Many-valued Polynomials

Multi-valued algebra promises to help with multi-valued problems such as: if x^2=1 and y = 2x then what is x+y? They aren't super popular as it is hard to make them into them into something consistent like a proper mathematical field. But never mind hey.

Let's define a multi-valued algebra as one where each element is a closed set of complex numbers, let's call the set of such closed sets M. Equality is set equality, summation is the Minkowski sum, and subtraction is the inverse of the Minkowski sum (which doesn't always have a solution).

Let's use capital letters for members of the multi-valued algebra, and lower case letters for single complex numbers.

In this system a 'complex linear multi-valued polynomial' means the complex-valued function:

y = aX + b

Which only occurs when X has a single element, so not very interesting.

a 'complex quadratic multi-valued polynomial' is:  

y = aX^2 + bX + c

which for the specific case a=1, b=-1, c=0, can be written as:

X = X^2 + y

which is reminiscent of the Julia set iteration for each y. 

Indeed the value of X is indeed a Julia set of that complex number, as it is the largest closed set that satisfies the equation. It also remains a Julia set when a,b,c are different values, just with a Euclidean transform applied.

So the function M->C is defined for Julia sets, and each Julia set is the preimage of its corresponding parameter y.

If we 'expand out' the domain into its individual complex elements then the function becomes a many-to-many map m(z): C->C, and m(0) is the Mandelbrot set. So the Mandelbrot set is the image of the number zero under this mapping. More generally m(x) are the generalised Mandelbrot sets that start at point x.  

Above: m(0.3i) and m(0.3-0.3i), these are variants of the Mandelbrot set.

We could approximate this idea by saying that the quadratic polynomial's "multivalue-range" is the generalised Mandelbrot sets, and its domain is the Julia sets.

While we have done a lot of hand waving here, the interesting result is that Julia sets and Mandelbrot sets are fundamental elements of multi-valued quadratics. Fixed y cross section is a Julia set and the fixed z cross section is a Mandelbrot set. 

I haven't explored higher order polynomials, but it seems probably that these produce higher order Julia sets and the higher order 'multibrots', and perhaps linear combinations of these at each order.

This is a nice idea as it suggests that Julia and Mandelbrot/Multibrot sets are the fundamental elements of this multi-valued algebra. If we make the leap that standard algebra is missing something in its inability to answer questions such as the initial one, then we may even consider Julia and Multibrot sets to be fundamental quantities more widely in algebra.

To add to this, the Multibrot sets are universal objects, which loosely means that you will find them in bifurcating mappings. 

The mapping m() can also be seen as a four dimensional shape. In this shape the Julia sets are the horizontal cross-sections, and the generalised Mandelbrot sets are the orthogonal cross-sections. It would be interesting to see what diagonal cross-sections look like.

An additional interesting link is between the Mandelbrot set and 2. I mean 2 in probably the most general sense as the fixed point of x^2-x. Just as 1 (or identity) is the fixed point of x^2 and 0 is the fixed point of 2x. These identities define 0 and 1 on other fields than just real numbers.

For many-valued quantities that would be the fixed point of X^2-X, which is X^2-X=X, which is probably the disk of radius 2. However, a slightly altered definition finds the set of complex values x such that X^2-X=x. This is the set of Julia set parameters that are in their own Julia set. This is the negative Mandelbrot set, a set that includes the number 2 in fact.