What have you tried? Did you calculate these for the first several values of n and look at the apparent growth rate? Did you try any heuristics based on the growth of primes? Did the results match?

Brahmagupta gave a parametrization of heronian triangles similar to Euclid's parametrization of right triangles. Have you tried playing with that?

I did not downvote the post.

What have you tried? Did you graph the function? Did you calculate a few terms of its Taylor series? If so, what did you see?

What have you tried? Did you plot the function for a few values of $n$, and if so what did you see?

Are you looking for a criterion or a classification? In the latter case, what are we classifying them up to? Isometry? quasi-Isometry? bi-Lipschitz homeomorphism?

Why have you written $\equiv$ here? Normally I'd think that denoted congruence modulo some number but there's no $\pmod{m}$ here...

1. According to the definitions I see online, the field of scalars of a pseudo Euclidean vector space is taken to be the reals by definition. 2. What to you mean by a pseudo Euclidean metric here other than the nondegenerate bilinear form itself?

Did you try it for some small values of $n$?

Let's call your initial sequence $a_i$ and the new sequence $b_i$. Two clarifications: (1) if $\sum_{i\in I} a_i = \sum_{j\in J} a_j$ must $\sum_{i\in I} b_i = \sum_{j\in J} b_j$? (2) If $\sum_{i\in I} a_i < \sum_{j\in J} a_j$ is the condition that $\sum_{i\in I} b_i < \sum_{j\in J} b_j$ or that $\sum_{i\in I} b_i \leq \sum_{j\in J} b_j$?

It seems like displaying the green lines doesn't convey any additional information, and for someone slightly colorblind like me having the lines be green and yellow makes it tough to see what's going on.

A guess: rewrite as $\det(D + i U^*BU)$ expand as a series in $i$, and collect