On the multiplicity estimates

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.