I want to verify that I understand the term "positive linear combination" here in the first sentence of the abstract correctly. Consider $a+b+1=0$ and $a-b+1=0$ where $a,b$ are variables in some polynomial ring, are the polynomials both positive or does $-b$ in the latter make it negative linear combination? This is important to understand the second example here.
What does particularly the part "positive linear combination" mean exactly?
If you have some vectors $\{\beta_i\}_{i = 1}^n$, then a positive linear combination is $$\sum_{i = 1}^n \lambda_i \, \beta_i,$$ where all $\lambda_i$ are non-negative. It does not mean that the $\beta_i$ satisfy some notion of positivity.