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Questions tagged [palindromic-polynomials]

This tag is used for both basic and advanced questions on polynomials that are palindromic; that is, reversing the coefficients (e.g., the highest and lowest degree coefficients swap) leaves the polynomial unchanged. This tag also covers polynomials that are antipalindromic, conjugate reciprocal, self-inversive, and similar variations.

A polynomial $f(X) = a_n X^n + \cdots + a_1 X + a_0$ is palindromic if $a_i = a_{n-i}$ for all $i$; or, equivalently, if the polynomials $f^\ast(X) = X^n f(1/X)$ and $f(X)$ coincide. Variations on this idea include Cohn's theorem, which concerns the number of roots on the unit disk of a polynomial satisfying $f^\ast = \alpha f$ for some fixed $\alpha\in \mathbb{C}$ with $|\alpha| = 1$.

14 questions
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If $F(x)=G(x)H(x)$ for polynomials $F$, $G$, $H$ with positive integer coefficients, and if $F$ is palindromic, then must $G$ and $H$ be palindromic?

Set $\mathbb{N}^*=\mathbb{N}\setminus \{0\}$. Let $F(x) = F_0 + F_1x + \dots + F_rx^r$ be a polynomial in $\mathbb{N}^*[x]$ and suppose that $F(x)$ is palindromic, i.e., $F_i = F_{r - i}$ for all $i=0,\dots,r$. Assume further that $F(x) = G(x)H(x)$,…
user969954
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Palindromic combinations of Chebyshev Polynomials share common roots?

Suppose that the real polynomial below $$p(x)=\sum_{k=0}^{2n}\alpha_{k}x^k$$ is a palindromic polynomial of even degree; that is, $p_{2n-k}=p_k$ for $0\leq k\leq 2n$ and $\alpha_0\neq 0$. Is it true that the combination of Chebyshev polynomials (of…
user123641
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Primes that can be represented as monic palindromic polynomials with integer coefficients: $f(x) = x^4+ ax^3+ bx^2+ ax + 1$?

Main question: Which are the primes that can be represented as monic palindromic polynomials with integer coefficients of degree 4: $$f(x) = x^4+ ax^3+ bx^2+ ax + 1.$$ If we take $x=2$, $$p = 16 + 8a + 4b + 2a + 1 = 10a + 4b + 17$$ If $p$ is an odd…
vvg
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Generalization of Palindromic number

Consider a number $n > 0$ in base $b ≥ 2$, where it is written in standard notation with $k+1$ digits $a_i$ as: $${\displaystyle n=\sum _{i=0}^{k}a_{i}b^{i}}$$ with, as usual, $0 ≤ a_i < b$ for all $i$ and $a_k ≠ 0$. Then n is palindromic if and…
Pruthviraj
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A question about a palindromic polynomial of even degree

In Wikipedia it is stated, If $p(x)$ is a palindromic polynomial of even degree $2d$, then there is a polynomial $q$ of degree $d$ such that $p(x) = x^d q(x + \frac{1}{x})$. My question is: How does one find the polynomial $q$ given $p$. The…
user276611
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Palindromic representation in base $x$ and Integer factoring

If $n \in \mathbb{Z}$ is representable as the product of two palindromic integers in base $x$. i.e., $$ \begin{align} n &= (x^4+px^3+qx^2+px+1)(x^4+rx^3+sx^2+rx+1) \newline &=x^8+x^7 (p + r)+x^6 (p r + q + s)+ x^5 (p (s + 1)+ q r + r) \newline & +…
vvg
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$\frac{1}{\alpha}$ is a root whenever $\alpha$ is a root of $p \iff p$ is a palindrome or an anti-palindrome

Let $$p(x)=\sum_{i=0}^{n}a_ix^i$$ be a polynomial in $\mathbb{R}[x]$ such that $0$ and $1$ aren't roots of $p.$ Prove that $\frac{1}{\alpha}$ is a root of $p$ whenever $\alpha$ is a root of $p,$ and they have the same multiplicity iff…
aqualubix
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Solvability criteria for a a monic almost palindromic quartic diophantine equation

My main question is: Is there a criteria for solving a quartic Diophantine equation of the form $$x^4 + ax^3 + bx^2 + ax + d = 0 \tag 1$$ We have the restriction $d \ne 1$. Here's my effort in obtaining a criteria and I would like the solution to be…
vvg
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Factoring palindromic polynomial

Q1. Is every sufficiently large composite integer a palindromic number in some non-trivial‡ base $B$? The trivial case of $n = 1(n-1) + 1 = (11)_{n-1}$ means it is palindromic in at least one base i.e., $n-1$. This MSE question and the sequence…
vvg
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Prove there does not exist an integer polynomial with roots $r^{\pm 1}e^{\pm \textrm{i}\theta}$

I am interested to know whether there is an intuitive/straightforward proof of the following result. There is no monic polynomial $p\in\mathbb{Z}[x]$ with roots $re^{\textrm{i} \theta}, re^{-\textrm{i} \theta}, r^{-1}e^{\textrm{i} \theta}$, and…
Zephos
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Existence of a self-reciprocal polynomial with certain root properties

Does there exist a monic self-reciprocal integer polynomial $p\in\mathbb{Z}[x]$ of degree 10 with roots $r\textrm{e}^{\pm\textrm{i}\theta_1}, r^{-1}\textrm{e}^{\pm\textrm{i}\theta_1}, r\textrm{e}^{\pm\textrm{i}\theta_2},…
Zephos
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Recovering a polynomial from its product with its reciprocal

Let $p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0\in\mathbb{Z}[x]$ be an integer polynomial. We can form the reciprocal polynomial $p^\ast(x)=a_0x^n+a_1x^{n-1}+\cdots+a_{n-1}x+a_n$. The product $p(x)p^\ast(x)$ is a palindromic polynomial. Can we…
Thomas Browning
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Is there a clever way to factor this special degree four polynomial?

Suppose that $f(z)=\alpha z^4+\beta z^3+\gamma z^2+\overline{\beta}z+\overline{\alpha}.$ Furthermore, suppose that two of the roots are complex and lie on a unit circle (and are conjugate to each other), but we don't know what they are. Is there a…
math314
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Palindromic Polynomial

What are the steps to solve this and what is the goal? $$ax^6 + bx^5 + cx^4 + dx^3 + cx^2 + bx + a$$
donutcrazy
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