Minimize the number of roots
Suppose \(p(x)\) and \(q(x)\) are polynomials of degree \(100\) with complex coefficients, having no common zeroes. Find the smallest possible total number of complex zeroes of the polynomials \(p,\) \(q,\) and \(p-q,\) counted without multiplicity.
Details and assumptions
When the zeroes are counted without multiplicity, the polynomial \(x^2(x-1)^3\) has two zeroes: \(x=0\) and \(x=1.\)