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On a conjecture of C. Bernstein and A. Yger. (English) Zbl 0866.13008

González-Vega, Laureano (ed.) et al., Algorithms in algebraic geometry and applications. Proceedings of the MEGA-94 conference, Santander, Spain, April 5-9, 1994. Basel: Birkhäuser. Prog. Math. 143, 17-28 (1996).
Let \({\mathfrak a}\) be an ideal in the ring of polynomials \(K[x_1,\dots, x_n]\) over a field \(K\), \(e\), \(d\) be positive integers and \(\phi_{\mathfrak a}(e,d)\) be the minimal integer \(D\) such that for all systems of generators \(\{f_1,\dots, f_m\}\) with \(\deg f_i\leq d\) and for all \(f\in{\mathfrak a}\), there exists a representation \(f^e= a_1f_1+\cdots+ a_mf_m\) with \[ \max\deg a_i\leq e\deg f+D. \] The author generalizes a result from his previous article [in: “Approximation diophantiennes et nombres transcendants”, Luminy 1990, 1-13 (1992; Zbl 0783.13015)], obtaining, in particular, that \(\phi_{\mathfrak a}(3^nd)\leq d^n+d\) (for \(d\geq 3)\), in accordance with a hypothesis of C. A. Berenstein and A. Yger [ibid., 15-37 (1992; Zbl 0765.32002)].
For the entire collection see [Zbl 0841.00016].

MSC:

13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13E15 Commutative rings and modules of finite generation or presentation; number of generators
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
68W30 Symbolic computation and algebraic computation
13A15 Ideals and multiplicative ideal theory in commutative rings
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