Let $x_1,\ldots,x_n\in [-1,1]^n$ and define the function $$f(x_1,\ldots,x_n):= \prod_{i=1}^n\prod_{j=i}^n\left(1-\prod_{k=i}^j x_k\right).$$ This is a positive function, and actually coincides with the determinant of a $(n+1)\times (n+1)$ matrix $M:=(a_{ij})_{i,j=1}^{n+1}$ where: $$a_{i,i}=1, \hspace{0.5cm}1\leq i\leq n+1,$$ $$a_{i+1,i}=1, \hspace{0.5cm} 1\leq i\leq n,$$ $$a_{i,j}=x_i^{j-i} \cdot x_{i+1}^{j-i-1}\cdots x_{j-1}, \hspace{0.5cm}j>i,$$ $$a_{i,j}= x_{j+1}\cdot x_{j+2}^2\cdots x_{i-1}^{i-j-1}, \hspace{0.5cm}j<i-1.$$ As an example, when $n=3$ the matrix $M$ has the form $$\left(\begin{matrix} 1 &x_1 &x_1^2x_2 & x_1^3x_2^2x_3\\ 1 & 1 & x_2 & x_2^2x_3\\ x_2 & 1 & 1 & x_3\\ x_2x_3^2 & x_3 & 1 &1 \end{matrix}\right).$$
Thanks to the definition of $M$, we immediately get $f\leq (n+1)^{(n+1)/2}$ by Hadamard's Lemma. But actually it is possible to get sharper estimates.
In 1977 M. Pohst https://www.sciencedirect.com/science/article/pii/0022314X77900075 proved $$f(x_1,\ldots,x_n)\leq 2^{[(n+1)/2]}\hspace{0.5cm}\forall n\leq 11,$$ where the brackets denote the floor integer part. His result was improved in 1996 by M. J. Bertin http://matwbn.icm.edu.pl/ksiazki/aa/aa74/aa7444.pdf who proved $$f(x_1,\ldots,x_n)\leq 2^{[(n+1)/2]}\hspace{0.5cm}\forall n.$$ However, while I have no problems with Pohst's proof, I have some troubles with Bertin's one.
The key idea of her proof is the following: the maximum of the function $f$, seen as determinant of $M$, is estimated by the maximum of determinants of matrices similar to $M$ where the $x_i's$ are in $\{-1,0,1\}.$ In other words, the maximum is attained pushing the $x_i's$ to the boundary of their defining intervals.
My problem is that I am not convinced by this argument: I would agree that the maximum of the determinant is attained by pushing the elements of $M$ to the boundary if all these elements were independent of each other (the determinant would be an harmonic function of its variables). But this is not the case for $M$: once you have settled $x_1,\ldots,x_n$ you immediately have all the remaining elements of the matrix. Just look at the first line, formed by $1, x_1, x_1^2x_2, x_1^3x_2^2x_3$ and so on.
In the end, I do not get an exhaustive explanation of the estimate from the paper: is there some step or detail that I am missing? Any suggestion is well accepted.
