"consider the collection of all polynomials (with complex coefficients) of degree less than N in x" okay so i'm considering a set of polynomials with complex coefficients and degree of the polynomial is less than N. what is this "in x" part that's being referred to? what's x?...
x is the variable in the polynomial. For example, a typical polynomial in this set is f(x) = a_{0} + a_{1}x + a_{2}x^{2} + ... + a_{N-1}x^{N-1}. The coefficients a_{0}, a_{1}, etc. are complex, and x can take on complex values.
As opposed to [itex]2- 3y+ y^2[/itex] or [itex]4- 7z^2+ 8z^9[/itex] which are "in" y and z rather than x!
Admittedly, that's a weird thing to say. I think it's likely that another set of polynomials is introduced, and so it is a matter of having a consistent notation to follow. It would be strange to just talk about one set of polynomials, and to care about what symbol the variable uses.
There are at least two reasons for the introduction of "in x" when describing these polynomials; 1) In a more formal setting, polynomials are treated like strings of symbols where some of the symbols (the coefficients) come from one set and the other symbols (the powers of the "variable(s)") from another. In order to avoid problems, it is usually stipulated that the variable(s) can't also be coefficient symbols, and so one must stipulate what the "variable" symbol is. In these settings the "variable" usually isn't meant to actually be a variable/place-holder, and the polynomial isn't intended to represent a "function" per se. This idea can be extended to formal "power series over __ in __", which are quite powerful in a somewhat "abstract nonesense" sorta way; one can develop quite a bit of complex analysis, for example, without even talking about complex numbers and functions. Heck, one particularly common way of developing the complex numbers uses formal polynomials. 2) To to those familiar with the "polynomial in __" terminology, one can talk about polynomials in ##e^x## ( ##e^{2x}+2e^x+1##), polynomials in ##\sin\theta## (##\sin^2\theta+2\sin\theta+1##), polynomials in ##y^2## (##y^4+2y^2+1##), etc. Methods for solving equations of these types then become more obvious; i.e. we can use techniques for solving polynomial equations to help us solve equations involving things that wouldn't normally be considered according-to-Hoyle polynomials.