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2021-08-04, 01:30   #29
jwaltos

Apr 2012
Oh oh.

22·113 Posts

Quote:
 Originally Posted by Nick The short answer is that if you are working with a single polynomial equation of degree at most 4 over the complex numbers, then you can choose any values you like ..
Thanks for responding Nick. Unfortunately it didn't really help me out. I'm a little more advanced than the books you recommended (or at least think I am) and where you state "you can choose any values you like" that is the crux of the matter. There must be a guiding principle which must limit my options (free will concept should not be in play) to specific numeric values of the parameters related to the value(s) I'm seeking. I'm looking initially at "real world" applications, here is mathstackexchange link: https://math.stackexchange.com/quest...rtic-equations

Theoretically any univariate polynomial can be considered as a base number system. Consider the single parameter as 10 and simplify the coefficients to represent a base 10 number..and so on. Multivariate systems..same deal. Extending this concept within base 10..consider digit sums and the picene hydrocarbon. Then consider the inverse tessellation of the same. Then consider the geometric form as an infinite sheet or surface...and so on down this particular rabbit hole.

Considering the difference of squares, sum of squares, primes that are and aren't the sum of two squares geometrically, such relations provide some simple figures. Embellishing these figures properly leads to reciprocity and a few other gems. Given that the general quartic can be solved via radicals is a particularly interesting limit point.

In any case, dimensional analysis and measurable objects/influences is what I'm looking at presently. Prior to this I was examining the general cubic parabola from which I have since elucidated some interesting facts.

Just to keep in line (pun intended) with the purpose of this thread here are a few "light read" books containing some of the above ideas:
Geometry and the Imagination, Hilbert and the other guy.
Explorations in Mathematical Physics, An Elegant Language...,Koks
Curves for the Mathematically Curious, Havil
Proofs from the Book, 6th Edition

..and not so light..
https://scholarworks.umb.edu/cgi/vie...masters_theses
https://uma.ensta-paris.fr/files/pub...imon-Petit.pdf

Last fiddled with by jwaltos on 2021-08-04 at 02:05