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Multiplicity of polynomials on trajectories of polynomial vector fields in \(\mathbb{C}^3\). (English) Zbl 0922.32023
Jakubczyk, Bronisław (ed.) et al., Singularities symposium – Łojasiewicz 70. Papers presented at the symposium on singularities on the occasion of the 70th birthday of Stanisław Łojasiewicz, Cracow, Poland, September 25–29, 1996 and the seminar on singularities and geometry, Warsaw, Poland, September 30–October 4, 1996. Warsaw: Polish Academy of Sciences, Institute of Mathematics, Banach Cent. Publ. 44, 109-121 (1998).
Let \(\delta\) be a vector field on \(\mathbb{C}^n\) with polynomial coefficients of degree \(d\) and \(f\) a polynomial of degree \(m\). The multiplicity of a zero of a restriction of \(f\) to a nonsingular trajectory of \(\delta\) is studied. There are bounds of this multiplicity, doubly exponential in terms of \(n\). For the case \(n=3\) a bound of the form \(m(1+2(m+d-1)^2)\) is given.
For the entire collection see [Zbl 0906.00013].

32S70 Other operations on complex singularities
93B25 Algebraic methods
93B05 Controllability
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