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Effective Bezout identities in $${\mathbb{Q}}[z_ 1,\dots ,z_ n]$$. (English) Zbl 0724.32002
Let $$p_ 1,...,p_ m\in {\mathbb{Z}}[z_ 11,...,z_ n]={\mathbb{Z}}[z]$$ without common zeros in $${\mathbb{C}}^ n$$. Hilbert’s Nullstellensatz says that there are $$\delta \in {\mathbb{Z}}^+$$ and polynomials $$q_ 1,...,q_ m\in {\mathbb{Z}}[z]$$ such that for every $$z\in {\mathbb{C}}^ n$$, $$\delta =p_ 1q_ 1+...+p_ mq_ m$$. Algorithms for resolution of this Bezout equation are due to Hermann, Seidenberg, and Buchberger. Denote by h(p) the logarithmic size of a polynomial $$p\in {\mathbb{Z}}[z]$$; $$h(p):=$$ the logarithm of the modulus of the coefficient of p of largest absolute value. Masser-Wüstholz, Brownawell, Kollar, Philippon, and Shiffman obtained estimates for the degree and the size of the polynomials $$q_ j$$ and the size of $$\delta$$. The purpose of this paper is to obtain better effective bounds for the size of $$\delta$$ and the $$q_ j.$$
The main result is the following Theorem: Let $$p_ 1,...,p_ N\in {\mathbb{Z}}[z]$$ without common zeros in $${\mathbb{C}}^ n$$, deg $$p_ j\leq D$$, $$D\geq 3$$, $$h(p_ j)\leq h$$. There is an integer $$\delta \in {\mathbb{Z}}^+$$, polynomials $$q_ 1,...,q_ N\in {\mathbb{Z}}[z]$$ such that $$p_ 1q_ 1+...+p_ Nq_ N=\delta$$, satisfying the estimates: $\deg q_ j\leq n(2n+1)D^ n,\quad h(q_ j)\leq \chi (n)D^{8n+3}(h+\log N+D \log D),$ $\log \delta \leq \chi (n)D^{8n+3}(h+\log N+D \log D),$ where $$\chi$$ (n) is an effective constant which can be computed explicitly following step by step the proof.
The ring $${\mathbb{Z}}$$ can be replaced by the ring of integers of any number field.
The proof depends on complex function theory, namely on explicit integral representation formulas of the Henkin type and on multidimensional residues, used as a tool in computations.

##### MSC:
 32B05 Analytic algebras and generalizations, preparation theorems 32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.) 32A27 Residues for several complex variables 32D15 Continuation of analytic objects in several complex variables 13P99 Computational aspects and applications of commutative rings
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