In standard polynomials, yes, all exponents are positive integers. If x is the input and y is the output, a polynomial of degree 2 is:
y = ax^2 + bx + c
So three parameters (a, b, c) specify the polynomial. Beware that some people write this as:
y = a + bx + cx^2
If we have (n) known correspondences between x and y, it is true that a polynomial of degree (n-1) can be found that passes exactly through those given points. But what does the curve do between those points? Sadly, it often does stupid things, going madly beyond what a human would consider sensible or desirable.
For that reason, we often prefer a series of joined paramaterised cubic curves (or Bezier curves or B-splines, they are all variations of the same thing). Each curve starts and ends at two of the desired points but its shape is influenced by two other points. Each curve joins to the next curve at one point, and has the same slope and curvature.
Bezier curves passing through given points more closely follow what a human would draw. Gimp and other editors use them in the “curve editor”. (Though I don’t like the Gimp curve editor because it can cause clipping.)
A power curve, in the form …
y = a*x^p + b
… has parameters a, p and b, and these have fairly intuitive meanings (“gain”, “gamma” and “lift” in some editors). They are easer to calculate than Beziers, and are better-behaved than polynomials.
EDIT: corrected the counting of degrees.