ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 2, Number 2, October 1996 S 1079-6762(96)00011-10

RESIDUES AND EFFECTIVE NULLSTELLENSATZ

CARLOS A. BERENSTEIN AND ALAIN YGER

(Communicated by Robert Lazarsfeld)

Abstract. Let K be a commutative field; an algorithmic approach to residue symbols defined on a Noetherian K-algebra R has been developed. It is used to prove an effective Nullstellensatz for polynomials defined over infinite factorial rings A equipped with a size. This result extends (and slightly improves) the previous work of the authors in the case A = Z.

Let p_{1}, …, p_{M} be polynomials in n variables with coefficients in an integral domain A, and respective degrees D_{1} ≥ D_{2} ≥ ⋯ ≥ D_{M}, with no common zeros in an integral closure of the quotient field K of A. It follows from the versions of the Hilbert Nullstellensatz in [B], [CGH], [Ko] that one can find an element r_{0} ∈ A\{0} and polynomials q_{j} ∈ A[x] such that

r_{0} = ∑_{j=1}^{M} q_{j}p_{j} (1)

with a priori estimates on the degrees

max_{j} deg q_{j} ≤ (3/2)^{ι}D_{1} ⋯ D_{μ},

where μ = min{n, M} and ι = max{j : 1 ≤ j < μ − 1, D_{j} = 2}. When A = Z, using analytic methods, and mainly integral representation formulas and multidimensional residues in C^{n}, one can show [BGVY, Section 5] that system (1) can be solved with the estimates

max_{j} deg q_{j} ≤ n(2n + 1)(3/2)^{ι} ∏_{j=1}^{μ} D_{j},

max h(q_{j}) ≤ κ(n)D_{1}^{4}( ∏_{j=1}^{μ} D_{j})^{8}(h + log M + D_{1} log D_{1}),

(2)

for the Mahler size h(q_{j}) in terms of the maximal Mahler size h of the original polynomials p_{j}. Recall that the Mahler size of p ∈ Z[x_{1}, …, x_{n}] is given by

h(p) = ∫_{[0,1]^{n}} log |p(e^{2πiθ_{1}}, …, e^{2πiθ_{n}})| dθ .

For A = Z, one can also recover the estimate for log |r_{0}|, implicit in (2), from the Arithmetic Bézout Theorem (see [Ph2],[BGS, Theorem 5.4.4]), which shows that

\begin{matrix}c \\ ◯ \end{matrix}1996 American Mathematical Society

Received by the editors April 15, 1996.

1991 Mathematics Subject Classification. Primary 14Q20; Secondary 13F20, 14C17, 32C30.

Key words and phrases. Effective Nullstellensatz, residues, arithmetic Bézout theory.

This research has been partially supported by grants from NSA and NSF.

the Faltings height H of the intersection of the arithmetic cycles X_{j} in P^{n}(Z) corresponding to the polynomials^{h}p_{j} (homogeneous versions of the original polynomials) has the bound

H ≤ c_{n}H(∏_{j=1}^{ν} D_{j})(1/H + ∑_{j=1}^{ν} 1/D_{j}),

for some constant c_{n}, where H := max_{j} H(X_{j}) and ν := min{n + 1, M}.

The origin of the results we announce here was the desire to obtain estimates similar to the Effective Nullstellensatz (2) for factorial rings A equipped with a size (in the sense of [Ph1]) and simultaneously eliminate Analysis from the proof of such results. Apart from the case where A = Z and the size is the Mahler size defined above, the natural example of such a ring is A = F_{p}[τ_{1}, …, τ_{q}], where p is a prime number and the size is the total degree in the parameters τ. The main idea is to keep the structure of the proof in [BY1], while eliminating all the analytic artifacts. In this process we develop the computational aspects for a Residue Calculus, where the residues are defined purely algebraically, following [H] and, especially, [L]. We are sure that this Residue Calculus will have many applications beyond the scope of this paper, in particular, the algorithms obtained permit the computation of residue symbols in the polynomial case, without appealing to elimination theory or using Gröbner bases. In particular, one should expect interesting geometric consequences of a generalized form of the Jacobi vanishing theorem obtained here, as one knows in the classical case [G],[Ku]. The reader will find complete proofs of these results in [BY2].

1. Residue calculus

Let R be a Noetherian K-algebra, where K is a commutative field. A sequence h_{1}, …, h_{n} generating an ideal I in R is said to be quasiregular [M] if whenever one has a relation of the form

∑_{k∈N^{n}:|k|=p} r_{k}h_{1}^{k_{1}} ⋯ h_{n}^{k_{n}} ∈ I^{p+1} , r_{k} ∈ R ,

for some p ∈ N, then all the r_{k} ∈ I. Given x_{1}, …, x_{n}, h_{1}, …, h_{n} in R such that (h_{1}, …, h_{n}) is a quasiregular sequence generating the ideal (h) = I and P := R/I is a finite-dimensional K-vector space, we follow Lipman [L, Chapter 3] to define the residue symbols as traces of certain K-linear operators from P into itself. Namely, let E = Hom_{K}(P, P) and σ : P → R be a K-linear section of the quotient map; then one can associate to any Q ∈ R the element Q^{♯} := ∑_{k∈N^{n}} q_{k}^{♯}h^{k} ∈ E[[h]], where the operators q_{k}^{♯} ∈ E are defined by the relations

Qσ(t) = ∑_{k∈N^{n}} σ(q_{k}^{♯}(t))h^{k} , t ∈ P ,

in the I-adic completion of R. Consider now the formal expansion in E[[h]],

Q^{♯} ∘ det (∂x_{i}^{♯}/∂h_{j}) = ∑_{k∈N^{n}} δ_{k}h^{k} ,

where the determinant is understood to take into account the noncommutativity of the operator product. Although the operators δ_{k} depend on the choice of the section σ, their traces do not. Now we can define the residue symbols as

Res [\begin{matrix}Qdx_{1} ∧ ⋯ ∧ dx_{n} \\ h_{1}^{k_{1}+1}, …, h_{n}^{k_{n}+1} \end{matrix}] = Res [\begin{matrix}Qdx \\ h^{k+\underline{1}} \end{matrix}] := Tr(δ_{k}), Q ∈ R, k ∈ N^{n} .

This definition extends the one given by Grothendieck [H]. In the polynomial case, computations of residues are usually performed by taking as a section σ the remainder with respect to a Gröbner basis of I. One can also show that in the analytic case (that is, R = C[x_{1}, …, x_{n}] or R = O, the ring of germs of holomorphic functions about 0 ∈ C^{n}) this definition of residue coincides with the usual one, defined, for instance, using Dolbeault cohomology [GH]. One can also see this using the following Transformation Law [L, Corollary 2.8].

Transformation Law. Let f = (f_{1}, …, f_{n}) and g = (g_{1}, …, g_{n}) be two quasiregular sequences in R, such that g = Af, where A is an n×n matrix with coefficients in R, and such that the quotients R/(f) and R/(g) are finite-dimensional K-vector spaces. Let Δ be the determinant of the matrix A; then, for any x_{1}, …, x_{n}, Q ∈ R,

Res [\begin{matrix}Qdx_{1} ∧ ⋯ ∧ dx_{n} \\ f_{1}, …, f_{n} \end{matrix}] = Res [\begin{matrix}QΔdx_{1} ∧ ⋯ ∧ dx_{n} \\ g_{1}, …, g_{n} \end{matrix}] .

Extending the base ring R with transcendental parameters, we can derive an extension of the Transformation Law.

Proposition 1. Let f = (f_{1}, …, f_{n}) and g = (g_{1}, …, g_{n}) be two quasiregular sequences in R, such that

g_{j} = ∑_{l=1}^{n} a_{jl}f_{l}, j = 1, …, n,

where the coefficients a_{jl} are in R and we let Δ be the determinant of the matrix A = [a_{jl}]. Then, for any x_{1}, …, x_{n}, Q ∈ R and any k ∈ N^{n},

Res [\begin{matrix}Qdx \\ f^{k+\underline{1}} \end{matrix}] = ∑_{\begin{matrix}|q_{;j}|=k_{j} \\ 1≤j≤n \end{matrix}} ∏_{i=1}^{n} (\begin{matrix}μ_{i} \\ q_{i;} \end{matrix}) Res [\begin{matrix}QΔ ∏_{1≤i,j≤n}(a_{ij})^{q_{i,j}}dx \\ g_{1}^{μ_{1}+1}, …, g_{n}^{μ_{n}+1} \end{matrix}] ,

where we have introduced the following notation for the matrix of indices q_{i,j} ∈ N:

q_{;j} = (q_{1,j}, …, q_{n,j}), q_{i;} = (q_{i,1}, …, q_{i,n}), μ_{i} = |q_{i;}| ,

and

(\begin{matrix}μ_{i} \\ q_{i;} \end{matrix}) = μ_{i}!/q_{i,1}! ⋯ q_{i,n}!.

This proposition was stated in the analytic case in [Ky]. The proof given there is not complete. One can also prove a very useful variant of the Transformation Law, which we have not been able to find in the literature.

Proposition 2. Let f_{0}, f_{1}, …, f_{n} be a regular sequence in R. Let g_{1}, …, g_{n} in R be such that the sequence (f_{0}, g_{1}, …, g_{n}) is quasiregular. Assume that there are nonnegative integers s_{1}, …, s_{n} and an n × n matrix A of elements in R such that

f_{0}^{s_{j}}g_{j} = ∑_{l=1}^{n} a_{jl}f_{l}, j = 1, …, n.

Let x_{0}, …, x_{n}, Q be elements of R; then, for any k_{0} ∈ N one has

Res [\begin{matrix}Qdx_{0} ∧ ⋯ ∧ dx_{n} \\ f_{0}^{k_{0}+1}, f_{1}, …, f_{n} \end{matrix}] = Res [\begin{matrix}QΔdx_{0} ∧ ⋯ ∧ dx_{n} \\ f_{0}^{k_{0}+1+|s|}, g_{1}, …, g_{n} \end{matrix}] ,

where |s| = s_{1} + ⋯ + s_{n} and Δ is the determinant of the matrix A.

In the case where R = K[x_{1}, …, x_{n}], which is our main interest here, one can extend the residue symbol to act on rational functions, as follows. Given a quasiregular sequence P_{1}, …, P_{n} in R, let Q_{1}/Q_{2} be a rational function such that the ideal (P_{1}, …, P_{n}, Q_{2}) is K[x_{1}, …, x_{n}]. Namely, we define

Res [\begin{matrix}(Q_{1}/Q_{2})dx_{1} ∧ ⋯ ∧ dx_{n} \\ P_{1}, …, P_{n} \end{matrix}] := Res [\begin{matrix}Q_{1}V dx_{1} ∧ ⋯ ∧ dx_{n} \\ P_{1}, …, P_{n} \end{matrix}] ,

where V is any polynomial such that for some U_{1}, …, U_{n} in R one has 1 = U_{1}P_{1} + ⋯ + U_{n}P_{n} + V Q_{2}. (This definition does not depend on the choice of V.)

In the previous transformation laws, one can replace Q by a rational function, provided the two residue symbols involved can be defined.

From now on, we deal only with the case R = K[x_{1}, …, x_{n}], where K is a field of arbitrary characteristic. Recall that a polynomial map P = (P_{1}, …, P_{n}) from K^{n} into itself is dominant if and only if K(x) is a finite-dimensional K(P)-vector space. It is proper if and only if K[x] is a finite type K[P]-module.

When P is dominant, the residue symbol

Res [\begin{matrix}Q(x)dx \\ P_{1} − u_{1}, …, P_{n} − u_{n} \end{matrix}]

(where u_{1}, …, u_{n} are n independent transcendental parameters) is an element of K(u). When P is proper, this symbol is a polynomial in u. In order to obtain more information about this polynomial, we shall assume that K is infinite, algebraically closed, and equipped with a nontrivial absolute value | · |. We consider the norm defined on K^{n} by |x| = max_{1≤i≤n} |x_{i}| . Recall that one can check properness by means of inequalities, namely, the map P is proper if and only if there exist three constants K, γ, δ > 0 such that

|x| ≥ K ⟹ |P (x)| ≥ γ |x|^{δ} . (3)

Any δ > 0 for which (3) holds is called a Łojasiewicz exponent for P.

Let P ∈ K[x_{1}, …, x_{n}]. We will denote as ^{h}P the homogeneous polynomial in n + 1 variables corresponding to the polynomial P in K[X_{0}, …, X_{n}], namely

hP (X_{0}, …, X_{n}) := X_{0}^{deg P} P (X_{1}/X_{0}, …, X_{n}/X_{0}).

As a consequence of the Lipman-Teissier theorem [LT] one has the following result.

Proposition 3. Let P = (P_{1}, …, P_{n}) be a proper polynomial map in K^{n} such that condition (3) holds for constants K, γ, δ, with δ ∈ N^{∗}. Assume that D = deg P_{j} for every j. Then, for any 1 ≤ j ≤ n, one can find a homogeneous polynomial R_{j} in two variables, with coefficients in K, distinguished in X_{j}, such that

(R_{j}(X_{0}, X_{j})X_{0}^{D−δ})^{n+1} ∈ I(^{h}P_{1}, …,^{h}P_{n}),

the homogeneous ideal generated by the ^{h}P_{j}.

This proposition, combined with the Transformation Law and residue calculus in one variable, leads to the following vanishing theorem for residue symbols.

Proposition 4. Let P_{1}, …, P_{n} be polynomials in K[x_{1}, …, x_{n}] with deg P_{j} = D, for 1 ≤ j ≤ n, which satisfy (3) with an integral exponent δ. Assume that

(1 − ϵ_{n})D < δ, for ϵ_{n} := 1/n(n + 1).

Then, for any (k_{1}, …, k_{n}) ∈ N^{n}, one has

deg Q ≤ n(n + 1)(|k| + n)(δ − (1 − ϵ_{n})D) − n − 1 ⟹ Res [\begin{matrix}Qdx \\ P^{k+\underline{1}} \end{matrix}] = 0.

Moreover, under the stronger hypothesis that

(1 − ϵ_{n}/(n + 1))D < δ,

one has

\begin{matrix}deg Q ≤ n(D − 1) \\ k ≠ 0 \end{matrix}\} ⟹ Res [\begin{matrix}Qdx \\ P^{k+\underline{1}} \end{matrix}] = 0. (4)

The case where D = δ corresponds to the situation where the P_{j} have no common zeros at ∞. In this case, (4) is a particular case (here all the polynomials have the same degree) of the Jacobi vanishing theorem. For the convenience of the reader, let us recall this theorem here [KK].

Jacobi vanishing theorem. Let P_{1}, …, P_{n} be elements of K[x_{1}, …, x_{n}] such that the homogeneous parts of highest degree define a power of the maximal ideal I(x_{1}, …, x_{n}). Then

deg Q ≤ ∑_{j=1}^{n} deg P_{j} − n − 1 ⟹ Res [\begin{matrix}Qdx \\ P_{1}, …, P_{n} \end{matrix}] = 0.

Using analytic methods, we were able to obtain in [BY1] a result similar to Proposition 4, without any restriction on the quotient δ/D. That result can be stated as follows. If P is a proper polynomial map from C^{n} to C^{n} such that the degrees D_{j} of the polynomials P_{j} are in decreasing order and δ is a Łojasiewicz exponent, then, for any polynomial Q ∈ C[x_{1}, …, x_{n}] and multiindex k, one has

(|k| + 2n − 1)δ > deg Q + D_{1} + ⋯ + D_{n−1} + n ⟹ Res [\begin{matrix}Qdx \\ P^{k+\underline{1}} \end{matrix}] = 0. (5)

This statement was crucial in the proof of the effective Nullstellensatz over C given in [BY1]. On the other hand, one can see that this vanishing theorem is not the best one could expect. For example, (5) implies that

deg Q < (2n − 1)δ − (D_{1} + ⋯ + D_{n−1}) − n ⟹ Res [\begin{matrix}Qdx \\ P_{1}, …, P_{n} \end{matrix}] = 0.

But a more careful analysis of the Bochner-Martinelli representation of the residue currents yields the statement

deg Q ≤ nδ − n − 1 ⟹ Res [\begin{matrix}Qdx \\ P_{1}, …, P_{n} \end{matrix}] = 0.

The point here is that this result depends on the Łojasiewicz exponent δ, but not on the degrees of the P_{j}. We do not know how to prove such result when K has positive characteristic, though it is possibly true. Nevertheless, Proposition 4, which holds for any characteristic, is enough to prove the effective Nullstellensatz theorem below. We use it now to obtain a global version of the Kronecker interpolation formula.

Proposition 5. Let P_{1}, …, P_{n} be n polynomials in K[x_{1}, …, x_{n}] of degree D, with the property that there exist strictly positive constants K, γ such that (3) holds for some integer δ > 0 satisfying

1 − 1/n(n + 1)^{2} < δ/D ≤ 1.

Suppose that g_{jl}, 1 ≤ j, l ≤ n, are elements in K[x_{1}, …, x_{n}, y_{1}, …, y_{n}], with degree less or equal than D − 1, and such that

P_{j}(x) − P_{j}(y) = ∑_{l=1}^{n}(x_{l} − y_{l})g_{jl}(x, y), 1 ≤ j ≤ n, x, y ∈ K^{n}.

Then, if Δ(x, y) := det[g_{jl}(x, y)]\begin{matrix}1≤j≤n \\ 1≤l≤n \end{matrix} (such Δ is called a Bézoutian for the map

P), the following polynomial identity holds:

1 = Res [\begin{matrix}Δ(x, y)dx \\ P_{1}(x), …, P_{n}(x) \end{matrix}] , y ∈ K^{n}.

The transformation laws are also used to compute the residue symbols

Res [\begin{matrix}Qdx \\ P^{k+\underline{1}} \end{matrix}]

from the relations of integral dependence of the coordinates over K(P) when P is a dominant map. For instance, one can use the following lemma.

Lemma 1. Let P_{1}, …, P_{n} be a quasiregular sequence in K[x_{1}, …, x_{n}]; then for any q ∈ N and for any α ∈ K^{n} the sequence (t^{q+1}, P_{1}(x) − α_{1}t, …, P_{n}(x) − α_{n}t) is a quasiregular sequence in K[x, t]. Moreover, we have the formula

Res [\begin{matrix}Q(x)dt ∧ dx \\ t^{q+1}, P_{1}(x) − α_{1}t, …, P_{n}(x) − α_{n}t \end{matrix}] = ∑_{|k|=q} Res [\begin{matrix}Qdx \\ P^{k+\underline{1}} \end{matrix}] α^{k}. (6)

The left-hand side of (6) can be computed using relations of the form

A_{j0}(u)x_{j}^{N_{j}} − ∑_{l=1}^{N_{j}} A_{jl}(u)x_{j}^{N_{j}−l} = ∑_{l=1}^{n} A_{j}^{l}(x_{j}, P, u)(P_{l} − u_{l}), j = 1, …, n,

so that it can be rewritten as

t^{s_{j}}(R_{j}(x_{j}, α) − tS_{j}(x_{j}, α, t)) = ∑_{l=1}^{n} A_{j}^{l}(x_{j}, P, αt)(P_{l} − α_{l}t), R_{j} ≢ 0.

Then, we use Proposition 2 and residue computations in one variable, to compute the left-hand side of (6). Note that such expression is, in principle, an element of K(α_{1}, …, α_{n}), say r_{1}(α)/r_{2}(α), while we know from (6) that it is a polynomial in α. If the P_{j} have coefficients in some integral domain A (with quotient field K) equipped with a size, then r_{1}(α) and r_{2}(α) have coefficients in A and a common denominator for all residue symbols on the right-hand side of (6) divides r_{2}(α) in A[α], that is, it divides all the coefficients of r_{2}(α). This remark is crucial with respect to size estimates for numerators or denominators of residue symbols. It is clear also that a good control on the size of the polynomials A_{jl}, for j = 1, …, n, 0 ≤ l ≤ N_{j}, and A_{j}^{l}, for 1 ≤ j, l ≤ n (which can be chosen with coefficients in A in this case) will provide good estimates for the sizes of r_{1}(α) and r_{2}(α).

2. Łojasiewicz inequalities

Another important tool in our proof of the Nullstellensatz is the Łojasiewicz inequality [JKS]. It will be used in the following form. Given n integers D_{1} ≥ D_{2} ≥ ⋯ ≥ D_{n} ≥ 1 we define, as in [JKS],

B := B(D_{1}, …, D_{n}) = (3/2)^{ι}D_{1} ⋯ D_{n},

where ι = #{j < n − 1 : D_{j} = 2}. Let P_{1}, …, P_{n} be polynomials with respective degrees deg P_{j} = D_{j}. Then there is a constant γ > 0 such that for any x ∈ K^{n}

|P (x)| ≥ γ(\frac{min\{1, dist(x, V (P))\}}{1 + |x|})^{B},

where V (P) denotes the set of common zeros of the polynomials P_{j} and the distance corresponds to the absolute value | · |.

Such an inequality, which is directly related to Kollár’s work [Ko] on the Nullstellensatz, has the following consequence, based on the arguments given in [BY1] and [BGVY, Propositions 5.7 and 5.8] when K = C. Small modifications are required by the fact that we are now working with fields of arbitrary characteristic.

Proposition 6. Let P_{1}, …, P_{n} be a quasiregular sequence in K[x_{1}, …, x_{n}]; then one can find n linear combinations (with coefficients in K) of the P_{j}, namely, P̃_{j}, for 1 ≤ j ≤ n, and n linearly independent K-linear forms L_{1}, …, L_{n}, as well as a positive constant K such that for any N ∈ N^{∗} and any x ∈ K^{n} with |x| > K one has

max_{1≤j≤n} |L_{j}(x)|^{NB} |P̃_{j}(x)| ≥ γ_{N} |x|^{(N−1)B}

for some constant γ_{N} > 0.

In fact, the linear combinations P̃_{j}, as well as the linear forms L_{j}, can be chosen with generic coefficients. Moreover, if we combine this result with Proposition 5 and the Arithmetic Bézout Theorem [Ph1, Theorem 4], we get the following technical but important result.

Proposition 7. Let P_{1}, …, P_{n} be a quasiregular sequence in K[x_{1}, …, x_{n}], with D_{1} ≥ D_{2} ≥ ⋯ ≥ D_{n}, with D_{j} := deg P_{j}. Then one can find a polynomial Φ in n(n + 1) + n^{2} variables u_{jl}, v_{jk}, for 1 ≤ j, k ≤ n, and 0 ≤ l ≤ n, with coefficients in K, with deg Φ ≤ 2^{n+1}(n + 1)^{4}D_{1}^{n}, such that, for any (U, V) ∈ K^{n×(n+1)} ⊕ M_{n}(K), with Φ(U, V) ≠ 0, the polynomials

Π_{U,V}^{j} (x) := U^{j}(x)⟨V^{j}, P (x)⟩ := (u_{j0} + ∑_{l=1}^{n} u_{jl}x_{l})(∑_{l=1}^{n} v_{jl}P_{l})

have degree exactly D_{1} + 1, define a quasiregular sequence in K[x_{1}, …, x_{n}], and moreover, if N ∈ N^{∗} is such that

(B + D_{1})/(NB + D_{1}) < 1/n(n + 1)^{2},

then the following polynomial identity holds in K[x_{1}, …, x_{n}]:

1 = Res [\begin{matrix}Δ_{N,U,V} (x, y)dx_{1} ∧ ⋯ ∧ dx_{n} \\ U^{1}(x)^{NB}⟨V^{1}, P ⟩, …, U^{n}(x)^{NB}⟨V^{n}, P ⟩ \end{matrix}] .

This formula, which is a polynomial identity in y, holds whenever Δ_{N,U,V} (x, y) is the determinant of an arbitrary n × n matrix whose coefficients δ_{jl} ∈ K[U, V, x, y] have degree in x, y at most NB +D_{1} −1 and for all j = 1, …, n satisfy the relations

U^{j}(x)^{NB}⟨V^{j}, P (x)⟩ − U^{j}(y)^{NB}⟨V^{j}, P (y)⟩ = ∑_{l=1}^{n}(x_{l} − y_{l})δ_{jl}(x, y).

3. Effective Nullstellensatz

In this section, A will be a unitary factorial regular integral domain with a size t; its quotient field will be denoted by K and assumed to be infinite. The size t can be extended to Pol(A) with values in {−∞} ∪ [0, +∞[; we refer to [Ph1] for this purpose. In the following theorem we need an auxiliary function defined for s ∈ N ∪ {+∞} by

ϑ(s) = inf{ξ ∈ [0, ∞[ : ϑ_{0}(ξ) ≥ s},

where

ϑ_{0}(ξ) = #{a ∈ A : t(a) ≤ ξ}, ξ ∈ [0, ∞[.

Let us state now our final result, that is, our solution of the Bézout identity with good degree and size estimates.

Effective Nullstellensatz. Let p_{1}, …, p_{M} ∈ A[x_{1}, …, x_{n}] and let A be an integral domain with infinite quotient field K. The ring A is assumed to be a factorial regular ring with Krull dimension κ and equipped with a size t (with corresponding c, c′, ϑ). The degrees D_{j} = deg p_{j} are assumed to be in decreasing order and h := max{t(p_{j}), c′ log(n+2)}. If p_{1}, …, p_{M} have no common zero in some algebraic closure \overline{K} of K, there exists r_{0} ∈ A, and q_{1}, …, q_{N} ∈ A[x], such that

r_{0} = ∑_{j=1}^{M} q_{j}p_{j},

with the estimates

deg(p_{j}q_{j}) ≤ n(n + 1)^{3}B(D_{1}, …, D_{n}) + n(D_{1} − 1),

t(p_{j}q_{j}) ≤ C_{0}ϖ^{4}2^{n}n^{17}c^{16}B^{4}D_{1}^{2}(h + ϑ[(γ_{0}D_{1})^{n}] + c′ log M + c′D_{1} log(2n + 2)), t(r_{0}) ≤ C_{0}ϖ^{4}2^{n}n^{17}c^{16}B^{4}D_{1}^{2}(h + ϑ[(γ_{0}D_{1})^{n}] + c′ log M + c′D_{1} log(2n + 2)),

where γ_{0}, C_{0} are absolute integral constants, B = B(D_{1}, …, D_{n}) is defined in Section 2, and the constants c, c′ depend only on the size t, while ϖ depends both on n and t.

In the two examples mentioned earlier, one can make the constants c, c′, ϖ and the function ϑ explicit. Namely, for A = Z, we have c = 3, c′ = 1, and ϖ = 9(n + 1)2^{n+2}(1 + 4 log(n + 1))^{n+2}. For A = F_{p}[τ_{1}, …, τ_{q}], we can take c = 1, c′ = 0, and ϖ = 2n + q + 1. Similarly, in the first example, ϑ(s) ≃ log s, and in the second, ϑ(s) ≃ (log s/ log p)^{1/q}.

Sketch of the proof. The key idea of the proof is to consider n affine forms (with coefficents in A), U_{1}, …, U_{n}, and n linear combinations ⟨V^{j}, P ⟩ (with coefficients in A) of the entries p_{j}, such that the hypotheses of Proposition 7 are fulfilled. Then, we start with the Kronecker identity

1 = Res [\begin{matrix}Δ_{N,U,V} (x, y)dx_{1} ∧ ⋯ ∧ dx_{n} \\ U^{1}(x)^{NB}⟨V^{1}, P ⟩, …, U^{n}(x)^{NB}⟨V^{n}, P ⟩ \end{matrix}] , (7)

which is valid for N = (n + 1)^{3}. From now on, we assume that the variables U, V have been fixed and so we will drop them, as well as N, from the notation, when convenient. In particular, we will denote by Θ_{j}(x) = U^{j}(x)^{NB}⟨V^{j}, P (x)⟩, for j = 1, …, n. Correspondingly, we let Δ(x, y) = Δ_{N,U,V} (x, y), and δ_{ij} be the entries of the corresponding matrix. We introduce a linear combination q of the p_{j} such that the ideal I(Θ_{1}, …, Θ_{n}, q) = K[x_{1}, …, x_{n}], and polynomials g_{n+1,l}, for 1 ≤ l ≤ n, in 2n variables such that q(x) − q(y) = ∑_{l=1}^{n} g_{n+1,l}(x, y)(x_{l} − y_{l}). Thus, we can rewrite the determinant Δ as

(1/q(x)) |\begin{matrix}δ_{11} & … & δ_{1n} & g_{n+1,1}(x, y) \\ ⋮ & ... & ⋮ & ⋮ \\ δ_{n1} & … & δ_{nn} & g_{n+1,n}(x, y) \\ Θ_{1}(y) − Θ_{1}(x) & … & Θ_{n}(y) − Θ_{n}(x) & q(y) \end{matrix}|

and develop this new (n + 1) × (n + 1) determinant along the last row to obtain

Δ(x, y) = (1/q(x))((∑_{j=1}^{n}(Θ_{j}(y) − Θ_{j}(x))Δ_{j}(x, y)) + q(y)Δ(x, y)).

Since the residue symbol is annihilated by the ideal, we can rewrite (7) as a Bézout identity

1 = ∑_{j=1}^{n} Res [\begin{matrix}Δ_{j}(x, y)dx/q(x) \\ Θ_{1}(x), …, Θ_{n}(x) \end{matrix}] Θ_{j}(y) + Res [\begin{matrix}Δ(x, y)dx/q(x) \\ Θ_{1}(x), …, Θ_{n}(x) \end{matrix}] q(y).

It is clear that this is an identity of the form 1 = ∑_{j=1}^{M} p_{j}(y)q_{j}(y), where the q_{j} are in K[x]. This is the formula that solves the Nullstellensatz with good estimates. Note that, up to this point, we already have the estimates for the degrees. The size estimates are obtained from the results in Section 1.

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Institute for Systems Research, University of Maryland, College Park, MD 20742

   E-mail address: carlos@src.umd.edu

Laboratoire de Mathématiques Pures, Université Bordeaux Sciences, 33405 Talence, France

   E-mail address: yger@math.u-bordeaux.fr