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Find all ints n for which n-independent polynomials exist

Source: IMO Shortlist 2000, A7

August 10, 2008
algebrapolynomialmodular arithmeticcoefficientsIMO Shortlist

Problem Statement

For a polynomial P P of degree 2000 with distinct real coefficients let M(P) M(P) be the set of all polynomials that can be produced from P P by permutation of its coefficients. A polynomial P P will be called n n-independent if P(n) \equal{} 0 and we can get from any Q∈M(P) Q \in M(P) a polynomial Q1 Q_1 such that Q_1(n) \equal{} 0 by interchanging at most one pair of coefficients of Q. Q. Find all integers n n for which n n-independent polynomials exist.
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