Imagine an n-simplex, the solution set for the expression: $a\_1$\*$x\_1$ + $a\_2$\*$x\_2$ + ... + $a\_n$\*$x\_n$ = S, where: 


1. $a\_1$ through $a\_n$ are positive bounded integers
2. $x\_1$ through $x\_n$ are positive bounded real numbers
3. '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.)




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Previous formulation of question: Imagine an expression of the form: $a\_1$\*$x\_1$ + $a\_2$\*$x\_2$ + ... + $a\_n$\*$x\_n$ = S, where: 


1. $a\_1$ through $a\_n$ are positive bounded integers
2. $x\_1$ through $x\_n$ are positive bounded real numbers
3. '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? 



Counting lattice points on an n-simplex
I am informed that you are "counting lattice points inside of a polyhedron."


[Here](http://www.math.ucdavis.edu/~deloera/.../manyaspectsofcountinglatpts.pdf) is a lecture on the subject - the picture on page six looks like the version of the problem you are interested in. To be honest, I found these notes by doing a google search. I am told that this is a huge field! 


It might help if you could narrow your problem even further. For example, you say that the $x\_i$ are bounded real numbers. Do you know these to some high precision? Or can you give some information on how the $x\_i$ are given? And can you say the same for $S$?


EDIT: 
[Here](http://www.math.ucdavis.edu/~deloera/RECENT_WORK/semesterberichte.pdf) is a survey paper by the same author, Jesús De Loera, covering the same material in greater detail. 


