STATUS: Draft


Solving Polynomial Equations (4)

Let g1 be solutions of $a_i$ for a general cubic from the notebook: Solving Polynomial Equations (3).

Create needed variables.

Recall solutions of general cubic where $ a_0 = 0$:

Introduction

Goal: find pattern in these numbers.

Observe: Patterns occuring in indices, powers, coefficients and signs.

Observe: All terms $a_k$ have the same total degree (k - 1) (e.g each term in $e_6$ is homogenous).

Observe: There appears to be some kind of relationship between indices and powers (note indices are related position).

Definition: The index degree of a term $c_1^m c_2^m c_3^m $ be the sum of the products of the indices and the degrees of each of the consituent variables and the claim is that this is constant in each of the values. It appears then that the index degree is always twice the usual degree (when indices are ignored

Observe: total usual degree is (k - 1) while the index degree is 2 (k - 1).

Strategy: Place all numbers in a matrix of coefficients (of size 7 x 7) , padded with with zeros to enable pattern exploration

Strategy: Prime decomposition

Strategy: Examine in the contdext of $c_1, c_2$ and $c_3$

Summary

Looking for patterns in solutions in is a powerful strategy to understand if a solution can be generalised or has some dependency on some kind of on-series or on-sequence. Rendering numbers in matrices tends to expose these patterns and also allows the use of linear algebra to seek patterns which may not be as obvious.