Effective Nullstellensatz for strictly regular sequences

by

Adelina FABIANO, Arkadiusz PŁOSKI and Piotr TWORZEWSKI

IMUJ PREPRINT 1999/21 KRAKÓW

Summary. Sharp versions of the Nullsellensatz for a class of polynomial sequences are given.

1. Main results. Let F_{1}, …, F_{k} : ℂ^{n} → ℂ, 1 ⩽ k ⩽ n be a sequence of nonconstant polynomials. Let d_{i} = deg F_{i} for i = 1, …, k.

Definition 1.1. The sequence F_{1}, …, F_{k} is strictly regular (s.r.) if there are linear forms L_{1}, …, L_{n−k} : ℂ^{n} → ℂ such that the mapping

(F_{1}, …, F_{k}, L_{1}, …, L_{n−k}) : ℂ^{n} → ℂ^{n}

is proper.

Remarks 1.2.

(1) If k = 1 then nonconstant F_{1} is a s.r. sequence.

(2) If k = n then F_{1}, …, F_{n} is a s.r. sequence if and only if (F_{1}, …, F_{n}) : ℂ^{n} → ℂ^{n} is a proper mapping.

(3) Let F_{i}^{∗} : ℂ^{n} → ℂ be a homogeneous form of degree d_{i} such that we have F_{i} = F_{i}^{∗} + (terms of degree < d_{i}) for i = 1, …, k. Suppose that the algebraic set (F_{1}^{∗}, …, F_{k}^{∗})^{−1}(0) is of dimension n − k, then F_{1}, …, F_{k} is a s.r. sequence.

(4) If F_{1}, …, F_{k} is a s.r. sequence then (F_{1}, …, F_{k})^{−1}(0) is of pure dimension n−k. In particular the algebraic cycle of zeroes Z_{(F_{1},... ,F_{k})} is defined (cf.[7]).

AMS Classification: 14E99, 32B99 Key words : proper polynomial mapping, geometric degree.

Work partially supported by KBN Grant 2 P03A 001 15.

Theorem 1.3. Let F_{1}, …, F_{k} : ℂ^{n} → ℂ be a s.r. sequence. If G : ℂ^{n} → ℂ is a nonconstant polynomial mapping such that G|(F_{1}, …, F_{k})^{−1}(0) = 0, then there is a μ > 0 such that

G^{μ} = A_{1}F_{1} + … + A_{k}F_{k},

with polynomials A_{1}, …, A_{k} : ℂ^{n} → ℂ such that deg(A_{i}F_{i}) ⩽ ( ∏_{i=1}^{k} d_{i})(deg G) for

i = 1, …, k

Remarks 1.4.

(1) Theorem 1.3 is a generalization to the case of s.r. sequence of the main result of [3].

(2) From the given formulation of the theorem it follows immmediately that

μ ⩽ ∏_{i=1}^{k} d_{i}.

Theorem 1.5. Let F_{1}, …, F_{k} : ℂ^{n} → ℂ be a s.r. sequence. If F_{k+1} : ℂ^{n} → ℂ is a polynomial of degree d_{k+1} > 0 such that (F_{1}, …, F_{k+1})^{−1}(0) = ∅, then

A_{1}F_{1} + … + A_{k+1}F_{k+1} = 1,

with polynomials A_{1}, …, A_{k+1} : ℂ^{n} → ℂ such that deg(A_{i}F_{i}) ⩽ d_{1} · . . . · d_{k+1} for i = 1, …, k + 1

Remarks 1.6.

(1) If degrees of polynomials in Theorem 1.5 are different than 2, then theorem follows from Kollar‘s theorem (see [5]).

(2) If k = n then [3] gives a better result.

2. Proofs. The proofs of Theorem 1.3 and Theorem 1.5 are based on the following proposition.

Proposition 2.1 (cf. [3], [6]). Suppose that H = (H_{1}, …, H_{n}) : ℂ^{n} → ℂ^{n} is a proper polynomial mapping of geometric degree μ. Then for every G : ℂ^{n} → ℂ of positive degree there exists a polynomial

P_{G}(W, T) = T^{μ} + P_{G}^{(1)}(W)T^{μ−1} + … + P_{G}^{(μ)}(W),

in n + 1 variables (W, T) = (W_{1}, …, W_{n}, T) such that:

(1) P_{G}(H_{1}, …, H_{n}, G) = 0

(2) For every w ∈ ℂ^{n} we denote by ζ^{(1)}(w), …, ζ^{(μ)}(w) the sequence of all points of F^{−1}(0) counted with multiplicities. Then

P_{G}(w, T) = ∏_{j=1}^{μ}(T − G(ζ^{(j)}(w))).

(3) weight(P_{G}) ⩽ ( ∏_{i=1}^{n} d_{i})(deg G), if weight(W_{i}) = deg H_{i} = d_{i} for i = 1, …, n

and weight(T) = deg G.

Now let F_{1}, …, F_{k} : ℂ^{n} → ℂ be a s.r. sequence of polynomials. Let us fix linear forms L_{1}, …, L_{n−k} : ℂ^{n} → ℂ such that the mapping

H = (F_{1}, …, F_{k}, L_{1}, …, L_{n−k}) : ℂ^{n} → ℂ^{n}.

is proper.

Proof of Theorem 1.3 Let P (W, T) = P_{G}(W, T). From Prop. 2.1 (2) and from the assumption G|(F_{1}, …, F_{k})^{−1}(0) = 0 it follows that P (0, w_{k+1}, …, w_{n}, T) = T^{μ} for every (w_{k+1}, …, w_{n}) ∈ ℂ^{n−k}. This means that P (0, W_{k+1}, …, W_{n}, T) = T^{μ}, and consequently P (W, T) = T^{μ} + W_{1}P_{1}(W, T) + … + W_{k}P_{k}(W, T) with

weight(W_{j}P_{j}(W, T)) ⩽ weight P (W, T) ⩽ (∏_{i=1}^{n} d_{i})(deg G) for j = 1, …, k .

Substitution F_{1}, …, F_{k}, G for W_{1}, …, W_{k}, T gives

G^{μ} = (−P_{1}(F, L, G))F_{1} + … + (−P_{k}(F, L, G))F_{k},

with

deg(−P_{i}(F, L, G)F_{i}) ⩽ weight(−P_{i}(W, T)W_{i}) ⩽ (∏_{i=1}^{k} d_{i})(deg G),

and the proof is complete.

Proof of Theorem 1.5 Let P (W, T) = P_{G}(W, T) and G = F_{k+1}. Since P (H, G) = 0 then we get

−(G^{μ} + P^{(1)}(F)G^{μ−1} + ⋯ + P^{(μ−1)}(F)G) = P^{(μ)}(H),

This gives

(∗) − F_{k+1}A_{k+1} = P^{(μ)}(H)

We have P^{(μ)}(w) = ± ∏_{j=1}^{μ} F_{k+1}(ζ^{(j)}(w)), consequently P^{(μ)}(0, w_{1}, …, w_{k}) ≠ 0 in ℂ^{n−k} and we may assume

P^{(μ)}(0, w_{k+1}, …, w_{n}) = −1 .

Then P^{(μ)}(W) = W_{1}A_{1}(W) + … + W_{k}A_{k}(W) − 1 with

weight(W_{k}A_{k}) ⩽ weight(P^{(μ)}) ⩽ d_{1} · . . . · d_{k} .

Substitution gives P^{(μ)}(H) = F_{1}A_{1}(F) + … + F_{k}A_{k}(F) − 1 with deg(F_{i}A_{i}(F)) ⩽ d_{1} · . . . · d_{k+1}, for i = 1, …, k. By (∗) we get A_{1}F_{1} + … + A_{k+1}F_{k+1} = 1 with deg(F_{i}A_{i}(F)) ⩽ d_{1} · . . . · d_{k+1}, for i = 1, …, k + 1. This ends the proof.

3. Examples. Let F = (F_{1}, …, F_{k}) : ℂ^{n} → ℂ^{k} be polynomial mapping. One can compare properties of the mapping F with the condition that F_{1}, …, F_{k} is a s.r. sequence.

It is easy to see that if F_{1}, …, F_{k} is a s.r. sequence then the mapping F is surjective and has all fibres of pure dimension n − k. In the next example we observe that this assumption with additional complete intersection of components of F is not enough for F_{1}, …, F_{k} to be a s.r. sequence.

Example 3.1. Consider polynomials:

F_{1} : ℂ^{3} ∋ (x, y, z) → x ∈ ℂ,

F_{2} : ℂ^{3} ∋ (x, y, z) → xy^{2} + y ∈ ℂ.

It is easy to see that F_{1}, F_{2} is a complete intersection and all fibres of F has dimension 1. We observe that F_{1}, F_{2} is not a s.r. sequence. Let us take a linear form L(x, y, z) = ax + by + cz and suppose that the mapping H = (F_{1}, F_{2}, L) is proper. Then obviously c ≠ 0 and we can assume c = 1. Taking the sequence h_{ν} = (ν^{−1}, −ν, bν − aν^{−1}), we have |h_{ν}| → +∞ (ν → +∞). Observe that we obtain H(h_{ν}) = (ν^{−1}, 0, 0) → 0 (ν → +∞), and so (F_{1}, F_{2}) is not a s.r. sequence.

Our next example says that an additional condition that all the fibres have the same degree (with multiplicity) is too strong in this situation.

Example 3.2. Consider polynomials:

F_{1} : ℂ^{3} ∋ (x, y, z) → y ∈ ℂ,

F_{2} : ℂ^{3} ∋ (x, y, z) → z + x^{2}y ∈ ℂ.

It is easy see that F_{1}, F_{2} is a s.r. sequence since the mapping (F_{1}, F_{2}, L) is a polynomial automorphism for L(x, y, z) = x. But deg F^{−1}((0, 0)) = 1 and deg F^{−1}((1, 0)) = 2.

DIPARTIMENTO DI MATEMATICA, UNIVERSITA DELLA CALABRIA, ARCAVACATA DI RENDE (CS) (AF) E-mail: fabiano@ccuws4.unical.it

DEPARTMENT OF MATHEMATICS, TECHNICAL UNIVERSITY, AL. TYSIA̧CLECIA PAŃSTWA POLSKIEGO 7, 25-314 KIELCE (AP) E-mail: mat-ap@srv1.tu.kielce.pl

INSTITUTE OF MATHEMATICS, JAGIELLONIAN UNIVERSITY, REYMONTA 4, 30-059 KRAKÓW (PT) E-mail: tworzews@im.uj.edu.pl

REFERENCES

[1] W. D. Brownawell, Bound for the degrees in the Nullstellensatz, Annals of Math., 126 (1987), 577–591.

[2] L. Caniglia, A. Galligo, J. Heinz, Borne simple exponentielle pour le degrés dans le théorème des zéros sur un corps de charactéristique quelconque, C. R. Acad. Sci. Paris, t. 307, Serié 1 (1988), 255–258.

[3] P. Cassou - Noguès, A. Płoski, Un théorème des zéros effectif, Bull. Pol. Ac. Math., 44 (1996), 61–70.

[4] A. Fabiano, G. Pucci, A. Yger, Effective Nullstellensatz and geometric degree for zerodimensional ideals, Acta Aritmetica 78.2 (1996), 165–187.

[5] J. Kollàr, Sharp effective Nullstellensatz, J. Amer. Math. Soc., 1 (1988), 963–975.

[6] A. Płoski, On the growth of proper polynomial mappings, Ann. Polon. Math., 45 (1985), 297–309.

[7] A. Płoski, P. Tworzewski, Effectve Nullstellensatz on Analytic and Algebraic Varieties Bull. Pol. Ac. : Math.46.1 (1998), 31–38.

[8] B. Shiffman, Deegre Bounds for the Division Problem in Polynomial Ideals, Michigan Math. J., 36 (1989), 163–171.

[9] B. Teissier, Résultats récents d’algébre commutative effective, Séminaire Bourbaki 1989-90 n^{o} 718, Astérisque 189-190 107–131.