  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.
