The maximum value of $N$ is 10. (1) Let $f(x)=(x-1)(x^3-1)(x^7-1)(x^{13}-1)(x^{25}-1)(x^{49}-1)(x^{99}-1)(x^{199}-1)(x^{399}-1)(x^{799}-1)x^{426}$, it's valid. (2) If $N>10$, then $N \geq 11$. It's obvious that the positive integer roots of $f(x)$ is $1$, or at least one of $c_i$ will be not less than 2. So, $(x-1)^{11}|f(x)$. Let $x=-1$, we have $2^{11}|f(-1)\implies |f(-1)|\geq 2048$. However, $|f(-1)|\leq 1+\sum_{i=0}^{2019} 1=2021<2048$. That's impossible.

Your example has only 2 roots, which are 1 and 0 isnt it?

Mikeglicker wrote: Your example has only 2 roots, which are 1 and 0 isnt it? You must pay attention to this: counting multiplicity