Multiplicity estimates: a Morse-theoretic approach. (English) Zbl 1334.14031
The author considers the problem of estimating the multiplicity of the zero of a degree \(d\) polynomial on \(\mathbb{C}^n\) when restricted to the trajectory of a nonsingular degree \(\delta\) polynomial vector field, at one or several points. By refining A. Gabrielov’s method [Math. Res. Lett. 2, No. 4, 437–451 (1995; Zbl 0845.32003)], the author improves simultaneously the estimates of Yu. V. Nesterenko [in: New advances in transcendence theory, Proc. Symp., Durham/UK 1986, 263–269 (1988; Zbl 0656.10034)] (which is of order \(d^n\) and doubly-exponential in \(n\)) and of A. Gabrielov [in: The Arnoldfest. Proceedings of a conference in honour of V. I. Arnold for his 60th birthday, Toronto, Canada, June 15–21, 1997. Providence, RI: American Mathematical Society. 191–200 (1999; Zbl 0948.32010)] (which is simply exponential in \(n\), but of order \(d^{2n}\)). The author’s result is \(n^n\) in \(n\) and of order \(d^n\).
Reviewer: Vladimir P. Kostov (Nice)
MSC:
| 14P05 | Real algebraic sets |
| 32M25 | Complex vector fields, holomorphic foliations, \(\mathbb{C}\)-actions |
| 11J81 | Transcendence (general theory) |
| 13F20 | Polynomial rings and ideals; rings of integer-valued polynomials |
| 11J99 | Diophantine approximation, transcendental number theory |
| 34M99 | Ordinary differential equations in the complex domain |
| 32B20 | Semi-analytic sets, subanalytic sets, and generalizations |
| 32A15 | Entire functions of several complex variables |
Keywords:
multiplicity estimates for vector fields; Bernstein-Kushnirenko-Khovanskii bound; Milnor fibers; polar varietiesReferences:
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