MathDB

Problem 3

Part of 2013 VJIMC

Problems(2)

Z[x] on set of integers, bounding closure measure of set

Source: VJIMC 2013 1.3

5/31/2021
Let SS be a finite set of integers. Prove that there exists a number cc depending on SS such that for each non-constant polynomial ff with integer coefficients the number of integers kk satisfying f(k)∈Sf(k)\in S does not exceed max⁡(deg⁡f,c)\max(\deg f,c).
Setsnumber theoryPolynomials
Z[x], P(\sqrt[3]5+\sqrt[3]25)=5+\sqrt[3]5

Source: VJIMC 2013 2.3

5/31/2021
Prove that there is no polynomial PP with integer coefficients such that P(53+253)=5+53P\left(\sqrt[3]5+\sqrt[3]{25}\right)=5+\sqrt[3]5.
algebranumber theoryPolynomials