10.1. Find all positive integers $n$, such that there exist non-zero real numbers $a, b, c, d$, satisfying the polynomial
$$
(a x+b)^{1000}-(c x+d)^{1000}
$$

after expansion and combining like terms, exactly $n$ non-zero terms remain.
10.1. $n=500,1000,1001$.

Obviously, there exist polynomials with 1001 and 1000 non-zero terms:
$$
\begin{array}{l}
(2 x+2)^{1000}-(x+1)^{1000}, \\
(2 x+1)^{1000}-(x+1)^{1000} .
\end{array}
$$

Assume the coefficients of the $x^{i}$ and $x^{j}$ terms $(i>j)$ are 0. Then
$a^{i} b^{1000-i}=c^{i} d^{1000-i}$,
$a^{j} b^{1000-j}=c^{j} d^{1000-j}$.
After rearranging, we get $\left(\frac{a d}{c b}\right)^{i}=\left(\frac{d}{b}\right)^{1000}=\left(\frac{a d}{b c}\right)^{j}$.
Solving this, we get $\left|\frac{d}{b}\right|=1,\left|\frac{a}{c}\right|=1$.
Clearly, when $a$ and $b$ are replaced by $-a$ and $-b$ respectively, the polynomial remains unchanged.
Therefore, we can assume $a=c$.
If $b=d$, then the polynomial is the zero polynomial;
If $b=-d$, then the polynomial
$$
(a x+b)^{1000}-(a x-b)^{1000}
$$

has exactly 500 non-zero terms.