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. 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 $v_i=a_ie_i\in\mathbb R^d$ 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 $v_i=a_ie_i\in\mathbb R^d$ and $w_i=b_ie_i\in\mathbb R^d$ for $i=1,\dots,d$ where $e_i$'s are unit vetcors. 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 $s$?

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. 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.