Criteria of algebraic independence with multiplicities and approximation by surfaces (Q2729302)

This paper proves a criterion of algebraic independence, which generalizes the well-known criterion of \textit{P. Philippon} [Publ. Math., Inst. Hautes Étud. Sci. 64, 5-52 (1986; Zbl 0615.10044)] by taking into account not only the values of families of polynomials at the point under study, but also the values of their derivatives at that point. It also generalizes previous work of the authors in the one variable case [Trans. Am. Math. Soc. 351, 1845-1870 (1999; Zbl 0923.11106)]. This criterion is motivated by applications to interpolation determinants. In combination with the construction based on the Dirichlet box principle, it also provides the existence of projective algebraic hypersurfaces of controlled degree and height whose distance to a given point is small. NEWLINENEWLINENEWLINEThe proof of the criterion uses a notion of height of algebraic cycles based on their Chow forms. For cycles of dimension \(t\), this height is attached to a choice of \(t+1\) adelic convex bodies, one for each generic homogeneous polynomial entering into the definition of their Chow forms. The construction is such that it behaves very simply with respect to specialization of one of these polynomials. It is also shown to be almost multi-linear with respect to a notion of factorization of these convex bodies. Estimations for these heights are given in specific instances.