Two invariants related to two conjectures due to Nagata [1]

Abstract : Seshadri’s constant is related to a conjecture due to Nagata. Another conjecture, al-

so due to Nagata and solved by Bombieri in 1970, is related with algebraic values of

meromorphic functions. The main argument of Bombieri’s proof leads to a Schwarz

Lemma in several variables, the proof of which gives rise to another invariant associa-

ted with symbolic powers of the ideal of functions vanishing on a finite set of points.

This invariant is an asymptotic measure of the least degree of a polynomial in seve-

ral variables with given order of vanishing on a finite set of points. Recent works on

the resurgence of ideals of points and the containment problem compare powers and

symbolic powers of ideals.