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Small degree implies linear factorisation

Source: Original RMM 2019 P6

June 21, 2020
polynomialalgebracontest problemmultivariate polynomial

Problem Statement

Let P(x)P(x) be a nonconstant complex coefficient polynomial and let Q(x,y)=P(x)−P(y).Q(x,y)=P(x)-P(y). Suppose that polynomial Q(x,y)Q(x,y) has exactly kk linear factors unproportional two by tow (without counting repetitons). Let R(x,y)R(x,y) be factor of Q(x,y)Q(x,y) of degree strictly smaller than kk. Prove that R(x,y)R(x,y) is a product of linear polynomials.
Note: The degree of nontrivial polynomial ∑m∑ncm,nxmyn\sum_{m}\sum_{n}c_{m,n}x^{m}y^{n} is the maximum of m+nm+n along all nonzero coefficients cm,n.c_{m,n}. Two polynomials are proportional if one of them is the other times a complex constant.

Proposed by Navid Safaie
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