What is that function? Polynomial?

Question

Is it a polynomial or rational polynomial or else?

$y = \dfrac{a}{x^4} + \dfrac {b}{x^2} + c$

I need to fit a curve to a discrete data of that form, so I need to know what fitting to use.

Answer 1

We have $$\frac{a}{x^4}+\frac{b}{x^2}+c=\frac{a}{x^4}+\frac{bx^2}{x^4}+\frac{cx^4}{x^4}=\frac{a+bx^2+cx^4}{x^4}$$ This is a rational function.

Answer 2

The equation $y-a/x^4-b/x^2-c=0$ yields $$ f(x,y)= - a - bx^2 - cx^4 + x^4y=0, $$ which is an affine algebraic curve, see here.

Answer 3

If we consider that the problem is to fit data using $(x_i,y_i)$ data points for a model $$y = \dfrac{a}{x^4} + \dfrac {b}{x^2} + c$$ define two variables $u_i=\dfrac{1}{x_i^4}$ and $v_i=\dfrac{1}{x_i^2}$ which make $$y=a u+b v +c$$ and the problem is just to perform a multilinear regression for which the normal equations are very simple to establish and use (otherwise, matrix formulation).