TITLE: Counting lattice points on an n-simplex
QUESTION [5 upvotes]: Imagine an n-simplex, the solution set for the expression: $a_1$*$x_1$ + $a_2$*$x_2$ + ... + $a_n$*$x_n$ = S, where: 

$a_1$ through $a_n$ are positive bounded integers
$x_1$ through $x_n$ are positive bounded real numbers  
'S' is the sum of the expression 

This n-simplex therefore has a single vertex on the origin, as well as a single vertex on each axis at some arbitrary (strictly positive) distance from the origin. 
What is the lattice integer-point count?  
Can one use Ehrhardt polynomials to compute the integer point count for the n-simplex, perhaps under the restriction that we have vertices strictly at integer coordinates?  

From "Geometry for N-Dimensional Graphics" (by Andrew J. Hanson, CS Dept., Indiana University) we know that the oriented volume for the n-simplex with vertices {$v_1$, ..., $v_n$}, or {$a_1$*$x_1$, ..., $a_n$*$x_n$} is: 

$V_n$ = $\dfrac{1}{n!}$ * det([($v_1$-$v_0$), ..., ($v_n$-$v_0$)])
(Problems writing LaTeX for matrices here, please see terms as column vectors to obtain square matrix.)

Previous formulation of question: Imagine an expression of the form: $a_1$*$x_1$ + $a_2$*$x_2$ + ... + $a_n$*$x_n$ = S, where: 

$a_1$ through $a_n$ are positive bounded integers
$x_1$ through $x_n$ are positive bounded real numbers  
'S' is the sum of the expression 

Can we say anything about the maximum value of 'S' (for a given $x_1$ through $x_n$) below which there is only one solution for positive integer coefficients $a_1$ through $a_n$?  For example, given the expression $a_1$*98 + $a_2$*99 = 'S', where coefficients $a_1$ and $a_2$ = [1 through 100], one finds that you can always exactly recover the original $a_1$ and $a_2$ if 'S' < 9899.  Is there an analytical or more elegant method for obtaining that bound?   
[Above such a bound, is there an efficient way to obtain all possible sets of integers $a_1$ through $a_n$ that satisfy the relation for a given 'S'? Can the LLL or PSLQ algorithms be used?]  <--  This seems to be a restricted/special case of the subset-sum problem, so existing dynamic programming algorithms would work.  Can one do better here?

REPLY [4 votes]: For a polynomial-time method of counting integer lattice points for the n-simplex (with fixed dimension):
Review article - Crites, A., Goff, M., Korson, M., Patrolia, L., Wolcott, L. "Counting Lattice Points in Polyhedra." 
Available here with references for Barvinok's 1994 & 1999 algorithms - http://www.math.washington.edu/~thomas/teaching/m583_s2008_web/Barvinok.pdf
For an implementation of Barvinok's algorithm, see J.A. De Loera's LattE program (hosted at UC Davis):
http://www.math.ucdavis.edu/~latte/group.htm