Polynomials with high multiplicity. (English) Zbl 0688.10034

Amoroso, Francesco

Acta Arith. 56, No. 4, 345-364 (1990).

Let S be a subset of \({\mathbb{C}}^ n\). For a positive integer M we define the quantity \(\omega_ M(S)\) as the minimum degree of an algebraic hypersurface having a singularity of order \(\geq M\) at any point of S. Several results of Waldschmidt, Masser, Wüstholz, Esnault and Viehweg give the inequality \[ (*)\quad (1/c_ n)\omega_ 1(S)\leq (1/M)\omega_ M(S) \] where \(c_ n\) is a positive constant depending only on n. In my paper, I work with the arithmetical equivalent of \(\omega_ M(S)\), namely the minimum size \({\bar \omega}{}_ M(S)\) of a polynomial with integer coefficients having a singularity of order \(\geq M\) at any point of S (as usual the size of a polynomial is defined as the maximum between its degree and its logarithmic height). The main result is to generalize the inequality (*) to the quantity \({\bar \omega}{}_ m(S)\). To do this I use the theory of Chow forms developed by Yu. V. Nesterenko and P. Philippon and a new definition of multiplicity, given in terms of the Chow form of an ideal.
In the second part, I give an application of the main result to the problem of comparing the transcendence type of an n-tuple of complex numbers with its approximation type.

Reviewer: F.Amoroso

MSC:

11J81	Transcendence (general theory)
13A15	Ideals and multiplicative ideal theory in commutative rings
13B25	Polynomials over commutative rings
14J70	Hypersurfaces and algebraic geometry
14J17	Singularities of surfaces or higher-dimensional varieties

Keywords:

minimum size; polynomial with integer coefficients; Chow forms; multiplicity; transcendence type; n-tuple of complex numbers; approximation type