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integer polynomials p(X, Y ) with minimal degree

Source: 2014 Saudi Arabia GMO TST III p2

July 26, 2020

Integer Polynomialpolynomialalgebra

Problem Statement

Let

S = \{f(a, b) | a, b = 1,2,3, 4

and

a \ne b\}

, and consider all nonzero polynomials

p(X,Y )

with integer coefficients such that

p(a, b) = 0

for every element

(a,b)

S

. (a) What is the minimal degree of such polynomial

p(X, Y )

? (b) Determine all such polynomials

p(X, Y )

with minimal degree.