General Systems Polynomial Equations

General Systems Polynomial Equations

We want to find a solution to the general problem of systems of polynomial equations. (There is a simple algorithm to solve these problems but maybe a neater way can be found?)

Firstly all systems of polynomial equations can be written in the following form:

∪_i=1..n{X^TA_iX = 0}

with X a vector, A a matrix with X₀=1. The solutions should all be written in terms of determinants {A_i+..}. Or using the antisymmetric tensor.

det(Q) = {Q}

One equation - n variables
For one equation a solution exists if:

 {Q} > 0

This can be written in terms of traces.
One variable: 2{Q} = [Q]²-[Q²]
Two variables: 6{Q} = [Q]³-3[Q][Q²]+2[Q³]
etc.
The general formula from traces comes from it's expression as a determinant.
QQQ...εε

Two Equations

Two equations - One(n?) Variable(s)
A solution exists (common root) if (each has a solution as above):
([Q]²-[Q²])([R]²-[R²]) - ([Q][R]- [QR])²  = 0

4{Q}{R} - ( {Q+R}-{Q}-{R} )² = 0

4 det(Q) det(R) - (det(Q+R)-det(Q)-det(R))² = 0 ??

(and condition they are not imaginary roots??)
Can this be written in terms of {Q}, {R} and {Q-R}?
2{Q} = [Q]²-[Q²]
2{R} = [R]²-[R²]
{Q-R} = [Q-R]²-[(Q-R)²] = ([Q]²-[Q²])  +  ([R]²-[R²])  +  2([QR]-[Q][R])
{Q+R} = [Q+R]²-[(Q+R)²] = ([Q]²-[Q²])  +  ([R]²-[R²])  +  2([Q][R]-[QR])
{Q-R} - {Q}-{R} = 2([QR]-[Q][R])
{Q+R} - {Q}-{R} = -2([QR]-[Q][R])

Two equations - Two Variables
(This is the intersection of ellipses)
(A 12D polynomial!) Can this be made from traces? e.g. [Q¹²]+[R]¹² ?
??? 4{Q}{R} - ( {Q-R}-{Q}-{R} )( {Q+R}-{Q}-{R} )  = 0  (6D polynomial)


Two equations - Three Variables
Is there a general formula made from traces?

Three Equations

Three equations - One Variable
They ALL share a common root if the determinant of the three equations is zero. (Can't write this in trace notation!!! in terms of Q, R & S ??)


= A¹¹_iA¹²_jA²²_kε^ijk   A=(Q,R,S)
Try{Q+R+S}

ε=((0,1),(-1,0)) (Doesn't generalise to more variables). (det(Q)=[QεQε]
[εQRS]+[εRSQ]+[εQRS]

N equations

N equations - One Variable
If any equation has a triple with all the others then there MUST be a common solution! So only need to test for triples using above determinant


N equations - Two Variables

N equations - N Variables
The maximal case