A New Formula for the Determinant

For context when I was like 6 years old I was writing code in BASIC on a “TRS-80 Model III” to do linear algebra for fun (trying to make an arcade game).

But never learned algebra until I was forced to at ~20 years old in university.

So my understanding of mathematics was always mechanical (algorithm) and never very abstract. And it always was extremely hard to read research paper because the Greek letter don’t even mean the same thing in a (Statistic/ machine learning) paper and in a (physics/ calculus paper). And the author never properly define all the function and variables and assume you are already an expert.

Would they still reject it if all you do is add a reference to a github page?

There’s a bit of a subtlety here, which is that the formula evaluates to zero whenever the partial partition contains a singleton. For example, the partial partitions of {1,2,3} are:

  {}
  {1}
  {2}
  {3}
  {1, 2}
  {1, 3}
  {2, 3}
  {1, 2, 3}
  {1, 2} | {3}
  {1, 3} | {2}
  {1} | {2, 3}
  {1} | {2} | {3}

  {}
  {1, 2}
  {1, 3}
  {2, 3}
  {1, 2, 3}

In general the number of partial partitions of n elements that have no singleton parts is the n’th Bell number[0]. The Bell numbers grow much more slowly than the factorials, but still much faster than any polynomial.

[0] https://en.wikipedia.org/wiki/Bell_number

They're certainly not above writing code. You seem to have missed the appendix, where they wrote some to prove that their bound on the tensor rank of 4x4 determinants over GF2 is exactly 12. https://gitlab.com/apgoucher/det4f2

Also, a formula is not an algorithm. You can oftentimes extract an algorithm from a formula, but the value of a formula in pure math is to be able to write something in a different way. Formulas reveal structure that (hopefully) can be exploited when solving other problems.

> People could even get down to running the code just to check that the formula actually works !

This is an interesting perspective, but you have to take into account that papers are written for some audience in mind -- this one is written for combinatorialists (hence the math.CO designation on the arXiv). It's not too hard of an exercise to plug this formula into Mathematica for this audience. But also, this audience would rather see a proof than computational evidence.

Huh? Some differences:

- A formula says nothing about evaluation order, sometimes not even about evaluation (example: where’s the algorithm in e^(iπ) -1 = 0?)

- the computations that an algorithm makes cannot always be expressed in a closed-form expression (example: what’s the formula for quicksort?)

A formula is “a concise way of expressing information symbolically” (https://en.wikipedia.org/wiki/Formula)

The higher math researchers usually pay for it being less accessible, because the first engineering paper that uses their works gets cited a lot more.

But that also leads to cargo-cult engineering, where the first implementation of the paper just gets copied around without understanding (e.g. if there are limitations/assumptions).

They are not the same - there's plenty of formulas for which there is no algorithm, and even proofs in complexity theory proving there can never be an algorithm.

Algorithms merely formalize a tiny subset of math as things which are computable via a step by step process, and modern computers grew out of exactly the question of what things are computable. Turns out not much compared to the space of all "things."

Start here to learn more https://en.wikipedia.org/wiki/Computable_function