Partially related to Non-negative Polynomials from Polynomial Ideal? where focus on graph theory but now I want to understand more generally how to determine non-negativity in more general case.

A. Let $P$ be the polytope where $p_i^{lowerBound}\leq p_i\leq p_i^{upperBound}$. How can you determine to which product of simplicial complexes $P$ is affinely equivalent?

Example Answer. The polytope spanned by the each entry in probability vector is affinely equivalent to some polytope in cases such as product of 1-dimensional hypercubes in the case of $p^{lowerBound}\leq p_i\leq p^{upperBound}$ for all $i$.

B. If the polytope is not equivalent, how does the paper In praise of order units Journal of Algebra and Its Applications discusses this?

C. If Gröbner basis needed for determining the non-negativity, how to determine the Gröbner basis to do computations and which monomial ordering? Reverse lexicographic monomial ordering?

D. How are the positivity results used for finding non-negativity of polynomial?

How can you find non-negative polynomials $f(p)$ from a polynomial ideal when $p\in P$ and $P$ is a compact polytope?

Partially related to Non-negative Polynomials from Polynomial Ideal? where focus on graph theory but now I want to understand more generally how to determine non-negativity in more general case.

A. Let $P$ be the polytope where $p_i^{lowerBound}\leq p_i\leq p_i^{upperBound}$. How can you determine to which product of simplicial complexes $P$ is affinely equivalent?

Example Answer. The polytope spanned by the each entry in probability vector is affinely equivalent to some polytope in cases such as product of 1-dimensional hypercubes in the case of $p^{lowerBound}\leq p_i\leq p^{upperBound}$ for all $i$.

B. If the polytope is not equivalent, how does the paper In praise of order units Journal of Algebra and Its Applications discusses this?

C. If Gröbner basis needed for determining the non-negativity, how to determine the Gröbner basis to do computations and which monomial ordering? Reverse lexicographic monomial ordering?

D. How are the positivity results used for finding non-negativity of polynomial?

How can you find non-negative polynomials $f(p)$ from a polynomial ideal when $p\in P$ and $P$ is a compact polytope?

Partially related to Non-negative Polynomials from Polynomial Ideal? where focus on graph theory but now I want to understand more generally how to determine non-negativity in more general case.

1. Let $P$ be the polytope where $p_i^{lowerBound}\leq p_i\leq p_i^{upperBound}$. How can you determine to which product of simplicial complexes $P$ is affinely equivalent?

1.1. If the polytope is not equivalent, how does the paper In praise of order units Journal of Algebra and Its Applications discusses this?

2. If Gröbner basis needed for determining the non-negativity, how to determine the Gröbner basis to do computations and which monomial ordering? Reverse lexicographic monomial ordering?

3. How are the positivity results used for finding non-negativity of polynomial?

How can you find non-negative polynomials $f(p)$ from a polynomial ideal when $p\in P$ and $P$ is a compact polytope?

Partially related to Non-negative Polynomials from Polynomial Ideal? where focus on graph theory but now I want to understand more generally how to determine non-negativity in more general case.

A. Let $P$ be the polytope where $p_i^{lowerBound}\leq p_i\leq p_i^{upperBound}$. How can you determine to which product of simplicial complexes $P$ is affinely equivalent?

Example Answer. The polytope spanned by the each entry in probability vector is affinely equivalent to some polytope in cases such as product of 1-dimensional hypercubes in the case of $p^{lowerBound}\leq p_i\leq p^{upperBound}$ for all $i$.

B. If the polytope is not equivalent, how does the paper In praise of order units Journal of Algebra and Its Applications discusses this?

C. If Gröbner basis needed for determining the non-negativity, how to determine the Gröbner basis to do computations and which monomial ordering? Reverse lexicographic monomial ordering?

D. How are the positivity results used for finding non-negativity of polynomial?

How can you find non-negative polynomials $f(p)$ from a polynomial ideal when $p\in P$ and $P$ is a compact polytope?