Lattice points in simplices - reference request

Question

I found this paper

http://homepages.math.uic.edu/~yau/35%20publications/An%20upper.pdf

which, in formulas (1.2) and (1.3), relates the number of non-negative and positive integer values that are contained in a simplex. See my related question here: Integer Points in Simplex .

However, no reference is given for formulas (1.2) and (1.3). Therefore, can anyone give me a reference or (if this is a trivial identity) explain it to me?

Thanks a lot!

Answer 1

Let's deduce $(1.3)$. Let $(x_1,…,x_n)$ be a positive integral solution of $$\frac{x_1}{a_1}+\cdots+\frac{x_n}{a_n}\leq1.$$Now put $y_i=x_i-1\geq0$ for all $i$. We claim that $(y_1,…,y_n)$ is a non-negative integral solution of $$\frac{y_1}{a_1(1-a)}+\cdots+\frac{y_n}{a_n(1-a)}\leq1.$$ In fact, $\frac{x_1}{a_1}+\cdots+\frac{x_n}{a_n}=\frac{x_1-1+1}{a_1}+\cdots+\frac{x_n-1+1}{a_n}=\frac{y_1}{a_1}+\frac{1}{a_1}\cdots+\frac{y_n}{a_n}+\frac{1}{a_n}=\frac{y_1}{a_1}+\cdots+\frac{y_n}{a_n}+a$. Hence, $$\frac{x_1}{a_1}+\cdots+\frac{x_n}{a_n}\leq1\Leftrightarrow \frac{y_1}{a_1}+\cdots+\frac{y_n}{a_n}\leq1-a\Leftrightarrow\frac{y_1}{a_1(1-a)}+\cdots+\frac{y_n}{a_n(1-a)}\leq1.$$ That gives us a 1-1 correspondence between the set of positive integral solutions of $\frac{x_1}{a_1}+\cdots+\frac{x_n}{a_n}\leq1$ and the set of non-negative integral solutions of $\frac{x_1}{a_1(1-a)}+\cdots+\frac{x_n}{a_n(1-a)}\leq1,$ which implies $P(a_1,…,a_n)=Q(a_1(1-a),…,a_n(1-a)).$