MHB Natural Solutions for Positive Variables?

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The discussion explores whether a solution exists in natural numbers for a finite set of positive variables given a non-strict order on their expressions. It begins by analyzing equalities, noting that they define a subspace in real numbers with a basis of rational vectors. The thread emphasizes that rational points are dense in this subspace, allowing for approximations. It also highlights that strict inequalities and positivity conditions create an open subset containing a specific point. Ultimately, it concludes that by adjusting a rational point, an integer solution can be derived.
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Hi everyone,
Consider a finite set of positive variables $P = \{x_1,x_2,\ldots, x_n \}$, and a non-strict order on the expressions $\Sigma_{x_i \subseteq P}x_i$. For example:
$$P = \{x_1,x_2,x_3\}$$
$$x_1 + x_2 + x_3 > x_1 + x_2 > x_2 + x_3 > x_1 = x_2 > x_3$$
Can we claim that if there is a solution in which $\forall i,x_i \in \mathbb{R}^+$, there must be a solution in which $\forall i, x_i \in \mathbb{N}^+$?
Thanks!
 
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You need to consider the equalities and the strict inequalities separately.

Start with the equalities. If there are $d$ of them, they will define an $(n-d)$-dimensional subspace $S$ of $\Bbb{R}^n$, which will have a basis $\{\mathbf{e}_1,\ldots,\mathbf{e}_{n-d}\}$ consisting of rational vectors (vectors with rational coordinates). [In your example, there is just one equality, $x_1 = x_2$. That defines a two-dimensional subspace of $\Bbb{R}^3$, with a basis $\{\mathbf{e}_1=(1,1,0),\mathbf{e}_2=(0,0,1)\}$.]

Every point in $S$ is of the form $\alpha_1\mathbf{e}_1 + \ldots + \alpha_{n-d}\mathbf{e}_{n-d}$, where $\alpha_1, \ldots, \alpha_{n-d}$ are real coefficients. By approximating these coefficients with rational numbers, you see that the rational points in $S$ are dense in $S$.

The strict inequalities in your set, together with the inequalities $x_i>0\ (1\leqslant i\leqslant n)$, define an open subset of $S$. You are told that this subset contains a point $\mathbf{x} = (x_1,\ldots,x_n)$. So by taking a rational point $\mathbf{r}$ in $S$ sufficiently close to $\mathbf{x}$, you can find a rational solution to the problem.

Finally, by multiplying $\mathbf{r}$ by the least common multiple of the denominators of all its coordinates, you get an integer solution to the problem.
 
Thanks!
 
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