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Find a certain polynomial

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December 19, 2019
algebrapolynomiallinear combination

Problem Statement

Let f1,f2,f3,f4 f_1,f_2,f_3,f_4 be four polynomials with real coefficients, having the property that f_1 (1) =f_2 (0),   f_2 (1) =f_3 (0),  f_3 (1) =f_4 (0),  f_4 (1) =f_1 (0) . Prove that there exists a polynomial f∈R[X,Y] f\in\mathbb{R}[X,Y] such that f(X,0)=f_1(X),  f(1,Y) =f_2(Y) ,  f(1-X,1) =f_3(X),  f(0,1-Y)=f_4(Y) .
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