Could someone provide pointers the literature that discusses polynomials with real coefficients in two real unknowns which answers questions such as:
1. For which polynomials is it true that P(x,y) => 0 for all x and y?
2. For which polynomials can P(x,y) = 0 be resolved into y = f(x), that is, a single valued function?
3. Same question as #2 in the case where f is a multi-valued function (that is, for each x there is a finite set of value that satisfy y = f(x))?
4. Which polynomials P and Q have a unique real solution for P(x,y)=0 and Q(x,y)=0?