Let $|D|$ be the linear system of degree $d$ hypersurfaces in $\mathbb{P}^n$ having multiplicity at least $m$ at $s$ general points.

Then $|kD|$ is the linear system of degree $kd$ hypersurfaces in $\mathbb{P}^n$ having multiplicity at least $km$ at $s$ general points.

The expected number of conditions imposed by the $s$ points of multiplicity $km$ is then $$s\binom{km-1+n}{n}.$$ Some of these conditions may be linearly dependent. Then the actual number of conditions is $$s\binom{km-1+n}{n}-P(k) = s\frac{(km+n-1)\cdot ... \cdot km}{n!}-P(k) = \frac{sm^n}{n!}k^n + Q(k)-P(k)$$ where $P(k)$ is a polynomial in $k$, and $Q(k) = s\binom{km-1+n}{n}-\frac{sm^n}{n!}k^n$. In particular, $\deg(Q) = n-1$.

Under which hypothesis may we conclude that $\deg(P)\leq n-1$? For instance, would it be enough to know that the linear system $|D|$ induces a birational map $\mathbb{P}^n\dashrightarrow X$?