Prove that if the polynomial can be written as the product of two monic polynomials with real non-negative coefficients, then those coefficients are all 0 or 1.
Proof by Andrei Stefanescu:
We call any polynomial symmetrical if and only if for all i. We will use the following lemma:
LEMMA: If P is a polynomial which can be written as , where Q and R are symmetrical polynomials, then P is symmetrical too.
Let and . We have that and . Since Q and R are symmetrical, it follows that and . Hence, for any i, and thus P is symmetrical.
Let and assume that , where g and h are nonconstant polynomials with non-negative coefficients. Let and . Since , all the complex roots of g and h have absolute value 1. Also, if , then . It follows that both g and h are a product of symmetric terms of the form or , hence by lemma, g and h are symmetrical.
If all the coefficients of g and h are in we are done. Otherwise, let k be the least number such that is not a subset of . Since , it follows that , thus . As g is symmetrical, . Computing the coefficient of gives us . Since and all the terms of the sum are non-negative, we obtain , thus one of and (say, )is 0. Computing the coefficient of , we have . As for and , all the terms but must be in . It follows that must be in , which leads to a contradiction with the definition of k.