Interesting how mathematicians simplify the the number system to test theories. In Fermat's Last Theorem, the author describes Andrew Wile using the same tool of a reduced set of numbers cycling as if on a clock face in order to test his hypotheses.
When I have a tricky bit of coding to get on with, I often just resort to prototyping with very simple test cases - e.g. just simple arrays like [[dog, 1], [cat, 2], [cat, 1], [bird, 3]] (OK, not that simple, but you get the idea). Yet I often find that others are unwilling to go back to basics as if it is too time consuming - they would rather plug away with whatever unwieldy data set and tangle of relationships that comprise the ultimate problem set of input data.
Breaking something down to very simple fundamentals can be key.
I supervise a lot of grad students writing numerical/machine learning code and my experience is if they’re not prorotyping their code on simple cases in your way, then it is almost certainly riddled with bugs they are unaware of.
As a lowbrow polynomial abuser whose career depends on surfacing the number of roots and their ballpark location —this article is inordinately well written.
To learn about the alchemy that saved my career: Eigenwillig, A. 2007. On multiple roots in descartes rule and their distance to roots of higher derivatives.
I could be totally wrong with this guess, but my fear is that it involves fitting high-order polynomials to some kind of scientific data. In my experience, a lot of applications of polynomials happen because physical scientists have never heard of, or are uncomfortable with, fitting data with anything but polynomials, even though splines are generally far superior for data-driven curve fitting.
Interesting.
New idea: measure rank in the ‘prime function’. Is it of low or incredibly high rank? And if the latter, does this ‘prove’ that there is no simple polynomial generating functon?
Interesting how mathematicians simplify the the number system to test theories. In Fermat's Last Theorem, the author describes Andrew Wile using the same tool of a reduced set of numbers cycling as if on a clock face in order to test his hypotheses.
When I have a tricky bit of coding to get on with, I often just resort to prototyping with very simple test cases - e.g. just simple arrays like [[dog, 1], [cat, 2], [cat, 1], [bird, 3]] (OK, not that simple, but you get the idea). Yet I often find that others are unwilling to go back to basics as if it is too time consuming - they would rather plug away with whatever unwieldy data set and tangle of relationships that comprise the ultimate problem set of input data.
Breaking something down to very simple fundamentals can be key.