On the multiplicity estimates

TL;DR

This paper establishes new multiplicity estimates for connected algebraic groups, including noncommutative ones, providing tools that enhance existing theorems and aid transcendence theory in algebraic group contexts.

Contribution

It introduces improved multiplicity estimates for algebraic groups, especially noncommutative, under specific conditions, advancing the theoretical framework and applications in transcendence theory.

Findings

Derived new obstruction varieties with specific properties.

Enhanced existing multiplicity estimates theorems.

Provided tools for transcendence theory in noncommutative groups.

Abstract

In this paper we show some multiplicity estimates theorems for a connected algebraic group (not necessarily commutative) $G$ over an algebraically closed subfield of $\mathbb{C}$. More specifically, under particular assumptions on the parameters and the points where the polynomial has high order with respect to a Lie subalgebra of the Lie algebra associated to $G$, we present a series of results where we find obstruction varieties with different properties. Some of the results obtained in this paper improve the multiplicity estimates theorem for arbitrary connected algebraic groups that already exist and also they shall be important tools in transcendence theory for noncommutative algebraic groups.