26 Removed 'second question', made first question easier to read / self-consistant.

[One can now state the first part of the question as: given

Imagine an n-simplex(, the solution set for the expressionbelow): $a_1$*$x_1$ + $a_2$*$x_2$ + ... + $a_n$*$x_n$ = S, with 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 .

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.)

] - Resolved (See answer below.)

To push my luck... the second part of my question is, given some bounds on [$a_1$, $a_2$, ..., $a_n$] what is the maximum sum, 'S', of the integer distances from the origin to the vertices on each axis that allows for only a single lattice integer point?

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?

25 Shortened question text.

Update/Reformulation of question (let me know if the title change is inappropriate). Thanks so much to everyone for bearing with me here, and thanks to Sam Nead for pointing out that this problem statement is equivalent to a particular integer-point enumeration problem for polytopes.

First part RESOLVED (See my answer below)

[One can now state the first part of the question as: given an n-simplex (the solution set for the expression below), with a single vertex on the origin, as well as on each axis at some arbitrary (strictly positive) distance from the origin, what is the lattice integer-point count?

Can 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.)

] - Resolved (See answer below.)

To push my luck... the second part of my question is, given some bounds on [$a_1$, $a_2$, ..., $a_n$] what is the maximum sum, 'S', of the integer distances from the origin to the vertices on each axis that allows for only a single lattice integer point?

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?

24 added 2 characters in body

Update/Reformulation of question (let me know if the title change is inappropriate). Thanks so much to everyone for bearing with me here, and thanks to Sam Nead for pointing out that this problem statement is equivalent to a particular integer-point enumeration problem for polytopes.

First part RESOLVED (See my answer below)

[One can now state the first part of the question as: given an n-simplex (the solution set for the expression below), with a single vertex on the origin, as well as on each axis at some arbitrary (strictly positive) distance from the origin, what is the lattice integer-point count?

Can 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.)

]

To push my luck... the second part of my question is, given some bounds on [$a_1$, $a_2$, ..., $a_n$] what is the maximum sum, 'S', of the integer distances from the origin to the vertices on each axis that allows for only a single lattice integer point?

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?

23 Resolved first part of question, provided note
22 added 99 characters in body; edited title
21 Changed title to properly name polytope 'n-simplex', provided formula for volume in n-dimensions given vertices {; deleted 7 characters in body; added 6 characters in body
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19 Changed title, reformulated question in terms of lattice integer point count for a particular kind of polytope; edited title; added 53 characters in body; added 13 characters in body; deleted 13 characters in body
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15 Updated question on equivalence to lattice integer point counting for polyhedra; deleted 9 characters in body
14 Changed "polynomial" to "expression" & disambiguated r_k / x_k issue
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6 Added tag 'polynomials', which seemed to be more appropriate.
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