TITLE: Does adding to degree n polynomial terms with negative exponents still have n roots?
QUESTION [6 upvotes]: A polynomial has n roots. You can add terms less than n and still have a polynomial with n roots. Does this continue if you take negative exponents?
Eg.
$$ax^2 + bx^1 + cx^0 + dx^{-1}...$$
Edit: I guess this isn't a polynomial per se, but does it still have n roots?

REPLY [8 votes]: The Laurent polynomal $P(x) = ax^n + bx^{n-1} + cx^{n-2} + ... + ux^{-m+2} + vx^{-m+1} + wx^{-m}$ can be equivalently expressed as the quotient $\frac{ax^{m+n} + bx^{m+n-1} + cx^{m+n-2} + ... + ux^2 + vx + w}{x^m}$, whose roots are the roots of its numerator, a polynomial of degree $n + m$.  The maximum number of roots is thus $n + m$, the sum of the most positive and most negative exponents.