Let $v_i=a_ie_i\in\mathbb R^d$ and $w_i=b_ie_i\in\mathbb R^d$ for $i=1,\dots,d$ where $e_i$'s are unit vetcors and $a_i,b_i$ are positive integers. Let $$S=conv\{0,rv_i+sw_i:i=1,\dots,d\}.$$ Can we express the cardinality of the set $S\cap\mathbb N_{\geq 0}^d$ as a bigraded polynomial of $r$ and $s$?
I am almost sure that it is a polynomial. But I can not prove it.
Let's see what happens in dim 2. You have $conv((0,0),(ra_1+sb_1,0),(0,ra_2+sb_2))$. The number of points in the closed triandle $(0,0),(A,0),(0,B)$ is $(A+1)(B+1)/2$ plus half the number of points on the diagonal, which is $gcd(A,B)+1$. So up to a polynomial in $r$ and $s$ you get $gcd(ra_1+sb_1,ra_2+sb_2)$. I am pretty sure it is not polynomial in $r,s$. For example, take $a_1=1$, $b_1=0$, $a_2=0$, $b_2=1$, so you get $gcd(r,s)$.
