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Bounds of traces in complete intersections and degrees in the Nullstellensatz. (English) Zbl 0844.14018

Authors’ abstract: In this paper we obtain an effective Nullstellensatz using quantitative considerations of the classical duality theory in complete intersections. Let \(k\) be an infinite perfect field and let \(f_1, \dots, f_{n - r} \in k[X_1, \dots, X_n]\) be a regular sequence with \(d : = \max_j \deg f_j\). Denote by \(A\) the polynomial ring \(k[X_1, \dots, X_r]\) and by \(B\) the factor ring \(k[X_1, \dots, X_n]/(f_1, \dots, f_{n - r})\); assume that the canonical morphism \(A \to B\) is injective and integral and that the Jacobian determinant \(\Delta\) with respect to the variables \(X_{r + 1}, \dots, X_n\) is not a zero divisor in \(B\). Let finally \(\sigma \in B^* : = \operatorname{Hom}_A (B,A)\) be the generator of \(B^*\) associated to the regular sequence.
We show that for each polynomial \(f\) the inequality \(\deg \sigma (\overline f) \leq d^{n - r} (\delta + 1)\) holds \((\overline f\) denotes the class of \(f\) in \(B\) and \(\delta\) is an upper bound for \((n - r)d\) and \(\deg f)\). For the usual trace associated to the (free) extension \(A \hookrightarrow B\) we obtain a somewhat more precise bound: \(\deg \text{Tr} (\overline f) \leq d^{n - r} \deg f\). From these bounds and Bertini’s theorem we deduce an elementary proof of the following effective Nullstellensatz: Let \(f_1, \dots, f_s\) be polynomials in \(k[X_1, \dots, X_n]\) with degrees bounded by a constant \(d \geq 2\); then \(1 \in (f_1, \dots, f_s)\) if and only if there exist polynomials \(p_1, \dots, p_s \in k[X_1, \dots, X_n]\) with degrees bounded by \(4n (d + 1)^n\) such that \(1 = \sum_i p_i f_i\). In the particular cases when the characteristic of the base field \(k\) is zero or \(d = 2\) the sharper bound \(4n d^n\) is obtained.

MSC:

14M10 Complete intersections
14A05 Relevant commutative algebra
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