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Addendum to the text of W. D. Brownawell: “Bounds for the degrees in the Nullstellensatz”. (A propos du texte de W. D. Brownawell: “Bounds for the degrees in the Nullstellensatz”.) (French) Zbl 0641.14002
In the paper which is mentioned in the title, W. D. Brownawell [ibid. 126, 577–591 (1987; Zbl 0641.14001)] proved a remarkable result which is the first sharp effective version of Hilbert’s Nullstellensatz. His main proposition uses elimination theory, and his final result combines this proposition with a result from complex analysis due to Skoda.
In the present note, the author shows that his previous paper [Publ. Math., Inst. Hautes √Čtud. Sci. 64, 5–52 (1986; Zbl 0615.10044)] enables him to get an explicit version of Brownawell’s main proposition for algebraic number fields.

MSC:
14A05 Relevant commutative algebra
11R09 Polynomials (irreducibility, etc.)
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
11C08 Polynomials in number theory
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