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THE NOETHER EXPONENT AND JACOBI FORMULA

arXiv:alg-geom/9305006v1 19 May 1993THE NOETHER EXPONENT AND JACOBI FORMULAby Arkadiusz P loskiAbstract. For any polynomial mapping F = (F1, .

. .

, Fn) of Cn with a finite num-ber of zeros we define the Noether exponent ν(F ). We prove the Jacobi formula forall polynomials of degree strictly less than Pni=1(deg Fi −1) −ν(F ).1.

The Noether exponentIf P = P(Z) is a complex polynomial in n variables Z = (Z1, . .

. , Zn) thenwe denote by ˜P = ˜P( ˜Z), ˜Z = (Z0, Z) the homogenization of P. If H is a set ofhomogeneous polynomials in n + 1 variables then we denote by V (H) the subset ofthe complex projective space Pn defined by equations H = 0,H ∈H.The polynomial mapping F = (F1, .

. .

, Fn) : Cn →Cn has a finite number ofzerosif the set V ( ˜F1, . .

. , ˜Fn) is finite.

We put V∞(F) = V ( ˜F1, . .

. , ˜Fn, Z0) andcall V∞(F) the set of zeros of F at infinity.

We identify Cn and Pn \ V (Z0). ClearyF has a finite number of zeros if and only if the sets F −1(0) ⊂Cn and V∞(F) ⊂Pnare both finite.Definition (1.1).

Let F = (F1, . .

. , Fn) be a polynomial mapping of Cn with afinite number of zeros.

By the Noether exponent of F we mean the smallest integerν ≥0 such that the homogeneous forms Zν0 , ˜F1, . .

. , ˜Fn satisfy Noether’s conditionat every point of the set V∞(F) (cf.

Appendix).If V∞(F) = ∅then ν(F) = 0.If V∞(F) ̸= ∅and the hypersurfaces meettransversally at any point of V∞(F), then ν(F) = 1. For any polynomial map-ping F = (F1, .

. .

, Fn) with a finite number of zeros we put µ(F) =Pz∈F −1(0)multz Fwhere multz F stands for the multiplicity of F at z. If F −1(0) = ∅then µ(F) = 0.Let di = deg Fi for i = 1, .

. .

, n.Proposition (1.2). If F has a finite number of zeros, then ν(F) ≤nQi=1di −µ(F).Proof.

We have ν(F) ≤max{ ( ˜F1, . .

. , ˜Fn)p : p ∈V∞(F) } (cf.

Appendix (A5)).On the other hand, by Bezout’s theoremPp∈V∞(F )( ˜F1, . .

. , ˜Fn)p =nQi=1di −µ(F) and(1.2) follows.Remark(1.3).

Let k = ♯V∞(F). Then a reasoning similar to the above shows thatν(F) ≤Qni=1 di −µ(F) −k + 1.Typeset by AMS-TEX1

2BY ARKADIUSZ P LOSKIProposition (1.4). Suppose that the polynomial mapping F = (F!, .

. .

, Fn) has afinite number of zeros and let P be a polynomial belonging to the ideal generatedby F1, . .

. , Fn in the ring of polynomials.

Then there exist polynomials A1, . .

. ,An such that P = A1F1 + · · · + AnFn with deg AiFi ≤deg P + ν(F) for i = 1,. .

. ,n.Proof.

The homogeneous forms Zν0 ˜P, ˜F1, . .

. , ˜Fn (ν = ν(F)) satisfy Noether’sconditions at every point of V ( ˜F1, .

. .

, ˜Fn), then by Noether’s Fundamental The-orem (cf. Appendix) there are homogeneous forms ˜A1, .

. .

, ˜An such that Zν0 ˜P =˜A1 ˜F1 + · · ·+ ˜An ˜Fn,deg( ˜Ai ˜Fi) = deg(Zν0 ˜P) = ν + deg P. We get (1.4) by puttingZ0 = 1.For any z = (z1, . .

. , zn) ∈Cn we put |z| = max(|z1|, .

. .

, |zn|). Recall that ifP : Cn →C is a polynomial of degree d then there exist a constant C > 0 such that|P(z)| ≤C|z|dfor |z| ≥1.Proposition (1.5).

Let F = (F1, . .

. , Fn) be a polynomial mapping with a finitenumber of zeros.

Then there exist positive constants C and R such that|F(z)| ≥C|z|min(di)−ν(F )for |z| ≥R.Proof. Since the fiber F −1(0) is finite then there are polynomials Pi(zi) ̸≡0 (i = 1,. .

. ,n) which belong to the ideal generated by F1, .

. .

, Fn in the ring of polynomials(cf. [4, p. 23]).

Let mi = deg Pi(zi). By (1.4) we can write Pi(zi) = Ai1F1 + · · · +AinFn,deg(AijFj) ≤mi + ν(F).

Hence there exist constants C > 0 and R ≥1such that|zi|mi ≤C|z|mi+ν(F )−min(di)|F(z)|if |zi| ≥R for some i ∈{ 1, . .

., n } and the proposition follows.Corollary (1.6). If ν(F) < minni=1(di) then F is proper i.e.lim|z|→∞|F(z)| = +∞.To end with let us note two corollaries of propositions (1.2), (1.4) and (1.5).Corollary (1.7).

Let F = (F1, . .

. , Fn) be a polynomial mapping with a finite num-ber of zeros.

Let µ = µ(F). Then(1.7.1) (cf.

[3], [10]) there is a constant C > 0 such that |F(z)| ≥C|z|µ−Q di+min(di) forlarge |z|. (1.7.2) (cf.

[11]) If P belongs to the ideal generated by F1, . .

. , Fn in the ring ofpolynomials, then P = A1F1 +· · ·+AnFn with deg(AiFi) ≤Qni=1 di −µ+deg Pfor i = 1, .

. .

,n.1. The Jacobi formulaLet F = (F1, .

. .

, Fn) be a polynomial mapping such that the fiber F −1(0) isfinite and let G : Cn →C be a polynomial. We denote by resF,z(G) the residue ofthe meromorphic differential formG(z)F1(z) .

. .

Fn(z)[dZ],[dZ] = dZ1 ∧· · · ∧dZn.The definition and all properties of residues we need are given in [5]. Let usrecall that if the Jacobian JF = det( ∂Fi∂Zj ) is different from zero at z ∈F −1(0), thenresF,z(G) = G(z)JF (z).

The main result of this note is

THE NOETHER EXPONENT AND JACOBI FORMULA3Theorem (2.1). Suppose that the polynomial mapping F = (F0, .

. .

, Fn) : Cn →Cn has a finite number of zeros. Then the Jacobi formula(J)Xz∈F −1(0)resF,z(G) = 0is satisfied for all polynomials G : Cn →C of degree strictly less than Pni=1(di −1)−ν(F).Before giving the proof of (2.1) let us make some remarks.

If F has no zeros atinfinity i.e. if V∞(F) = ∅then ν(F) = 0 and (2.1) is reduced to the Griffiths–Jacobitheorem (cf.

[5]). In [1], [2], [6] and [7] there are given another generalisations of theJacobi theorem.

However, these results does not imply ours. If ν(F) ≥Pni=1(di−1)then the unique polynomial satisfying the assumption of (2.1) is G ≡0.Proof of (2.1).

Let Ωbe the meromorfic form in Pn given in Cn by formulaΩ=G(z)F1(z) . .

. Fn(z)[dZ].By Residue Theorem for Pn we getXz∈F −1(0)resF,z(G) = −Xp∈V∞(F )resp Ω.It suffices to show, that resp Ω= 0 for all p ∈V∞(F).Let W = (W1, .

. .

, Wn) be an affine system of coordinates in an affine neigh-bourhood of p such that W1 = 0 is the hyperplane at infinity and p has co-ordinates (0, . .

. , 0).Without loss of generality we may assume that Z1 =1W1 ,Z2 = W2+c2W1, .

. .

, Zn = Wn+cnW1.Let P ∗(W) = W d1 P( 1W1 , W2+c2W1, . .

. , Wn+cnW1) for any polynomial P(Z) of degree d.A simple calculation shows that near p ∈Pn:Ω=−W ν1 G∗(W)F ∗1 (W) .

. .

F ∗n(W)[dW],ν =nXi=1(di −1) −1 −deg G.By assumptions ν ≥ν(F), therefore W ν1 G∗(W) belongs to the local ideal generatedby F ∗1 (W), . .

. , F ∗n(W).

Consequentlyresp Ω= res0−W ν1 G∗(W)F ∗1 (W) . .

. F ∗n(W)[dW]= 0and we are done.Corollary 2.1.

If F −1(0) ̸= ∅then ν(F) ≥Pni=1(di −1) −deg JF .Proof. If F −1(0) ̸= ∅thenPz∈F −1(0)resF,z(JF ) = µ(F) ̸= 0, consequently we cannothave deg JF < Pni=1(di −1) −ν(F).

4BY ARKADIUSZ P LOSKICorollary 2.2.(cf. [2]) If the hypersurfaces ˜Fi = 0 (1 ≤i ≤n) meet transversallyat infinity then (J) holds for any polynomial of degree strictly less than Pni=1(di −1) −1.Proof.

If ˜Fi = 0 (1 ≤i ≤n) meet transversally then ν(F) ≤1 and (2.2) followsimmediately from (2.1).Corollary 2.3. Suppose that for any p ∈V∞(F):(i) the hypersurfaces ˜Fi = 0 (1 ≤i ≤n) have distinct tangent cones at p,(ii) ordp ˜Fi ̸= di(1 ≤i ≤n).Then (J) holds for any polynomial of degree ≤n −2.Proof.

By (A6) we have ν(F) ≤max{nPi=1(ordp ˜Fi −1) + 1 : p ∈V∞(F) } ≤nPi=1(di−2)+1 because ordp( ˜Fi) ≤di−1 by (ii). Consequently Pni=1(di−1)−ν(F) ≤n −1 and it suffices to use (2.1).Example.

Let F(Z1, Z2) = (Zd11−1, Z1Z2 + Zd22 )(d1 ≥1, d2 ≥2).Thencondition (i) is satissfied but (ii) fails.We havePz∈F −1(0)resF,z(1) = −1, hencecondition (ii) is essential.Appendix: Noether’s Conditions.Let H0, H1, . .

. , Hn be homogeneous forms of n + 1 variables such that theset V = V (H1, .

. .

, Hn) is finite. We denote by Op the ring of holomorfic germsat p ∈Pn.

Let di = deg Hi for 0 ≤i ≤n.Max Noether’s Fundamental Theorem. The following two conditions are equiv-alent:(A1) There is an equation H0 = A1H1 + .

. .

AnHn (with Ai forms of degree d0 −di). (A2) For any p ∈V there is an linear form L such that V ∩V (L) = ∅and H0Ld0 ∈ H1Ld1 , .

. .

, HnLdnOp.The proof of Noether’s theorem follows easily (cf. [4, p. 120]) from the affineversion of the theorem (cf.

[12]) and from the following(A3) Property. If H is a homogeneous form of n + 1 variables such that V ∩V (H) = ∅, then H is not a zero-divisor modulo ideal generated by H1, .

. .

, Hn inthe ring of polynomials.Proof of (A3). If V ∩V (H) = ∅then H1, .

. .

, Hn, H form the sequence of param-eters in the local ring O of holomorfic functions at 0 ∈Cn+1, consequently H is nota zero-divisormod (H1, . .

. , Hn)O. Whence follows easily (A3).We say that the sequence H0, H1, .

. .

, Hn satisfies Noether’s conditions at p ∈Vif (A2) holds true. Let (H1, .

. .

, Hn)p denotes the intersection number of H1, . .

. ,Hn at p and let ordp H be the order of H at p. We have the followingCriteria for Noether’s conditions.

The sequence H0, . .

. , Hn satisfiesNoether’s conditions at p ∈V if any of the following are true:(A4) H1, .

. .

, Hn meet transversally at p and p ∈V (H0),(A5) ordp H0 ≥(H1, . .

. , Hn)p,

THE NOETHER EXPONENT AND JACOBI FORMULA5(A6) H1, . .

. , Hn have distinct tangent cones at p and ordp H0 ≥Pni=1(ordp Hi−1)+1.Proof.

(A5) follows from the Mutiplicity theorem (cf. [8, p. 258]), (A6) is provedin [9] (Theorem 2.3), (A4) is a special case both of (A5) and (A6).References1.

A. Berenstein, A. Yger, Une formule de Jacobi et ses cons´equences, Ann. Scient.

´Ec. Norm.Sup.

4e s´erie 24 (1991), 363–377.2. G. Biernat, On the Jacobi–Kronecker formula for a polynomial mappping having zeros atinfinity, (to appear).3.

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Acad. Math.

31 (1983), 115–120.4. W. Fulton, Algebraic Curves (An Introduction to Algebraic Geometry), New York–Amster-dam, 1969.5.

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Math. 35 (1976), 321–390.6.

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Math. 381 (1987), 181–204.8.

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Letters XL (2), 72 (1990), 23–29.10., An inequality for polynomial mappings, Bull. Pol.

Acad. Math., (to appear).11.

B. Schiffman, Degree Bounds fot the Division Problem in Polynomial Ideals, Michigan Math.J. 36 (1989), 163–171.12.

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Sci. Letters 7 (1989), 1–3.Wyk ladnik Noethera i formu la JacobiegoStreszczenie.Dla ka˙zdego odwzorowania wielomianowego F = (F1, .

. .

, Fn) przestrzeni Cn o sko´nczonejliczbie zer definiujemy wyk ladnik Noethera ν(F ) a nast ιepnie dowodzimy formu ly Jacobiego dlawielomian´ow stopnia mniejszego od Pni=1(deg Fi −1) −ν(F ).Department of Mathematics,Technical University,Al. 1000 LPP 7, 25–314 Kielce,Poland

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