˇ p i ` j q d ” k p mod p q , i, j ą i ` j ă p ) “ ! p i, j q ˇˇ p i ` j q ” kd p mod p q , i, j ą i ` j ă p ) “ p ´ k “ p ; kd % p ´ . (cid:3) The following proposition is the first stepstone of Theorem 5.11.
Proposition 5.24.
For every integer k with | k | ě p , we have D k X L “ H . The proof of the proposition will be given after some lemmas.
Lemma 5.25.
Let ` P , P , . . . , P p ˘ be a sequence of consecutive points in D k X Y for some k . (1) There are exact t p u points in this sequence belonging to Y .(2) In particular, if ℓ is an integer with r p s ă ℓ ď p , then at least ℓ ´ r p s points inthe set t P , P , . . . , P ℓ u belong to Y .Proof. (1) It follows directly from Lemma 5.16 (1)-(3).(2) Combining (1) with Pigeonhole principle, we complete the proof of (2). (cid:3) ENERIC NEWTON POLYGON FOR EXPONENTIAL SUMS IN TWO VARIABLES WITH TRIANGULAR BASE37
Lemma 5.26.
There do not exist two points P P L and Q P m p L q such that ÝÝÝÑ Q P is adiagonal vector of weight less or equal to .Proof. Suppose the lemma were false. Then there exists an integer k such that ! p P, Q q P p L X D k q ˆ p m p L q X D k q ˇˇ w pÝÝÑ QP q ď ) is not empty. We put p P , Q q to be a pair of points in this set which maximize the weight w pÝÝÝÑ Q P q . By the inductive definition of r β , we can define r β on P by r β p P q “ Q , whichcontradicts to the assumption that P does not belong to L . (cid:3) Lemma 5.27.
For any integer k , there do not exist two points P P p Y z L q X D k and P P L X D k such that (5.3) w p P ´ r β p P qq ă w p P ´ r β p P qq ď . Proof.
Suppose that P and P are two points which satisfy conditions in this lemma. We eas-ily see that inequality (5.3) violates the requirement in construction of r β that p P , r β p P qq maximizes w p P ´ r β p P qq , a contradiction. (cid:3) Lemma 5.28.
For an arbitrary point P in D k X Y X W p d , d s , if it satisfies (5.4) ! Q P D k X m p Y q ˇˇ w p P ´ Q q ď ) ě ! Q P D k X L ˇˇ w p Q q ą w p P q ) ` , then it belongs to L .Proof. Let P be a point which satisfies conditions in this lemma. Suppose that P does notbelong to L . Then by Lemma 5.26, each element in ! Q P D k X m p Y q ˇˇ w p P ´ Q q ď ) is equal to r β p P q for some P P L zt P u . From equality (5.4), we know that at least one ofthese P does not belong to ! Q P D k X L ˇˇ w p Q q ě w p P q ) . Then we obtain a contradiction directly from Lemma 5.27. (cid:3)
Corollary 5.29.
For an arbitrary point P in D k X Y X W p d , d s , if it satisfies (5.5) ! Q P D k X m p Y q ˇˇ w p P ´ Q q ď ) ě ! Q P D k X Y ˇˇ w p Q q ě w p P q ) , then it belongs to L .Proof. Let P be a point satisfying condition in this lemma. Suppose that P does not belongto L . Then we have ! Q P D k X Y ˇˇ w p Q q ě w p P q ) ě ! Q P D k X L ˇˇ w p Q q ą w p P q ) ` . Combining it with Lemma 5.28, we get P P L , a contradiction. (cid:3) Therefore, in order to show that each point P in D k X Y X W p d , d s for | k | ě p belongsto L , it is enough to prove that P satisfies inequality (5.5). The following functions give alower bound for cardinality of the first set in (5.5) and an upper bound for the second one. Definition 5.30.
We define g p p i ` j q : “ i t p u if 0 ď j ď p i t p u ` t j u ´ r p s if p ă j ă p . Lemma 5.31.
For any integers B , B and k with ă B ă B ă d and | k | ď B , byLemma 5.25, we have ´ Q P D k ˇˇ B ď d ¨ w p Q q ď B ( X m p Y q ¯ ě g p B ´ B q . Definition 5.32.
Define g p p i ` j q : “ i t p u ` t j u if 0 ď j ď p p i ` q t p u if p ă j ă p . Lemma 5.33.
For any integers B , B and k with d ă B ă B ă d , by Lemma 5.25, wehave ´ Q P D k ˇˇ B ď d ¨ w p Q q ď B ( X Y ¯ ď g p B ´ B q . Lemma 5.34.
Both g and g are non-decreasing and g p k ` p q ě g p k q for every k ą .Proof. It follows from their definitions. (cid:3)
Proof of Proposition 5.24 . Consider a point P in D k X Y X W p d , d s . We have that ! Q P D k X m p Y q ˇˇ w p P ´ Q q ď ) “ ! Q P D k ˇˇ d ¨ w p P q ´ d ď d ¨ w p Q q ď d ) X m p Y q . By Lemma 5.31, we know that(5.6) ! Q P D k X m p Y q ˇˇ w p P ´ Q q ď ) ě g p d ´ w p P q d q . On the other hand, since ! Q P D k X Y ˇˇ w p Q q ě w p P q ) “ ! Q P D k ˇˇ d ¨ w p P q ď d ¨ w p Q q ď d ´ | k | ) X Y , by Lemma 5.33, we know that(5.7) ! Q P D k X Y ˇˇ w p Q q ě w p P q ) ď g ` d ´ | k | ´ d ¨ w p P q ˘ . By Lemma 5.34 and | k | ě p , the terms on the right side of these inequalities abovesatisfy g ` d ´ | k | ´ w p P q d ˘ ď g ` d ´ p ´ w p P q d ˘ ď g ` d ´ w p P q d ˘ . Hence, we have(5.8) ! Q P D k X m p Y q ˇˇ w p P ´ Q q ď ) ě ! Q P D k X Y ˇˇ w p Q q ě w p P q ) . Combining it with Corollary 5.29, we prove this proposition. (cid:3)
Proposition 5.35.
The intersection W r d, d ´ p s X L is empty.Proof. By Lemma 5.28, it is enough to prove that each point P in W p d, d ´ p s , say P P D k ,satisfies(5.9) ! Q P D k X L ˇˇ w p Q q ą w p P q ) ` ď ! Q P D k X m p Y q ˇˇ w p P ´ Q q ď ) . Now we estimate the size of the two sets in (5.9) as follows.By Proposition 5.24 and the assumption of P , we are reduced to proving (5.9) for P in D p´ p ,p q X W p d, d ´ p s , which guarantees us a point P in C such that(5.10) P “ P ` p ip , ip q ENERIC NEWTON POLYGON FOR EXPONENTIAL SUMS IN TWO VARIABLES WITH TRIANGULAR BASE39 for some integer i ě
1. Assume that P belongs to W j . Put B p P q : “ ! P P C X L X D k ˇˇ w p P q ě w p P q ) . Then we give the following estimations.
1. Estimation of ! Q P D k X m p Y q ˇˇ w p P ´ Q q ď ) . By definition of B p P q , we know that r β p B p P qq is contained in ! Q P D k X m p Y q ˇˇ w p P ´ Q q ď ) X W r j ´ d,d q . On the other hand, by (5.10), we know easily that D k X m p Y q X W r j ´ d ´ ip ,j ´ d q Ă ! Q P D k X m p Y q ˇˇ w p P ´ Q q ď ) . Therefore, r β p B p P qq and D k X m p Y q X W r j ´ d ´ ip ,j ´ d q are two disjoint subsets of ! Q P D k X m p Y q ˇˇ w p P ´ Q q ď ) . By Lemma 5.25, we know that ´ D k X m p Y q X W r j ´ d ´ ip ,j ´ d q ¯ “ i t p u , which implies ! Q P D k X m p Y q ˇˇ w p P ´ Q q ď ) ď B p P q ` i t p u .
2. Estimation of ! Q P D k X L ˇˇ w p Q q ą w p P q ) . Consider the disjoint decomposition(5.11) ! Q P D k X L ˇˇ w p Q q ą w p P q ) “ ´ D k X L X W p j ´ ip ,j s ¯ Y ´ B p P qz P ¯ . We need consider the following two cases:Case 1: When P belongs to L , we have ´ B p P qz P ¯ “ B p P q ´
1, which implies ! Q P D k X L ˇˇ w p Q q ą w p P q ) ď ´ D k X W p j ´ ip ,j s ¯ ` ´ B p P qz P ¯ “ i t p u ` B p P q ´ . Case 2: When P does not belong to L , we have ´ D k X L X W p j ´ ip ,j s ¯ ď ´ D k X W p j ´ ip ,j s ¯ ´ , which implies ! Q P D k X L ˇˇ w p Q q ą w p P q ) ď ´ D k X W p j ´ ip ,j s ¯ ´ ` ´ B p P qz P ¯ “ i t p u ` B p P q ´ . In either case, it is easy to check (5.9), which completes this proposition. (cid:3)
Definition of r β . We next construct a map β : L Ñ m p L q . Put r J “ t d ´ p , d ´ p ` , . . . , d ´ u . Write(5.12) K : “ ! P ˇˇ P P W r J X D “ r d s , r d s ` p ˘) and K : “ K X m p Y q . The general idea of constructing β is to map L to disjoint sets p K ` p i k p , ´ i k p qq X m p Y q , where p i , . . . , i L q is a certain sequence of numbers in some range, such that forany two points P and P in L if β p P q and β p P q belong to p K ` p i k p , ´ i k p qq X m p Y q for a same k , then P ´ β p P q “ P ´ β p P q . Remark 5.36.
An easy computation shows that W d ´ X m p Y q is not empty, say that Q is a point in it. The most naive construction of β is to make an injection from L to t Q ` p ip , ´ ip qu L i “ . However, the construction requires a very stronge condition that d “ O p p q . In order to weaken this condition, we need a more detailized construction (seeConstruction 5.41).We start the construction of β with giving more details of its codomain. Recall thatwe defined the numbers d , d and d in Notation 5.17. Lemma 5.37.
We have p K X W d ´ i q “ p ´ if i ” d p mod p q ; ` i p p ´ d q ˘ % p otherwisefor all ď i ď p ´ .Proof. Since Y and m p Y q are symmetric about y “ d ´ x , we have ´ K X W d ´ i ¯ “ ´ W d ` i X Y X D “ r d s , r d s ` p ˘¯ . Find p ă j ď p such that 2 d ´ j ” d ` i p mod p q . By Corollary 5.15, we have ´ W d ` i X Y X D “ r d s , r d s ` p ˘¯ “ ´ W d ´ j X D r´ p ,p q X Y ¯ . From Corollary 5.14, we know ´ W d ´ j X W r´ p ,p q ¯ “ ´ p W d ´ j Y W d ´ p ´ j q X C ¯ . Therefore, by Proposition 5.18, if j “ p , then we have pp W d ´ j Y W d ´ p ´ j q X C q “ p ´ . It is not hard to see from the relation between i and j that i ” d p mod p q . Combiningall these equlities above, we get p K X W d ´ i q “ p ´ j ‰ p , we have pp W d ´ j Y W d ´ p ´ j q X C q“ jd % p ´ “p d ´ i q d % p ´ “ i p p ´ d q % p . By a similar argument, we complete the proof immediately. (cid:3)
In order to support our construction of β , we need several technical lemmas. ENERIC NEWTON POLYGON FOR EXPONENTIAL SUMS IN TWO VARIABLES WITH TRIANGULAR BASE41
Notation 5.38.
For any subset K of K and any integer k P r , p ´ s , we put K p k q : “ P ` p´ k, k q if P ` p´ k, k q P K ; P ` p p ´ k, ´p p ´ k qq otherwise. Lemma 5.39.
Let J be a subset of r J . Suppose that there are at least n points in W j X K for each j P J . Then p q for every subset S of W J X K of cardinality m , there exists at least an integer i in r , p ´ s such that ´ S X K p i q ¯ ě P mn p T . p q For the set of lattice points W J X K , there exists a subset I of t , , . . . , p u of cardinalityless than or equal to P ´ log p ´ n p q ` p p J q ˘T such that (5.13) ď i P I K p i q X W J “ W J X K . Proof. (1). Since p ´ Ţ i “ p W J X K p i qq ‹ covers S at least n times, by Pigeonhole principle, thereexists some i such that S X K p i q ě P mn p T .(2). By (1), we can choose a sequence p i , i , . . . q from t , , . . . , p u such that(5.14) m k ď m k ´ ´ P nm k ´ p T ď m k ´ p ´ n p q , where m k : “ p W J X K ´ k Ť j “ K p i j q X W J q .Write t “ Y ´ log ` ´ n p q p p p J q ˘] `
1. Repeated application of (5.14) gives m t ď m p ´ n p q t “ p p J qp ´ n p q t ă . It implies m t “
0. Therefore, the length of this sequence cannot be longer than t ´
1, whichcompletes the proof of (2). (cid:3)
Let u be a real number in p , q . Depending on u , we decompose r J into three groups:(1) J p u q “ ! j P r J ˇˇ j ą d ´ p u p ´ d ) ,(2) J p u q “ ! j P r J ˇˇ p W j X K q ě p u ) , and(3) J p u q : “ r J H´ J p u q Y J p u q ¯ .By Lemma 5.37, we know that(5.15) 3 p ´ p u ď J p u q ď p . Notation 5.40.
Set h “ log p p p ´ d q . Construction 5.41 (Construction of β ) . We construct β in three steps: Step 1.
Lemma 5.37 shows that K X W d ´ is not empty, say that it contains a point Q .We put W J ` d X L : “ t P , P , . . . , P t u , where t is its cardinality, and define β on W J ` d X L as β : W J ` d X L Ñ m p L q P i ÞÑ Q ` ` p p i ´ q , ´ p p i ´ q ˘ . Namely, β maps W J ` d X L into a disjoint union of K ` p p i, ´ p i q for 0 ď i ď t ´ J , we know that J “ t p u p ´ d u . Since W J ` d X L is in an isoscelesright triangle with side lengths J , we have t ď p t p u p ´ d u ` q t p u p ´ d u . It is easily checkthat t ď p p p u ´ h q ` p u ´ h q . Step 2.
We denote by θ the unique map from K to K given by parallel transform. ByLemma 5.39 (2), there is a sequence p i , i , . . . , i t q such that t ď k “ K p i k q X W J “ K X W J and t ď X ´ log p ´ pu p q ` p p J q ˘\ . Since t Ť k “ K p i k q depends only on the elements in set t i , i , . . . , i t u , we can in fact require p i , i , . . . , i t q to be increasing.It is easily seen that t ď X ´ log p ´ pu p q ` p p J q ˘\ ď X ´ log p ´ pu p q p p q \ . For each point P in W J ` d X L , we put k p P q to be the smallest number such that K p i k p P q q X W J contains θ p P q . Then we define β p P q : “ θ p P q ` p x p P q , ´ x p P qq , where x p P q “ t p ` i k p P q ` k p P q p .x Namely, β maps W J ` d X L into a disjoint union of K ` “ t p ` i k ` kp ‰ p , ´ q for 1 ď k ď t . Step 3.
Write J p u q “ t j , j , . . . , j s u . By (5.15), we know that s ď p u . Let d be thelargest number in r J such that p W j X K q ě p . By Lemma 5.37, we have J P p d ´ p , d s and p ď p d ´ d qp p ´ d q ă p . Replacing J in Lemma 5.39 by t d u , we obtain a sequence p i , i , . . . , i t q from t , , . . . , p u such that t ď k “ K p i k q X W d “ K X W d and t ď X ´ log p ´ p { p q p p q \ “ X log p p q \ . Similar to
Step 2 , we assume that p i , i , . . . , i t q is increasing. Then we define β on W J ` d X L as follows:Consider each P P W J ` d X L . Suppose that P belongs to W j l ` d for some j l P J . Let k p P q be the smallest number such that K p i k p P q q X W d contains θ p P q ` p d ´ j l , d ´ j l q .Then we define β p P q : “ θ p P q ` p d ´ j l ` x p P q , d ´ j l ´ x p P qq , where x p P q “ p r k p P q ` p l ´ qp t ` q ` t ` t ` s ` i k p P q . Namely, β maps W J ` d X L into a disjoint union of K ` p r k ` p l ´ qp t ` q ` t ` t ` s ` i k ( p , ´ q for 1 ď k ď t and 1 ď l ď s .Notice that the codomain of β is both a disjoint union of shifts of K and a subset of (cid:3) ∆ . Then for a fixed d , the residue p of p modulo d cannot be too large. The followingcomputation gives p an upper bound such that the construction for β above is realizable.In fact, the complicated conditions in Theorem 4.2 are also from this computation. ENERIC NEWTON POLYGON FOR EXPONENTIAL SUMS IN TWO VARIABLES WITH TRIANGULAR BASE43 xy y “ xy “ p x ´ q y “ p x ´ q y “ p x ´ q y “ ´ x Figure 5.
Determine the optimizer of h . From the constructing above, we know that the image of β is included in a union ofdisjoint shifts of K . Moreover, the number of these shifts, denoted by N , is counted andestimated as follows: N “ t ` t ` t ˆ s ď p p p u ´ h q ` p u ´ h q ` X ´ log p ´ pu p q p p q T ` \ log p p q T ˆ p u ď p p u ´ h q ` ` p p q p ´ u ` ln 3 p ´ u ` log p p q ˆ p u . (5.16)Recall K “ ! P ˇˇ P P W r J X D “ t d u , t d u ` p ˘) . It is easy to see that the largest x -coordinateof points in the codomain of β is equal to d ` p p N ` s ` q , which obvious is controledby d . Then we get a necessary condition:(5.17) d ě p p N ` s ` q . We write G p h, u q “ max t p u ´ h q , ´ u, u u . Recall that h “ log p p p ´ d q is fixed by p and d . Therefore, our next goal is to determine the minimum of G p h, u q by varying thevalue of u inside r , s . Definition 5.42.
We call u an optimizer of h if G p h, u q “ min u Pr , s ` G p h, u q ˘ .In order to get an optimizer of a given h P p , s . We need to consider two cases:. Case 1.
When ď h ď
1. From Figure 5, the optimizer u of h is the x -coordinate of thepoint of intersection of lines y “ x and y “ ´ x . Therefore, we know that u “ is theoptimizer of this h . Plugging u “ into equations (5.16) and (5.17), we have4 p p N ` s ` q ď p ” ln 3 ` ` p ` q ln p ı ` p . Case 2.
When 0 ă h ă . Based on the same observation of Figure 5, an optimizer u of h is the x -coordinate of the point of intersection of lines y “ p x ´ h q and y “ ´ x . An easycomputation shows that u “ ` h . Combining it with (5.16) and (5.17), we have4 p p N ` s ` q ď p ´ h ” ln 3 ` ` p ` q ln p ı ` p . Notation 5.43. (1) We write(5.18) V p β q : “ ! P ´ β p P q ˇˇ P P L ) . (2) The reflection of a vector ~v through a diagonal line y “ x is denoted by ~v _ . Let V be a set of vectors. We put V _ : “ t ~v _ | ~v P V u . Lemma 5.44.
We know that Im p r β q X ! P ´ ~v ˇˇ P P L and ~v P V p β q Y V p β q _ ) is empty.Proof. It is easy to check that any point Q in ! P ´ ~v ˇˇ P P L and ~v P V p β q Y V p β q _ ) belongs to D k for some | k | ą d . Then this lemma follows simply from the definition of r β . (cid:3) Construction 5.45 (Construction of r β ) . Step 1 . Write E : “ β : L ã Ñ m p Y q ˇˇ P ´ β p P q P V p β q Y V p β q _ for all P P L ( . We know that E is non-empty, for β is automatically contained in it. Step 2 . We line up the elements in V p β q to form a sequence, denoted by p ~v , ~v , . . . , ~v N q . Step 3 . Define a partial order over E as follows:For any two maps β , β P E , we denote β ă β , if there exists an integer 1 ď k ď N such that ‚ t P | P ´ β p P q “ ~v i or ~v _ i u “ t P | P ´ β p P q “ ~v i or ~v _ i u for all 1 ď i ď k ´
1, and ‚ t P | P ´ β p P q “ v k or v _ k u ă t P | P ´ β p P q “ v k or v _ k u . Step 4 . Let r β be a maximal element in E .By Hypothesis 5.1, we know that m p L q X r β p L q “ H . Combining it with Lemma 5.4,we can simply prove the existence of s p r β q . Since maps r β and s p r β q are symmetric, wedefine r β : Y I´ Dom p r β q Y Dom p s p r β qq ¯ Ñ m p Y q I´ Im p r β q Y Im p s p r β qq ¯ P ÞÑ m p P q . Putting r β , s p r β q and r β together, we define a bijection r β : Y Ñ m p Y q such that(5.19) r β p P q “ $’&’% r β p P q if P P L ; s p r β qp P q if P P Dom p s p r β qq ; r β p P q otherwise . Since Y I´ Dom p r β q Y Dom p s p r β qq ¯ Ă L , we know that w p P ´ m p P qq ď P P L . Combining it with the constructive definition of r β and s p r β q , we easily checkthat r β is a special bijection (see Definition 4.21). ENERIC NEWTON POLYGON FOR EXPONENTIAL SUMS IN TWO VARIABLES WITH TRIANGULAR BASE45
Completion of the proofs.Notation 5.46.
For a bijection β : Y Ñ m p Y q , we write L β : “ L Y β ´ p m p L qq . Recall that we defined the meaning of two bijections β, β : Y Ñ m p Y q to be relatedin Definition 4.26. Proposition 5.47. p a q A bijection β : Y Ñ m p Y q is related to r β if and only if p q P ´ β p P q “ P ´ r β p P q or p P ´ r β p P qq _ for all P in L ; p q β ˇˇ L “ r β ˇˇ L ; p q β p P q “ m p P q for P P Y zp L Y L β q ; p q β is symmetric. p b q The number of bijections related to r β is equal to k , where k “ P P L | P ´ p P ´ r β p P qq _ P m p Y q ( . Proof of Proposition 5.47. “ ùñ ”. It is straightforward.“ ðù ”. By the construction of r β , we know that P ´ β p P q is not diagonal for each point P in L ; and β p Q q´ Q is not diagonal for each point Q in m p L q . On the other hand, since β and r β are related, there are exact 2 ‚ L non-diagonal vectors in ! P ´ β p P q ˇˇ P P Y ) .Therefore, we have(5.20) ! P ´ β p P q ˇˇ P P L β ) ‹ “ ! P ´ r β p P q ˇˇ P P L r β ) ‹ . Recall that we denote V p β q “ t ~v , ~v , . . . u . Assume that β does not satisfy Prop-erty (1). We put i be the smallest number such that there exists some point P whichsatisfies P ´ r β p P q “ ~v i or ~v _ i and P ´ β p P q ‰ ~v i or ~v _ i . It is easy to see that for each point Q in m p Y q X D k , there exists at most one vector ~v in V p β q Y V p β q _ such that P ` ~v belongs to L . Combining it with Lemma 5.44 allows usto induce a injection β : L Ñ m p Y q from r β such that β p P q “ β p P q if P “ P ; β p P q else . It is easy to check that β is greater than r β with respect to “ ă ”, a contradiction. Therefore, β satisfies Property (1).Apply the same argument to m ˝ β ˝ m , we know that β p Q q ´ Q “ r β ´ p Q q ´ Q or p r β ´ p Q q ´ Q q _ for all Q in m p L q . Recall that we define the partial order “ ă ” and V ‹ p β q in Definitions 5.6 and 5.7. One cancheck that V ‹ p r β | Y z L r β q is actually the only maximal element in the set ! V ‹ p β q ˇˇ β : Y z L r β Ñ m p Y z L r β q and β is diagonal ) with respect to the partial order “ ă ”.As a corollary of Lemma 5.44, we know that for each k , there does not exist two pointsin D k such that one is from L and the other is from β p L q . Therefore, if we put β to bea bijection from Y to m p Y q such that β is an element in ! β : Y Ñ m p Y q | L β “ L β & β is diagonal ) , which maximizes V ‹ p β q with respect to “ ă ”, then we have β | L “ r β | L . Moreover, we can check that V ‹ p β q ĺ V ‹ p r β q and the equation hold if and only if L β isweakly symmetric and β p P q “ m p P q for each P P Y zp L Y L β q . On the other hand, since β and r β are related, we know that V ‹ p β q “ V ‹ p r β q . Hence,we have β | Y z L β “ β | Y z L β , which implies that β satisfies Property (2) and (3).Then we are left to show that β | L β is symmetric. First, from the argument above,we know that it is weakly symmetric. Therefore, for any point P in L , if we put P : “ m p β p P qq , we know that β p P q P m p L q . Since we checked Property (1) already, we knowthat P ´ β p P q P V p β q Y V p β q _ . As the argument above, there are at most one vector ~v in V p β q Y V p β q _ such that P ´ ~v P m p L q , which obviously is P ´ m p P q . Therefore,we know that β is symmetric.(b) It follows directly from (a). (cid:3) Proof of Proposition 4.28. (1) By Proposition 5.47, we check that r β constructed in (5.19)is exactly the needed r β in this proposition. Moreover, we know that the integer i in thisproposition is equal to P P L | P ´ p P ´ r β p P qq _ P m p Y q ( . (2) From Lemma 4.27, we know that two special bijections contribute a same monomialto r v sp h p T q in Lemma 4.24 if and only if they are related. By Corollary 4.27 and part (1) ofthis proposition, we have(5.21) r v h p T q “ i ś P P T x ś i “ b P,τ p r β q ,i ! x ź i “ r a ř P P T b P,τ p r β q ,i Q i ` “other terms” , where “other terms” is a power series in Z p r r a s which contains no term like r a ř P P T b P,τ p r β q ,i Q i .Since for any P P T and any 1 ď i ď x , we know that b P,τ p r β q ,i in (5.21) is less than p , wecomplete the proof of this proposition. (cid:3) References [AS] A. Adolphson and S. Sperber, Exponential sums and Newton polyhedra: Cohomology and estimates.
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