F. Halter-Koch

Władysław Narkiewicz

Journal de Théorie des Nombres de Bordeaux

JOURNAL DE THÉORIE DES NOMBRES DE BORDEAUXF. HALTER-KOCHWŁADYSŁAW NARKIEWICZPolynomial mappings defined by forms witha common factorJournal de Théorie des Nombres de Bordeaux, tome 4, no 2 (1992),p. 187-198<http://www.numdam.org/item?id=JTNB_1992__4_2_187_0>© Université Bordeaux 1, 1992, tous droits réservés.L’accès aux archives de la revue « Journal de Théorie des Nombresde Bordeaux » (http://jtnb.cedram.org/) implique l’accord avec les condi-tions générales d’utilisation (http://www.numdam.org/conditions). Toute uti-lisation commerciale ou impression systématique est constitutive d’uneinfraction pénale. Toute copie ou impression de ce fichier doit conte-nir la présente mention de copyright.Article numérisé dans le cadre du programmeNumérisation de documents anciens mathématiqueshttp://www.numdam.org/187-Polynomial mappings definedby forms with a common factorpar F. HALTER-KOCH AND W. NARKIEWICZ1. Introduction and PreliminariesFor a field Ii, we denote by K a fixed algebraic closure of K. For a pointx = (x1, ... , xn~ E kn, we setA subset V C K’~ is called K-homogeneous if there exist homogeneouspolynomials Hl,..., Hr E ~~~X1, ... , Xn~ such thatFor m , n e N and polynomials E .~~ ~X 1, ... , X ~ ~, we definethe polynomial mappingand we setWe consider the following finiteness property, which a field Il may haveor not:Manuscrit reçu le 29 novembre 1991, r6vis6 le 3 d6cembre 1992.188An algebraically closed or real closed field clearly never has the property(PO). As the main result of this paper, we shall prove that every finitelygenerated field has the property (PO) (Theorem 3). This is well known ifFo = 1 and additionally either global field, cf. [5], [6].In [1] we also dealt with the case Fo = 1; there we considered a propertywhich is somewhat weaker than (PO) and proved it for a larger class offields. We also discussed the various hypotheses stated in (PO), we outlinedthe history of the problem, and we gave several further references.The case 1 was up to now only considered in a rather special case,cf. [7]. The following example (due to the referee) shows that the inequalityin (2) cannot be weakened in a trivial way: let K be an infinite field,m = n = 3, V = k3, F = (hence Fo = X3), G -(XI,X2,X3) and X = I x, z E K" ~ C V; then (1), (3) and (5)are obviously satisfied, and (4) holds since (x, y, z) = F(x-1 z, x-lz, x3z-2 );however, deg~(F) = deg*(G) + 2deg(Fo), and X is infinite.The main tools in proving that a field has the property (PO) are thetheory of height functions (to be explained in § 2), the theory of placesof algebraic function fields (§ 3) and the following set-theoretical Lemma,already proved in [1].MAIN LEMMA. Let X, 3) be sets and F, G : ~ -&#x3E; ~ mappings such thatG(~). Let J : X --t R be a mapping such that C R is discrete,and let C &#x3E; 0 be a real constant with the property that x , y E 3C, f (x) &#x3E; Cand F(x) = G(y) implies f (y) &#x3E; f (x). If189then D G(Xc). If moreover 3lc is finite and G is injective, thenX=Xc.2. HeightsIn this section, we collate the necessary facts from the theory of heightfunctions, following [8].Let I( be a field, equipped with a family of absolute valuessatisfying the product formula. We regard M¡( as a set of places suchthat, for every v E is an absolute value defining v; we denote byIiv the completion of K at v.For a finite extension field L of let ML be the set of all places of Lextending those of M¡(. If v E ML and w v, thenwhere Lw is a finite-dimensional local lev-algebra with residue field L.We define an absolute value )’ w of L byThen (I - satisfies the product formula.The family M = gives rise to a height function H : f,7n --, R,defined for every n E N, as follows: for x = (xl, ... , E l’(n, let L be afinite extension of K such that x E Ln, and setThanks to the normalization this definition does not depend on thechoice of the field L. We call H = HM the height function associated withM.For a monic polynomialof degree d E fi1, we set190LEMMA 1. Let K be a field, equipped with a family of absolute valuessatisfying the product formula, and let H be the associated height function.If n e N, x E .Kn and z E then we haveassertion is equivalent withand hence also withSince obviouslyand (by the product formula)it is sufficient to observe that 0  a  1 and b &#x3E; 0 implies L~The next Lemma gives information about the behaviour of heights underpolynomial mappings. ,LEMMA 2. Let il be a field, equipped with a family of absolute valuessatisfying the product formula, and let H be the associated height function.Let F = (Fl , ... , Fm ) ; Î(n --~ ¡em be a polynomial mapping, defined byhomogeneous polynomials Fl,. - ., F~ E ~~~X1, ... , Xn~, d* = deg*(F) andd* = deg*(F).191i) There exists a constant C1 E such that, for all x E 7~~~ii) Let V C be a I( -homogeneous set such that F1, ... , F~.,.L do nothave a common non-trivial zero in V. Then there exists a constantC2 E such that, for all x E V,Proof.i) follows almost immediately from the definition of the height function,cf. [8, 2.3].ii) is proved in [5, Proposition 2]. 0PROPOSITION 1. be a field, equipped with a family of absolutevalues satisfying the product formula, and let H be the associated heightfunction.For m, n (Fl,... 1 Fm) 7 G - (Gi Gm) : k- , Î(mbe polynomial mappings defined by homogeneous polynomials Fj , Gj ELet Fo E be a common factor of Fl,..., say FOF§ where Fj E .~[Xl, ... , Xn~, and suppose that deg*(F) &#x3E;deg*(G) + 2 deg(Fo). Let V C Pi’ be a K-homogeneous set such thatFi, ... , F,~ do not have a common non-trivial zero in V.i) There exists a constant Co E R&#x3E;o such that x , y E V, H(x) &#x3E; Coand F(x) = G(y) implies H(y) &#x3E; H(x).ii) Let Co be a constant satisfying i) , ~ C V a subset such that G(X), GIX is injective, and Xo = f x I H(x)  Co} is finite.Then X is finite,.and Lemma 1 implies192for all x E By Lemma 2, there exist constants Cl , C2 E R&#x3E;o suchthat x , y E ~ and F(x) = G(y) impliesand hence H(y) &#x3E; H(x), provided that H(x) is large enough.ii) follows immediately from i) and the Main Lemma, applied for F , G :~E-~~t’n and f =H:~-~I~. C7THEOREM 1. Let il be a field, equipped with a family of absolute valuessatisfying the product formula, and let H be the associated height function.Suppose that for every C E R&#x3E;o and n E fiT, the setis finite. Then Ii has the property (PO). In particular, every global fieldhas the property (PO).Proof. The first assertion follows immediately from Proposition 1. If Il isa global field (i. e., either an algebraic number field or an algebraic functionfield in one variable over a finite field), then we use the finiteness resultsfrom [8, 2.5]. 0We close this section with two further Lemmas concerning heights tobe used in § 3. For simplicity, we restrict ourselves to the case of non-archimedian absolute values.LEMMA 3. Let I~ be a field, equipped with a family of non-archimedeallabsolute values satisfying the product formula, and let H be the associatedheight function.i) For monic polynomials f , g E we have H( fg)  H( f )H(g~.ii) then H(x - y)  H(x)H(y).193Proof.i) is obvious; we even have equality, cf. [2, ch. 3, Prop. 2.4].ii) Set x = (x1, ... , xn ), y = ( yi , ... , yn ), and let L be a finite extensionof Ii such that x, y E Ln. Then we haveLEMMA 4. be a field, equipped with a family of non-archimedeanabsolute values satisfying the product formula, and let H be the associatedheight function.Let d , n E h1, and let x = (x1, ... , xn) E be such that dI~(x)  d.For i E ~l, ... , n~, let fi E be the minimal polynomial of xi, andf = fi -...-fn E Then we havewhere E Ie are the conjugates of xi (counted with multiplicity in theinseparable case) and di ~ d. Since H(X - = H(xi ), weobtain, using Lemma 3,1943. Algebraic function fieldsWe start with some preliminaries on height functions and places of alge-braic function fields; for the general theory of places cf. [3, ch. I].Let Kl = Ko(t) be a rational function field over an arbitrary field Iio ,and let P be the set of all monic irreducible polynomials in Forp E P, we define an absolute value j’ jp as follows: Ifwhere c E ~~ vp 6 Z and vp = 0 for almost all thenThen () ’ is a family of non-archimedean absolute values sat-isfying the product formula. The corresponding height function will bedenoted by Ha; we call it the canonical height function for If x e 1(1’ then x = (al,8-I,...,an,B-I), where E {3 is monic and an , ,8) = 1; this impliesand hencewhereEvery r E ko defines a place ~T : K, --~ ko by 0,(h) = 1~ (T ) . Aplace 0: - is called a I(o- place if OIK, = §r for some 7 E -f!Co.Obviously, every (T E Î(o) extends to a I(o-place of K(I. For a ICO-place0 of I(l, we denote by R~ _ C the local ring associated with 0.We extend the ring homomorphism 0: R -- to homomorphisms of thepolynomial rings 0 : (by acting on thecoefficients) and to mappings 15: (by acting on the components).195PROPOSITION 2. Let Iio be a field which is neither algebraically closed norreal closed, let K, = be a rational function field over and Ha thecanonical height function.For every there exists a ~~1 -~ Pio with thefollowing property:and Z C a subset satisfyingthen we have Z C R’, and ~~,~ : .~ ~ injective.Proof. Let r E I(o be an element satisfyingand let 0 : Kl 2013 ko U {oo} a place extending 0,. We assert that 0 has therequired property.Let m e N and Z C K’’n be a subset such that Ha(z) + + m  Mfor all z E .~. Puttingit is sufficient to prove that ,~* C Rm, and ~(x) ~ 0 for all 0 ~ x E ~*.For x E Z*, we have M2, and Ha(x)  M2 by Lemma 3. Forx = (xl, ... , x."i ), we denote by Ii E the minimal polynomial of ~2,and we setwhere ao, ... , ~3 E is monic and gcd,Lemma 4 implies  - -and therefore196integral over Rp, and since Ro is integrally closed, we obtain x E If x # 0, then we 0 for some j m~ and 0 forsome i E (0, ... , d -1 ~. Let I E f 0,..., d -1 } be minimal such tha,t 0;then we obtainDividing by x~ and multiplying with 0 yields an equationSince ~(al ) - = 0, we obtain 0 and hence ~(x) ~0. DTHEOREM 2. Let 1(0 be a fields, and suppose that every finite extensionfield o~’ I(o has the property (PO). Then every finitely generated extensionfield has the property (PO).Proof. It is sufficient to prove that every finitely generated extension fieldof transcendence degree 1 over the property (PO); the general casefollows by an obvious induction on the transcendence degree.Let = Ilo(t) be a rational function field over and let 7~ be a finiteextension of We shall prove that K has the property (PO); obviously,we have k = kl. Let Ho be the canonical height function associated with~~o (t). ,Let F = (F1, ... , Fm ), G = (G1, ... , Gm ) : Pi,’ --~ lem be polynomialmappings defined by homogeneous polynomials Fj, Gj E 7~[Xi,...,~].Let Fo E .~~~X1, ... , X~~ be a common factor of say Fj =where F~ E ~i ~X1, ... , Xn~, and suppose that deg*(F) &#x3E; deg*(G) +2 deg(Fo). Let V C be a ]( - homogeneous set such that F1, ... , F£ donot have a common non-trivial zero in V, and let X C V be a subset suchthat and GIX is injective. By Proposition 1, there exists aconstant Co E R&#x3E;o such that x , y Ha(x) &#x3E; Co and F(x) = G(y)implies Ha(y) &#x3E; Ha(x). Putting197the Main Lemma yields F(%o) D G(Xo)? and according to Proposition 1 itis sufficient to prove that Xo is finite. By Lemma 2, there exists a constantCl E R&#x3E;o such thatfor all x ~ Xo.Let H1, ... , Hr E ]([XI,... , Xn] be homogeneous polynomials such thatnI .~l~x~ - ... - Hrtx~ - 0~. Since Fl’,..., F., HI,...,Hrdo not have a common non-trivial zero in Pi n it follows from Hilbert’s Null-stellensatz [9, §130] that there exist polynomials Xnsuch that - -for some exponent q E N and all v E {I,... , n}.Let D be the (finite) set of all coefficients of the polynomials Fo , Fj, Hj , Fjv and Since has the property (PO), it is neither alge-braically closed nor real closed, and Proposition 2 applies: there exists a.~1’ -~ U such that D C RO, Xo C Rn, ol D U ~0~ isinjective and is injective.There exist elements yl , ... , yk E Ro such that ICO (t, YI, Yk)and consequently ~(~i ) = L U {oo}, where L = 1(0 ( Ø( t), 0(yl),.. - , L is a finite extension of Ilo, thus it has the property (PO). We use theplace 0 to shift the whole situation from K to L. We have i = thepolynomials Fo , F) , Hj , Fjv and H~v have coefficients in andconsequently their 0-images have coefficients in L.We consider the polynomial mappings ~(F) - (~(F1 ), ... , ~(F~ )) and; by construction one has:then Vo C kg is a K-homogeneous set, and sincethe polynomials ~(Fi ), ... , ~(F;L ) do not have a common non-trivial zeroin VO.198Obviously, we havecontend thatand weare both injective. Indeed, if x, y Ethen 4&#x3E;(G(x)) = §(G(y)) and henceboth injective.Since L has the property (PO), we conclude that is finite, andhence Xo is also finite. 0THEOREM 3. Every finitely generated field has the property (PO).Proof. By Theorem 2, it suffices to prove that every finite extension of aprime field has the property (PO). For finite fields, this is obvious; foralgebraic number fields, it follows from Theorem 1. 0REFERENCES[1] F. Halter-Koch, W. Narkiewicz, Finiteness properties of polynomial mappings, toappear. [2] S. Lang, Fundamentals of Diophantine Geometry, Springer 1983.[3] S. Lang, Introduction to algebraic geometry, Interscience Publ. 1958.[4] D. J. Lewis, Invariant sets of morphisms in projective and affine number spaces, J.Algebra 20 (1972), 419-434.[5] P. Liardet, Sur les transformations polynomiales et rationnelles, Sém. Th. Nomb.Bordeaux exp. n° 29, 1971-1972.[6] P. Liardet, Sur une conjecture de W. Narkiewicz, C. R. Acad. Sc. Paris 274 (1972),1836-1838.[7] W. Narkiewicz, On transformations by polynomials in two variables, II, Coll. Math.13 (1964), 101-106.[8] J.-P. Serre, Lectures on the Mordell-Weil-Theorem, Aspects of Mathematics, Braunschweig 1989.[9] B. L. van der Waerden, Algebra, 2. Teil, Springer 1967.Franz Halter-KochInstitut fur MathematikKarl- Franz ens- Universi tåtHehlridhstrafle 36/IVA-8010 Graz, 4sterreichWladyslaw NarkiewiczInstytut MatematycznyUniwersytet WroclawskiPlac Grunwaldzki 2/4PL-50-384 Wroclaw, Polen

Polynomial mappings defined by forms with a common factor

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Polynomial mappings defined by forms with a common factor

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Journal de Théorie des Nombres de Bordeaux