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Polynomials with same degree

Source: 2023 Taiwan TST Round 1 Independent Study 2-A

March 16, 2023
algebrapolynomialTaiwan

Problem Statement

Given some monic polynomials P1,…,PnP_1, \ldots, P_n with real coefficients, for any real number yy, let SyS_y be the set of real number xx such that y=Pi(x)y = P_i(x) for some i=1,2,...,ni = 1, 2, ..., n. If the sets Sy1,Sy2S_{y_1}, S_{y_2} have the same size for any two real numbers y1,y2y_1, y_2, show that P1,…,PnP_1, \ldots, P_n have the same degree.
Proposed by usjl
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