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Factoring Multivariable Polynomials

Source: 2020 USOJMO Problem 6

June 21, 2020
AMCUSA(J)MOUSAJMO

Problem Statement

Let n≥2n \geq 2 be an integer. Let P(x1,x2,…,xn)P(x_1, x_2, \ldots, x_n) be a nonconstant nn-variable polynomial with real coefficients. Assume that whenever r1,r2,…,rnr_1, r_2, \ldots , r_n are real numbers, at least two of which are equal, we have P(r1,r2,…,rn)=0P(r_1, r_2, \ldots , r_n) = 0. Prove that P(x1,x2,…,xn)P(x_1, x_2, \ldots, x_n) cannot be written as the sum of fewer than n!n! monomials. (A monomial is a polynomial of the form cx1d1x2d2…xndncx^{d_1}_1 x^{d_2}_2\ldots x^{d_n}_n, where cc is a nonzero real number and d1d_1, d2d_2, …\ldots, dnd_n are nonnegative integers.)
Proposed by Ankan Bhattacharya
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