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Registered Member #72
Joined: Thu Feb 09 2006, 08:29AM
Location: UK St. Albans
Posts: 1659
It's neither science nor computing, but it is general, so I guess here's the least worst place to ask
Polynomials are defined for both real and complex numbers, a third order polynomial in x is for instance y=a0.x^3 + a1.x^2 + a2.x + a3, also writen as y=a0.x.x.x + a1.x.x + a2.x + a3
Where x is complex, the cubed term is easiest to visualise in polar coordinates, so writing complex x as R@theta, x=mag(x)@ang(x), x^3 is mag(x)^**link**(x).
Now the question. If instead of x.x.x, I want x'.x.x, where x' is the complex conjugate of x, then this product in polar terms is mag(3)^3@ang(x), as x'.x has the same magnitude as x.x, but is a real number, so the final angle is just ang(x), not 3.ang(x).
What is an equation of the form y=a0.x'x.x + a1.x'.x + a2.x + a3 called, is it a polynomial? I don't think so, but what is it called? The mathematics of polynomials is well defined, with all sorts of theorems about their properties and their roots, but is there any branch that studies equations of the form I have written?
Without a name for them, it's a bit tricky to search for them.
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Mathematics Question on not Polynomials
