If f is a polynomial with all roots of multiplicity one, then the derivative does not have a zero in common with f. To see this, express f factored, use product rule for derivatives.
A lot of modern analytic number theory advances are being made using hardy's major and minor circle (arc?) Technique, but most advances just seem to be bounds improvements
The book linear algebra done right avoid the use of determinants until the very end. The proofs are done without t he determinant. If det makes you uneasy, check it out!
Has any one set up a generating function yet? Usually those make easy work of counting problems. I have to take the girl to the dentist but I'll check it out later for you. I wonder what your marbles are : p