NORMAL FAMILIES OF MEROMORPHIC MAPPINGS OF SEVERAL COMPLEX VARIABLES FOR MOVING HYPERSURFACES IN A COMPLEX PROJECTIVE 3 SPACE 1 0 2 GERDDETHLOFFAND DO DUC THAI AND PHAMNGUYENTHU TRANG n a J 4 Abstract. The main aim of this article is to give sufficient con- V] ditions for a family of meromorphic mappings of a domain D in Cn into PN(C) to be meromorphically normal if they satisfy only C someveryweakconditionswithrespecttomovinghypersurfacesin . h PN(C), namely that their intersections with these moving hyper- t a surfaces, which may moreover depend on the meromorphic maps, m are in some sense uniform. Our results generalize and complete [ previous results in this area, especially the works of Fujimoto [2], Tu [19], [20], Tu-Li [21], Mai-Thai-Trang [6] and the recent work 1 of Quang-Tan [10]. v 7 8 6 1. Introduction. 0 . 1 0 Classically, a family of holomorphic functions on a domain D C F ⊂ 3 is said to be (holomorphically) normal if every sequence in contains 1 F a subsequence which converges uniformly on every compact subset of : v D to a holomorphic map from D into P1. i X In 1957 Lehto and Virtanen [5] introduced the concept of normal r a meromorphic functions in connection with the study of boundary be- haviour of meromorphic functions of one complex variable. Since then normal families of holomorphic maps have been studied intensively, re- sultinginanextensive development intheonecomplexvariablecontext and in generalizations to the several complex variables setting (see [22], [3], [4], [1] and the references cited in [22] and [4]). The first ideas and results on normal families of meromorphic map- pings of several complex variables were introduced by Rutishauser [11] and Stoll [14]. The research of the authors is partially supported by a NAFOSTED grant of Vietnam (Grant No. 101.01.38.09). 1 2 GERD DETHLOFF ANDDO DUCTHAI AND PHAMNGUYEN THU TRANG The notion of a meromorphically normal family into the N-dimen- sional complex projective space was introduced by H. Fujimoto [2] (see subsection 2.5belowforthedefinitionoftheseconcepts). Also in[2], he gave some sufficient conditions for a family of meromorphic mappings of a domain D in Cn into PN(C) to be meromorphically normal. In 2002, Z. Tu [20] considered meromorphically normal families of mero- morphic mappings of a domain D in Cn into PN(C) for hyperplanes. Generalizing the above results of Fujimoto and Tu, in 2005, Thai-Mai- Trang [6] gave a sufficient condition for the meromorphic normality of a family of meromorphic mappings of a domain D in Cn into PN(C) for fixed hypersurfaces (see section 2 below for the necessary definitions): Theorem A. ([6, Theorem A]) Let be a family of meromorphic mappings of a domain D in Cn intoFPN(C). Suppose that for each f , there exist q 2N +1 hypersurfaces H (f),H (f),...,H (f) in 1 2 q PN∈(CF) with ≥ inf D(H (f),...,H (f));f > 0 and f(D) H (f)(1 i N+1), 1 q i ∈ F 6⊂ ≤ ≤ (cid:8) (cid:9) where q is independent of f, but the hypersurfaces H (f) may depend i on f, such that the following two conditions are satisfied: i) For any fixed compact subset K of D, the 2(n 1)-dimensional − Lebesgue areas of f 1(H (f)) K (1 i N + 1) with counting − i ∩ ≤ ≤ multiplicities are bounded above for all f in . F ii) There exists a closed subset S of D with Λ2n 1(S) = 0 such that − for any fixed compact subset K of D S, the 2(n 1)-dimensional − − Lebesgue areas of f 1(H (f)) K (N + 2 i q) with counting − i ∩ ≤ ≤ multiplicities are bounded above for all f in . F Then is a meromorphically normal family on D. F Recently, motivated by the investigation of Value Distribution The- ory for moving hyperplanes (for example Ru and Stoll [12], [13], Stoll [15], and Thai-Quang [16], [17]), the study of the normality of families of meromorphic mappings of a domain D in Cn into PN(C) for moving hyperplanes or hypersurfaces has started. While a substantial amount ofinformationhasbeenamassedconcerning thenormalityoffamiliesof meromorphic mappings for fixed targets through the years, the present knowledge of this problem for moving targets has remained extremely meagre. There are only a few such results in some restricted situations (see [21], [10]). For instance, we recall a recent result of Quang-Tan [10] which is the best result available at present and which generalizes Theorem 2.2 of Tu-Li [21]: NORMAL FAMILIES OF MEROMORPHIC MAPPINGS 3 Theorem B.(see [10, Theorem1.4])Let be a family of meromorphic mappings of a domain D Cn into PN(FC), and let Q , ,Q (q 1 q 2N+1) be q moving hypersu⊂rfaces in PN(C) in (weakly) gen·e·r·al positio≥n such that i) For any fixed compact subset K of D, the 2(n 1)-dimensional − Lebesgue areas of f 1(Q ) K (1 j N +1) counting multiplicities − j ∩ ≤ ≤ are uniformly bounded above for all f in . F ii) There exists a thin analytic subset S of D such that for any fixed compact subset K of D, the 2(n 1)-dimensional Lebesgue areas of − f 1(Q ) (K S) (N + 2 j q) regardless of multiplicities are − j ∩ − ≤ ≤ uniformly bounded above for all f in . F Then is a meromorphically normal family on D. F We would like to emphasize that, in Theorem B, the q moving hy- persurfaces Q , ,Q in PN(C) are independent on f (i.e. they 1 q ··· ∈ F are common for all f .) Thus, the following question arised na- ∈ F turally at this point: Does Theorem A hold for moving hypersurfaces H (f),H (f),...,H (f) which may depend on f ? The main aim of 1 2 q ∈ F this article is to give an affirmative answer to this question. Namely, we prove the following result which generalizes both Theorem A and Theorem B: Theorem 1.1. Let be a family of meromorphic mappings of a do- main D in Cn into PFN(C). Suppose that for each f , there exist q 2N + 1 moving hypersurfaces H (f),H (f),...,H∈ F(f) in PN(C) 1 2 q ≥ such that the following three conditions are satisfied: i) For each 1 6 k 6 q, the coefficients of the homogeneous polyno- mials Q (f) which define the H (f) are bounded above uniformly on k k compact subsets of D for all f in , and for any sequence f(p) , F { } ⊂ F there exists z D (which may depend on the sequence) such that ∈ infp N D(Q1(f(p)),...,Qq(f(p)))(z) > 0. ∈ (cid:8) (cid:9) ii) For any fixed compact subset K of D, the 2(n 1)-dimensional − Lebesgueareas of f 1(H (f)) K (1 i N+1) counting multiplicities − i ∩ ≤ ≤ are bounded above for all f in (in particular f(D) H (f) (1 i i F 6⊂ ≤ ≤ N +1)). iii) There exists a closed subset S of D with Λ2n 1(S) = 0 such − that for any fixed compact subset K of D S, the 2(n 1)-dimensional − − Lebesgue areas of f 1(H (f)) K (N+2 i q) ignoring multiplicities − i ∩ ≤ ≤ are bounded above for all f in . F 4 GERD DETHLOFF ANDDO DUCTHAI AND PHAMNGUYEN THU TRANG Then is a meromorphically normal family on D. F In the special case of a family of holomorphic mappings, we get with the same proof methods: Theorem 1.2. Let be a family of holomorphicmappings of a domain D in Cn into PN(CF). Suppose that for each f , there exist q 2N + 1 moving hypersurfaces H (f),H (f),...,H∈ (Ff) in PN(C) suc≥h 1 2 q that the following three conditions are satisfied: i) For each 1 6 k 6 q, the coefficients of the homogeneous polyno- mials Q (f) which define the H (f) are bounded above uniformly on k k compact subsets of D for all f in , and for any sequence f(p) , F { } ⊂ F there exists z D (which may depend on the sequence) such that ∈ infp N D(Q1(f(p)),...,Qq(f(p)))(z) > 0. ∈ (cid:8) (cid:9) ii) f(D) H (f) = (1 6 i 6 N +1) for any f F . i ∩ ∅ ∈ iii) There exists a closed subset S of D with Λ2n 1(S) = 0 such − that for any fixed compact subset K of D S, the 2(n 1)-dimensional − − Lebesgue areas of f 1(H (f)) K (N+2 i q) ignoring multiplicities − i ∩ ≤ ≤ are bounded above for all f in . F Then is a holomorphically normal family on D. F Remark 1.1. There are several examples in Tu [20] showing that the conditions in i), ii) and iii) in Theorem 1.1 and Theorem 1.2 cannot be omitted. We also generalise several results of Tu [19], [20], [21] which allow not to take into account at all the components of f 1(H (f)) of high − i order: The following theorem generalizes Theorem 2.1 of Tu-Li [21] from the case of moving hyperplanes which are independant of f to moving hypersurfaces which may depend on f (in fact observe that for mov- ing hyperplanes the condition H , ,H in T N is satisfied by 1 ··· q S { i}i=0 taking T ,...,T any (fixed or moving) N + 1(cid:0)hyperpl(cid:1)anes in general 0 N e position). Theorem 1.3. Let be a family of holomorphic mappings of a do- main D in Cn into PFN C . Let q > 2N +1 be a positive integer. Let m , ,m be positive (cid:0)int(cid:1)ergers or such that 1 q ··· ∞ q N 1 > N +1. (cid:18) − m (cid:19) Xj=1 j NORMAL FAMILIES OF MEROMORPHIC MAPPINGS 5 Suppose that for each f , there exist N + 1 moving hypersurfaces T f , ,T f in PN∈CFof common degree and there exist q mov- 0 N ··· ing(cid:0) h(cid:1)ypersurfa(cid:0)ces(cid:1) H f (cid:0), (cid:1) ,H f in T f N such that the 1 ··· q S { i }i=0 following conditions a(cid:0)re(cid:1)satisfied:(cid:0) (cid:1) (cid:0) (cid:0) (cid:1) (cid:1) e i) For each 0 6 i 6 N, the coefficients of the homogeneous poly- nomials P (f) which define the T (f) are bounded above uniformly on i i compact subsets of D, and for all 1 6 j 6 q, the coefficients b (f) ij of the linear combinations of the P (f), i = 0,...,N which define the i homogeneous polynomials Q (f) which define the H (f) are bounded j j above uniformly on compact subsets of D, and for any fixed z D, ∈ inf D Q f , ,Q f (z) : f > 0. 1 q ··· ∈ F (cid:8) (cid:0) (cid:0) (cid:1) (cid:0) (cid:1)(cid:1) (cid:9) ii) f intersects H f with multiplicity at least m for each 1 j q j j ≤ ≤ (see subsection 2.6 fo(cid:0)r (cid:1)the necessary definitions). Then is a holomorphically normal family on D. F The following theorem generalizes Theorem 1 of Tu [20] from the case of fixed hyperplanes to moving hypersurfaces (in fact observe that forhyperplanestheconditionH (f), ,H (f)in T (f) N issat- 1 ··· q S { i }i=0 isfied by taking T (f),...,T (f) any N +1 hyperpla(cid:0)nes in gene(cid:1)ral po- 0 N e sition). Theorem 1.4. Let be a family of meromorphic mappings of a do- main D in Cn intoFPN C . Let q > 2N + 1 be a positive integer. Suppose that for each f (cid:0) (cid:1) , there exist N + 1 moving hypersurfaces T f , ,T f in PN∈CFof common degree and there exist q mov- 0 N ··· ing(cid:0) h(cid:1)ypersurfa(cid:0)ces(cid:1) H f (cid:0), (cid:1) ,H f in T f N such that the 1 ··· q S { i }i=0 following conditions a(cid:0)re(cid:1)satisfied:(cid:0) (cid:1) (cid:0) (cid:0) (cid:1) (cid:1) e i) For each 0 6 i 6 N, the coefficients of the homogeneous poly- nomials P (f) which define the T (f) are bounded above uniformly on i i compactsubsets of D, and for all 1 6 j 6 q, the coefficients b (f) of the ij linear combinations of the P (f), i = 0,...,N which define the homo- i geneous polynomials Q (f) which define the H (f) are bounded above j j uniformly on compact subsets of D, and for any sequence f(p) , { } ⊂ F there exists z D (which may depend on the sequence) such that ∈ infp N D(Q1(f(p)),...,Qq(f(p)))(z) > 0. ∈ (cid:8) (cid:9) ii) For any fixed compact K of D, the 2(n 1)-dimensional Lebesgue − areas of f 1 H (f) K (1 k N +1) counting multiplicities are − k ∩ ≤ ≤ (cid:0) (cid:1) 6 GERD DETHLOFF ANDDO DUCTHAI AND PHAMNGUYEN THU TRANG bounded above for all f (in particular f D H f (1 k k ∈ F 6⊂ ≤ ≤ N +1)). (cid:0) (cid:1) (cid:0) (cid:1) iii) There exists a closed subset S of D with Λ2n 1(S) = 0 such that − for any fixed compact subset K of D S, the 2(n 1)-dimensional − − Lebesgue areas of z Suppν f,H (f) ν f,H (f) (z) < m K (N +2 k q) k k k ∈ ∩ ≤ ≤ (cid:8) (cid:0) (cid:1)(cid:12) (cid:0) (cid:1) (cid:9) ignoringmultiplicities fo(cid:12)rallf are boundedabove, where m q ∈ F { k}k=N+2 are fixed positive intergers or with ∞ q 1 q N +1 < − . m (cid:0)N (cid:1) X k k=N+2 Then is a meromorphically normal family on D. F The following theorem generalizes Theorem 1 of Tu [19] from the case of fixed hyperplanes to moving hypersurfaces. Theorem 1.5. Let be a family of holomorphic mappings of a do- main D in Cn intoFPN C . Let q > 2N + 1 be a positive integer. Suppose that for each f (cid:0) (cid:1) , there exist N + 1 moving hypersurfaces T f , ,T f in PN∈CFof common degree and there exist q mov- 0 N ··· ing(cid:0) h(cid:1)ypersurfa(cid:0)ces(cid:1) H f (cid:0), (cid:1) ,H f in T f N such that the 1 ··· q S { i }i=0 following conditions a(cid:0)re(cid:1)satisfied:(cid:0) (cid:1) (cid:0) (cid:0) (cid:1) (cid:1) e i) For each 0 6 i 6 N, the coefficients of the homogeneous poly- nomials P (f) which define the T (f) are bounded above uniformly on i i compactsubsets of D, and for all 1 6 j 6 q, the coefficients b (f) of the ij linear combinations of the P (f), i = 0,...,N which define the homo- i geneous polynomials Q (f) which define the H (f) are bounded above j j uniformly on compact subsets of D, and for any sequence f(p) , { } ⊂ F there exists z D (which may depend on the sequence) such that ∈ infp N D(Q1(f(p)),...,Qq(f(p)))(z) > 0. ∈ (cid:8) (cid:9) ii) f(D) H (f) = (1 6 i 6 N +1) for any f . i ∩ ∅ ∈ F iii) There exists a closed subset S of D with Λ2n 1(S) = 0 such that − for any fixed compact subset K of D S, the 2(n 1)-dimensional − − Lebesgue areas of z Suppν f,H (f) ν f,H (f) (z) < m K (N +2 k q) k k k { ∈ }∩ ≤ ≤ (cid:0) (cid:1)(cid:12) (cid:0) (cid:1) (cid:12) NORMAL FAMILIES OF MEROMORPHIC MAPPINGS 7 ignoringmultiplicities forallf in areboundedabove, where m q F { k}k=N+2 are fixed positive intergers and may be with ∞ q 1 q N +1 < − . m (cid:0)N (cid:1) X k k=N+2 Then is a holomorphically normal family on D. F Let us finally give some comments on our proof methods: The proofs of Theorem 1.1 and Theorem 1.2 are obtained by gen- eralizing ideas, which have been used by Thai-Mai-Trang [6] to prove TheoremA,tomovingtargets, whichpresentsseveralhighlynon-trivial technical difficulties. Among others, for a sequence of moving targets H(f(p)) which at the same time may depend of the meromorphic maps f(p) : D PN C , obtaining a subsequence which converges locally → uniformly on D(cid:0)is(cid:1)much more difficult than for fixed targets (among others we cannot normalize the coefficients to have norm equal to 1 everywhere like for fixed targets). This is obtained in Lemma 3.6, after having proved in Lemma 3.5 that the condition D(Q ,...,Q ) > δ > 0 1 q forces a uniform bound, only in terms of δ, on the degrees of the Q , i 1 i q (in fact the latter result fixes also a gap in [6] even for the ≤ ≤ case of fixed targets). The proofs of Theorem 1.3, Theorem 1.4 and Theorem 1.5 are ob- tained by combining methods used by Tu [19], [20] and Tu-Li [21] with the methods which we developed to prove our first two theorems. How- ever, in order to apply the technics which Tu and Tu-Li used for the case of hyperplanes, we still need that for every meromorphic map f(p) : D PN C , the Q (f(p)),...,Q (f(p)) are still in a linear system 1 q → given by N +(cid:0)1 s(cid:1)uch maps P (f(p)),...,P (f(p)). The Lemmas 3.11 to 0 N Lemma 3.14 adapt our technics to this situation (for example Lemma 3.14 is an adaptation of our Lemma 3.6) 2. Basic notions. 2.1. Meromorphic mappings. Let A be a non-empty open subset of a domain D in Cn such that S = D A is an analytic set in D. Let f : A PN(C) be a holomorphic map−ping. Let U be a non-empty → ˜ connected open subset of D. A holomorphic mapping f 0 from U into CN+1 is said to be a representation of f on U if f(z6≡) = ρ(f˜(z)) for all z U A f˜ 1(0), where ρ : CN+1 0 PN(C) is the − canonical∈proje∩ction−. A holomorphic mapping−f {: A} →PN(C) is said → 8 GERD DETHLOFF ANDDO DUCTHAI AND PHAMNGUYEN THU TRANG to be a meromorphic mapping from D into PN(C) if for each z D, ∈ there exists a representation of f on some neighborhood of z in D. 2.2. Admissible representations. Letf beameromorphic mapping of a domain D in Cn into PN(C). Then for any a D, f always has ∈ ˜ an admissible representation f(z) = (f (z),f (z), ,f (z)) on some 0 1 N ··· neighborhoodU ofainD,whichmeansthateachf (z)isaholomorphic i function on U and f(z) = (f (z) : f (z) : : f (z)) outside the 0 1 N ··· analytic set I(f) := z U : f (z) = f (z) = ... = f (z) = 0 of 0 1 N { ∈ } codimension 2. ≥ 2.3. Moving hypersurfaces in general position. Let D be a do- main in Cn. Denote by the ring of all holomorphic functions D H on D, and [ω , ,ω ] the set of all homogeneous polynomials D 0 N H ··· Q [ω , ,ω ] such that the coefficients of Q are not all iden- D 0 N ∈ H ·e·· tically zero. Each element of [ω , ,ω ] is said to be a moving D 0 N hypersurface in PN(C). H ··· e Let Q be a moving hypersurface of degree d > 1. Denote by Q(z) the homogeneous polynomial over CN+1 obtained by evaluating the coefficients of Q in a specific point z D. We remark that for generic z D this is a non-zero homogenous p∈olynomial with coefficients in C. Th∈e hypersurface H given by H(z) := w CN+1 : Q(z)(w) = 0 (for genericz D)isalsocalledtobeamov{ing∈hypersufaceinPN(C)w}hich ∈ is defined by Q. In this article, we identify Q with H if no confusion arises. We say that moving hypersurfaces Q q of degree d (q > N+1) { j}j=1 j in PN(C) are located in (weakly) general position if there exists z D ∈ such that for any 1 6 j < < j 6 q, the system of equations 0 N ··· Q (z) ω , ,ω = 0 ji 0 ··· N (cid:26) 0 6 i 6(cid:0) N (cid:1) has only the trivial solution ω = 0, ,0 in CN+1. This is equivalent ··· to (cid:0) (cid:1) 2 2 D(Q ,...,Q )(z) := inf Q (z)(ω) + + Q (z)(ω) > 0, 1 q 1≤j0<Y···<jN≤q||ω||=1(cid:18)(cid:12)(cid:12) j0 (cid:12)(cid:12) ··· (cid:12)(cid:12) jN (cid:12)(cid:12) (cid:19) where Q (z)(ω) = a (z).ωI and ω = ω 2 1/2. j I =dj jI || || | j| | | P (cid:0)P (cid:1) NORMAL FAMILIES OF MEROMORPHIC MAPPINGS 9 2.4. Divisors. Let D be a domain in Cn and f a non-identically zero holomorphic function on D. For a point a = (a ,a ,...,a ) D we 1 2 n ∈ expand f as a compactly convergent series ∞ f(u +a ,.....,u +a ) = P (u ,...,u ) 1 1 n n m 1 n mX=0 on a neighborhood of a, where P is either identically zero or a homo- m geneous polynomial of degree m. The number ν (a) := min m;P (u) 0 f m { 6≡ } is said to be the zero multiplicity of f at a. By definition, a divisor on D is an integer-valued function ν on D such that for every a D there ∈ are holomorphic functions g(z)( 0) and h(z)( 0) on a neighborhood 6≡ 6≡ U of a with ν(z) = ν (z) ν (z) on U. We define the support of the g h − divisor ν on D by Supp ν := z D : ν(z) = 0 . { ∈ 6 } We denote +(D) = ν : a non-negative divisor on D . D { } Let f be a meromorphic mapping from a domain D into PN C . For each homogeneous polynomial Q [ω , ,ω ], we define(cid:0)th(cid:1)e D 0 N ∈ H ··· divisor ν f,Q on D as follows: For each a D, let f = f , ,f e ∈ 0 ··· N be an ad(cid:0)missib(cid:1)le representation of f in a neighborhood U(cid:0)of a. Then(cid:1) e we put ν f,Q (a) := ν (a), Q(f˜) (cid:0) (cid:1) ˜ where Q(f) := Q f , ,f . 0 N ··· (cid:0) (cid:1) Let H be a moving hypersurface which is defined by the homoge- neous polynomial Q [ω , ,ω ], and f be a meromorphic map- D 0 N ping of D into PN C∈.HAs abo·v·e· we define the divisor ν(f,H)(z) := e ν f,Q (z). Obviou(cid:0)sly(cid:1), Suppν(f,H) is either an empty set or a pure (n(cid:0) 1)(cid:1) dimensional analytic set in D if f(D) H (i.e., Q(f˜) 0 on − − 6⊂ 6≡ U). We define ν(f,H) = on D and Suppν(f,H) = D if f(D) H. ∞ ⊂ Sometimes we identify f 1(H) with the divisor ν(f,H) on D. We can − rewrite ν(f,H) as the formal sum ν(f,H) = n X , where X are i i i iP∈I the irreducible components of Suppν(f,H) and n are the constants i ν(f,H)(z) on X Reg(Suppν(f,H)), where Reg( ) denotes the set of i ∩ all the regular points. We say that the meromorphic mapping f intersects H with multi- plicity at least m on D if ν(f,H)(z) m for all z Suppν(f,H) and ≥ ∈ 10GERD DETHLOFF ANDDO DUCTHAI AND PHAMNGUYEN THU TRANG inparticular that f intersects H withmultiplicity onD if f(D) H ∞ ⊂ or f(D) H = . ∩ ∅ 2.5. Meromorphically normal families. Let D be a domain in Cn. i)(See [1]) Let bea familyof holomorphicmappings ofD into acom- F pact complex manifold M. is said to be a (holomorphically) normal F family on D if any sequence in contains a subsequence which con- F verges uniformly on compact subsets of D to a holomorphic mapping of D into M. ii) (See [2]) A sequence f(p) of meromorphic mappings from D into PN(C) is said to conver{ge m}eromorphically on D to a meromorphic mapping f if and only if, for any z D, each f(p) has an admissible ∈ representation f˜(p) = (f(p) : f(p) : ... : f(p)) 0 1 N (p) on some fixed neighborhood U of z such that f converges uni- { i }∞p=1 formly on compact subsets of U to a holomorphic function f (0 i ≤ ˜ i N) on U with the property that f = (f : f : ... : f ) is a 0 1 N ≤ representation of f on U (not necessarily an admissible one ! ). iii) (See [2]) Let be a family of meromorphic mappings of D into PN(C). is saidFto be a meromorphically normal family on D if any F sequence in has a meromorphically convergent subsequence on D. F iv) (See [14]) Let ν be a sequence of non-negative divisors on D. i { } It is said to converge to a non-negative divisor ν on D if and only if any a D has a neighborhood U such that there exist holomorphic ∈ functions h ( 0) and h( 0) on U such that ν = ν , ν = ν and h i 6≡ 6≡ i hi h { i} converges compactly to h on U. 2.6. Other notations. Let P , ,P be N +1 homogeneous poly- 0 N nomials of common degree in C[ω··,· ,ω ]. Denote by P N the 0 ··· N S { i}i=0 N (cid:0) (cid:1) set of all homogeneous polynomials Q = b P (b C). i i i ∈ i=0 P N Let Q := b P q be q (q > N +1) homogeneous polynomials { j ji i}j=1 i=0 P in P N . We say that Q q are located in general position in S { i}i=0 { j}j=1 P(cid:0) N if(cid:1) S { i}i=0 (cid:0) (cid:1) 1 6 j < < j 6 q, det b = 0. ∀ 0 ··· N jki 06k,i6N 6 (cid:0) (cid:1)