Trajectories of polynomial vector fields and ascending chains of polynomial ideals

Novikov, Dmitri; Yakovenko, Sergei

Annales de l'Institut Fourier, Tome 49 (1999) no. 2, pp. 563-609.

Nous donnons une borne supérieure complètement explicite pour le nombre d’intersections isolées entre une courbe intégrale d’un champ de vecteurs polynomial et une hypersurface algébrique dans l’espace euclidien de dimension quelconque. La borne est polynomiale par rapport à la hauteur des polynômes et la taille de la courbe, l’exposant étant une fonction explicite dépendant seulement du degré et de la dimension.

Le problème est alors très étroitement lié au problème de la longueur des chaînes ascendantes des idéaux polynomiaux, engendrées par les dérivations successives.

We give an explicit upper bound for the number of isolated intersections between an integral curve of a polynomial vector field in ${ℝ}^n$ and an algebraic hypersurface. The answer is polynomial in the height (the magnitude of coefficients) of the equation and the size of the curve in the space-time, with the exponent depending only on the degree and the dimension.

The problem turns out to be closely related to finding an explicit upper bound for the length of ascending chains of polynomial ideals spanned by consecutive derivatives.

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