over which the scheme $M(P)_{x_m}$ will have a $L$-valued point. This bound is obtained using the following lemma’s:

Let $f: Y\rightarrow X$ be a morphism of finite-type scheme over $\bar K$. Then there exists an integer $n$ such that the radical index of the schematic fiber of $x$ is less than or equal to $n$ for all closed points $x \in X$.

Let $A$ be an affine $K_s$-algebra with radical index $\leq p^n$. Then $A$ admits a $K_s^{p^{-n}}$-rational point.

Bounds for the field of definition of the instability parabolic and its consequences
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[\[Bounds for the instability parabolic\]]{} In this section we give explicit bounds for the field of definition of the instability parabolic associated to non-semistable points for the action of a reductive algebraic group $G$ acting on a vector space $V$ defined over an arbitrary field $K$. We do this by giving explicit bounds for the field of definition for the instability 1-PS associated to these points. We first do this $G=Sl(2)$, where we can get much better bounds than for a general $G$, then for the tensor power representation of $Sl(n)$, then for an arbitrary representation of $Sl(n)$ and then for an arbitrary representation of any arbitrary 