MathDB
Pretty Polynomials

Source: 2021 MEMO T-2

September 5, 2021
polynomialalgebrafloor functionmemoMEMO 2021

Problem Statement

Given a positive integer nn, we say that a polynomial PP with real coefficients is nn-pretty if the equation P(⌊x⌋)=⌊P(x)⌋P(\lfloor x \rfloor)=\lfloor P(x) \rfloor has exactly nn real solutions. Show that for each positive integer nn
[*] there exists an n-pretty polynomial; [*] any nn-pretty polynomial has a degree of at least 2n+13\tfrac{2n+1}{3}.
(Remark. For a real number xx, we denote by ⌊x⌋\lfloor x \rfloor the largest integer smaller than or equal to xx.)
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