MathDB
Polynomials

Source: Serbia TST 2022, Problem 1

May 31, 2023
polynomialalgebra

Problem Statement

For a non-constant polynomial P(x)=anxn+an−1xn−1+…+a1x+a0∈R[x],an≠0,n∈NP(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots+a_{1} x+a_{0} \in \mathbb{R}[x], a_{n} \neq 0, n \in \mathbb{N}, we say that PP is symmetric if ak=an−ka_{k}=a_{n-k} for every k=0,1,…,⌈n2⌉k=0,1, \ldots,\left\lceil\frac{n}{2}\right\rceil. We define the weight of a non-constant polynomial P∈R[x]P \in \mathbb{R}[x], denoted by t(P)t(P), as the multiplicity of its zero with the highest multiplicity.
a) Prove that there exist non-constant, monic, pairwise distinct polynomials P1,P2,…,P2021∈R[x]P_{1}, P_{2}, \ldots, P_{2021} \in \mathbb{R}[x], none of which is symmetric, such that the product of any two (distinct) polynomials is symmetric.
b) What is the smallest possible value of t(P1⋅P2⋅…⋅P2021)t\left(P_{1} \cdot P_{2} \cdot \ldots \cdot P_{2021}\right), if P1,P2,…,P2021∈R[x]P_{1}, P_{2}, \ldots, P_{2021} \in \mathbb{R}[x] are non-constant, monic, pairwise distinct polynomials, none of which is symmetric, and the product of any two (distinct) polynomials is symmetric?
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