**A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode**

“_....the hyper-Catalan numbers 𝐶𝐦 count the number of subdivisions of a polygon into a given number of triangles, quadrilaterals, pentagons, etc. (its type 𝐦), and we show that their generating series solves a polynomial equation of a particular geometric form. This solution is straightforwardly extended to solve the general univariate polynomial equation._”

Wildberger, N. J. and Rubine, D. (2025) ‘A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode’, The American Mathematical Monthly, pp. 1–20. doi: https://doi.org/10.1080/00029890.2025.2460966.

One day, one decomposition
A235034: Numbers whose prime divisors, when multiplied together without carry-bits (as encodings of GF(2)[X]-polynomials, with A048720), produce the original number; numbers for which A234741(n) = n

3D graph, threejs - webGL https://decompwlj.com/3Dgraph/A235034.html
2D graph, first 500 terms https://decompwlj.com/2Dgraph500terms/A235034.html

I ran across the Wikipedia article on Littlewood polynomials. It has a plot of all the roots of the degree 15 polynomials, that looks very nice. I thought I would create an animation showing the roots for degree 1, degree 2, etc. I also thought maybe I'd add a plot of the roots for something with degree higher than 15. Here is the degree 16 plot (this is reduced to 25% of the original image). It took 2 hours in Sage on my laptop, so I might try 17, even 18 - who knows? I have the thought that I ought to be able to reduce the precision, and this ought to speed things up a lot (since for plotting much lower precision than the default is needed). I don't particularly like blue, though: I'll have to try other colors.
https://en.wikipedia.org/wiki/Littlewood_polynomial

In #mathematics, a polynomial is a #mathematicalExpression consisting of #indeterminates (also called #variables) and #coefficients, that involves only the operations of #addition, #subtraction, #multiplication and #exponentiation to #nonnegativeInteger powers, and has a finite number of #terms. An example of a polynomial of a single indeterminate x is x2 − 4x + 7. An example with three indeterminates is x3 + 2xyz2 − yz + 1. #Polynomials appear in many areas of mathematics and science.

`We show that the Weierstrass method, like the well known #Newton method, is not generally convergent: there are open sets of #polynomials p of every degree d≥3 such that the dynamics of the Weierstrass method applied to p exhibits attracting periodic orbits.`

https://arxiv.org/abs/2004.04777

@rwxrwxrwx
I wrote this function #'LAMBDAISE that turns a cl-buchberger:polynomial into an unevaluated lambda form at run time. I feel like this is going to have a more elegant expression, but I figure if
the lambdaiseing is happening offline it's okay. What do you think? What do other #CommonLisp #lisp users think? #polynomials will use for synth later

#100daystooffload on codes for turning symbolic polynomials into lambda forms
https://gopher.tildeverse.org/tilde.club/0/~screwtape/synthember-100days-tooffload/008.lisp.txt

@82mhz @thankfulmachine @AlgoCompSynth

Learn how to symbolically #solve #math equations using computer #algebra in #Python: With the release of #SymPy 1.12, the symbolic #equation #solving guide pages I wrote are all live in the latest version of the documentation. I wrote ten pages on topics such as systems of #equations, ordinary #differential equations, #polynomials, #inequalities, #matrix equations, and Diophantine equations, and #numerical solving. https://docs.sympy.org/latest/guides/solving/index.html

Since I was teaching orthogonal #polynomials (OPs), I asked my class whether they’d read Great Expectations by #Dickens, which you may remember had APs :)