Valuations compatible with a projection

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1 ADERNOS DE MATEMÁTIA 05, October (2004) ARTIGO NÚMERO SMA# 205 Valuatons compatble wth a projecton Fuensanta Aroca * Departamento de Matemátca, Insttuto de êncas Matemátcas e de omputação, Unversdade de São Paulo - ampus de São arlos, axa Postal 668, São arlos SP, Brazl E-mal: fuen@cmc.usp.br Gven an N-dmensonal germ of analytc hypersurface H, a fnte projecton π : H N and a valuaton ν on the rng the rng of convergent seres n N varables, we study the valuatons on the rng O H that extend π ν. All these valuatons are descrbed when ν s a monomal valuaton whose weght vector s not orthogonal to any of the faces of the Newton Polyhedron of the dscrmnant of the projecton π. Ths descrpton s done n terms of the Puseux parameterzatons of H wth exponents n a cone. October, 2004 IM- USP. INTRODUTION Let H be an rreducble germ of analytc hypersurface at the orgn n N+, let π : H ( N, 0) be a fnte projecton and denote by R the rng of convergent seres n N varables. Gven a valuaton ν : R R 0 { }, we want to descrbe all the valuatons ν : O H R 0 { } that extend ν. That s, that make the dagram O H π ν R ν R 0 { } () commute. In ths note all these valuatons are descrbed when ν s a monomal valuaton whose weght vector s not orthogonal to any of the faces of the Newton Polyhedron of the dscrmnant of the projecton π. Ths descrpton s done n terms of the Puseux parameterzatons of H. The queston was posed to me by Bernard Tesser at the congress Sngularty theory and Applcatons held at Sapporo n I thank Danel Levcovtz for frutful dscussons durng the preparaton of ths note. * The research was carred out whle the author was proftng from a DTI fellowshp gven by IM-AGIMB processo 3846/2004- (NV) 237 Sob a supervsão Pq/IM

2 238 F. AROA In all what follows H wll be a germ of analytc hypersurface embedded n ( N+, 0) defned by F (x,..., x N, y) = 0, where F = y d + a d y d + a 0 s an rreducble polynomal of degree d n y and a R. The projecton π : H N (x,..., x N, y) (x,..., x N ) (2) wll be supposed to be fnte. The dscrmnant of the projecton π wll be denoted by δ. That s δ(x,..., x N ) = Resultant y (F, y F ). The deal of R[y] generated by F wll be denoted by I. So that O H = R[y] I the natural ncluson R R[y] I. and π s 2. MONOMIAL VALUATIONS Gven a Laurent seres ϕ = α Z N a α x α, the set of exponents of ϕ s the set E(ϕ) := {α Z N a α 0}. When E(ϕ) s fnte, ϕ s a Laurent polynomal. When E(ϕ) s contaned n the frst orthant, ϕ s a seres wth non-negatve exponents. A subset of R N of the form σ = {λ v + + λ r v r λ R 0 } for some v,..., v r Q N s called a (ratonal convex polyhedral) cone. A cone s sad to be strongly convex when t contans no non-trval lnear subspace. Let σ R N be a strongly convex cone. The set of formal Laurent seres wth exponents n σ s a rng wth the natural sum and product. [[σ]] = {ϕ E(ϕ) σ} Defnton 2.. The Newton polyhedron of a seres φ R s the convex hull n R N of the set E(φ) + R N 0 and s denoted by NPφ. Let V be a vertex of NPφ. The cone of NPφ assocated to V s the cone σ V = {v R N (V + λv) NPφ for some postve real number λ}. Remark 2.. The cone σ V s always strongly convex and contans the postve orthant. Sob a supervsão da Pq/IM

3 LIFTING VALUATIONS 239 Now for k N consder the rng of seres [[σ]] := k α ( k Z)N σ a α x α a α. When k dvdes k there s a natural ncluson [[σ]] [[σ]]. So, t makes sense to k k consder the rng of formal Puseux seres wth exponents n σ [[σ]] := k N [[σ]] k, We wll work only wth cones σ contanng the frst orthant, for such σ we have the followng dagram of nclusons: R [[σ]] Fr([[σ]]) [[σ]] Fr([[σ]] ) where Fr(A) stands for the feld of fractons of A. The dual of a cone σ s the cone σ := {v R N u v 0, u σ}. A cone σ R N s strongly convex f and only f the nteror of σ (as a subset of R N ) s not empty. Defnton 2.2. Let σ be a strongly convex cone. Gven w σ the map ν w : [[σ]] R { } defned by ν w φ = s a valuaton called the monomal valuaton wth weght w. mn w α (3) α E(φ) By restrcton, ν w nduces a valuaton n all subrngs of [[σ]] that we wll stll denote by ν w. The extenson of ν w from R to [[σ]] s not unque. Anyhow the followng holds: Lemma 2.. There exsts a unque way to extend ν w from the rng of seres wth exponents n σ to the rng of Puseux seres wth exponents n σ. Proof: The feld extenson [[σ]] = [[σ]][x [[σ]][t,..., t N ] k k,..., xn k ] = ({t k x } =,...,N ) Fr ([[σ]]) Fr ([[σ]]) (x k,..., xn k ) Sob a supervsão Pq/IM

4 240 F. AROA s a normal algebrac extenson. Its Galos group s formed by the automorphsms g (j,...,j N ) : x k ξ j x k ; ξ k =, (j,..., j N ) N N. by a theorem of Ostrowsk and Krull [4, F,Theorem ] two extensons of a gven valuaton are conjugate. Snce ν w (g (j,...,j N )(φ)) = ν w φ we have the result. Remark Let [σ] denote the rng of Laurent polynomals wth exponents n σ. That s [σ] = {ϕ [[σ]] #E(ϕ) < }. Then [[σ]] s the completon of [σ] wth respect to the valuaton ν w for any w n the nteror of σ. The feld of fractons of [σ] s the same as the feld of fractons of [x,..., x N ]. Then, [[σ]] s a rng of dmenson N. 3. PARAMETRIZATIONS WITH EXPONENTS IN A ONE. J. McDonald showed n [3] that gven a polynomal P n [x,..., x N ][y], for any w R N of ratonally ndependent coordnates, there exsts a strongly convex cone σ, wth w σ such that P has a root n [[σ]]. Moreover he gves an algorthm to compute such seres. Then, P. González Pérez showed n [2] that σ may be chosen to be a cone of the Newton polyhedron of the dscrmnant of P wth respect to y. Defnton 3.. s the set where τ(z,..., z N ) = ( z,..., z N ). Let σ R N be a cone. For ϱ (R >0 ) N, the σ-wedge of polyradus ϱ W(σ, ϱ) := {z } ( ) N ; τ(z) u ϱ u, u σ Z N Denote by δ the dscrmnant of F wth respect to y. For each vertex V of the Newton Polyhedron of δ let σ V be the cone of NPδ assocated to V (Defnton 2.). By [, Proposton 5.], there exsts ϱ V (R >0 ) N such that the σ V -wedge of polyradus ϱ V does not ntersect the zero locus of δ. Defnton 3.2. A connected component of π (W(σ V, ϱ V )) H wll be called a σ V - branch of H. Gven a σ V -branch of H the degree of the coverng π : W(σ V, ϱ V ) wll be denoted by. Remark 3.. Let B V of F n y. be the set of σ V -branches of H, d = B V, d beng the degree The followng proposton s proved n []: Sob a supervsão da Pq/IM

5 LIFTING VALUATIONS 24 Proposton 3.. Let V be a vertex of NPδ and let σ V be the cone of NPδ assocated to V. Gven a σ V -branch of H, there exst a σ V -wedge W and a seres ϕ [[σ V ]], convergent on W such that parameterzes. That s: Φ : W N+ (x,..., x N ) d (x d,..., x N, ϕ (x)) { (x,..., x N, ϕ (x)) (x,..., x N ) W } =. (4) Remark Let ζ be a prmtve -root of unty, the set of functons wth property (4) s { ϕ (ζ x,..., ζ N x N ) j {,..., } } } = {ϕ (),..., ϕ() and t has exactly elements. Remark The rreducblty of H mples that a polynomal h R s an element of I f and only f t vanshes on. That s h(x,..., x N, ϕ (x,..., x N )) = 0 whch s equvalent to h(x,..., x N, ϕ (x,..., x N )) = σ-branhes AND PRIMARY DEOMPOSITION. Let V be a vertex of NPδ and let be a σ V -branch of H. We wll denote by J the kernel of the morphsm [[σ V ]][y] [[σ V ]] h(x,..., x N, y) d h(x d,..., x N, ϕ (x)). where ϕ s as n proposton 3.. Snce [[σ V ]] s an ntegral doman, J s a prme deal of [[σ V ]][y]. By remark 3.3 we have I = J R[y] for any B V. Proposton 4.. Let I σ V be the extenson of I to the rng [[σ V ]][y] va the natural ncluson. Then I σ V = J. B V Sob a supervsão Pq/IM

6 242 F. AROA Proof: Set φ () 3.2. Each ϕ () := ϕ() (x d,..., x N ), =,..., where the ϕ () are as n remark d s a root of F as a polynomal n y. Then (y φ () ) dvdes F as an element of [[σ V ]] [y]. Ths, together wth remark 3., mples Let K () F = be the kernel of the morphsm B V = = (y φ () ). (5) By remark 3.3 we have [[σ V ]] [y] [[σ V ]] h(x, y) h(x, φ () (x)). J = K () [[σ V ]][y] for any {,..., }. (6) Let I σ V be the extenson of I to the rng [[σ V ]] [y] va the natural ncluson. Equaton (5) mples I σ V = K (), and the concluson follows from (6). B V = Proposton 4.2. The only prme deals P of [[σ V ]][y] wth the property P R[y] = I are of the form J wth B V. Proof: It follows from proposton 4. and remark THE THEOREM Gven w R >0 N, take a vertex V of the Newton polyhedron of δ such that w σ. Each σ V -branch of H nduces a valuaton that extends ν w. ν : O H R h(x,..., x N, y) ν w h(x,..., x N, ϕ ) Theorem 5.. Let w R >0 N be a vector non-orthogonal to any of the faces of the Newton Polyhedron of δ, and let V be the only vertex of NPδ such that w belongs to the dual of σ V. Then all the valuatons that extend ν w are the ones nduced by the σ V -branches of H. We start by provng a lemma: Sob a supervsão da Pq/IM

7 LIFTING VALUATIONS 243 Lemma 5.. Let w be a vector n the nteror of σ V, and let ν : O H R 0 { } be a valuaton that extends ν w. There exsts a σ V -branch of H and a valuaton ν : R { } that makes the dagram [[σ V ]][y] J commutatve. [[σ V ]] [[σ V ]][y] J O H ν w ν ν R { } Proof: An element h [[σ]][y] s wrtten as h = deg h =0 ψ y, where ψ = α Z N σ for some A Z N. For each {0,..., deg h} and j Z set := a () α x α. α Z N σ j w α < j + a () α x α Snce w s n the nteror of σ, for all j, the set {α Z N σ j w α < j + } s fnte and then the s are Laurent polynomals. We have ψ = j=0 for all {,..., deg h}. Let R[y] I be the localzaton of R[y] wth respect to I. Snce I R = {0}, we have R[y] I, j Z and {,..., deg h} Let ˆν : R I R be the extenson of the morphsm gven by the composton Snce ν extends ν w and ˆν {,..., deg h} R[y] O H R. ( ) ( Fr(R), we have ˆν = ν w ψ (K) y ν ) j. Then K + mn{0, ν(y) deg h}, K. (7) Sob a supervsão Pq/IM

8 244 F. AROA Set K τ K := ˆν j=0 {,..., deg h} Inequalty (7) mples that ether τ K K + mn{0, ν(y) deg h} for all K or there exsts K such that τ l = τ K for all l > K. Then t makes sense to defne: νh := lm τ K. K By constructon, ν(hh ) = ν(h) + ν(h ) and ν(h + h ) mn{ ν(h), ν(h )}, then ν nduces a valuaton ν : [[σ]][y] ν R. ( ) The deal ν ( ) s prme and ν ( ) R[y] = I. Then, by proposton 4.2 there exsts B V such that ν ( ) = J. Proof of theorem: Let and ν : [[σ]][y] J R { } be as n the lemma. Let K () be as defned n the proof of proposton 4., and let ν be an extenson of ν to [[σ V ]] [y] So, Gven h R[y], y. h(x, y) = h(x, φ () ) + (y φ() )g(x, y), where g(x, y) [[σ]]. K (). ν(h) = ν(h) = ν(h(x, φ () lemma 2. )) = ν w h(x, φ () )) = ν h. REFERENES. Fuensanta Aroca. Puseux parametrc equatons of analytc sets. Proc. Amer. Math., 32: , P. D. González Pérez. Sngulartés quas-ordnares torques et polyèdre de Newton du dscrmnant. anad. J. Math., 52(2): , John McDonald. Fber polytopes and fractonal power seres. J. Pure Appl. Algebra, 04(2):23 233, Paulo Rbenbom. Théore des valuatons. Sémnare de Mathématques Supéreures de l Unversté de Montréal, (Été 964). Les Presses de l Unversté de Montréal, Montreal, Que., 965. Sob a supervsão da Pq/IM