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Questions tagged [polynomials]

Questions in which polynomials (single or several variables) play a key role. It is typically important that this tag is combined with other tags; polynomials appear in very different contexts. Please, use at least one of the top-level tags, such as nt.number-theory, co.combinatorics, ac.commutative-algebra, in addition to it. Also, note the more specific tags for some special types of polynomials, e.g., orthogonal-polynomials, symmetric-polynomials.

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34 views

Squares in $\prod_{m=1}^{M}\left(1+(\Phi_n(m))^2\right)$, where $\Phi_n(x)$ is the the $n$th cyclotomic polynomial, and a related sequence

In this post we denote the $n$th cyclotomic polynomial as $\Phi_n(x)$, as reference we've for example the Wikipedia Cyclotomic polynomial. And for each integer $m>1$ we denote the product of its ...
157 views

What generalizes symmetric polynomials to other finite groups?

Multivariate polynomial indexed by ${1, \ldots, n}$ are acted on by $S_n$: for $\sigma \in S_n$, define $\sigma(x_i) = x_{\sigma(x_i)}$, etc. Symmetric polynomials are those polynomials which are ...
306 views

Collinear Galois conjugates

This is inspired by this old question, which may provide a bit more background. But the two present questions seem somewhat more fundamental to me. Let $p$ be an irreducible polynomial with integer ...
130 views

85 views

Finding the number of real roots of a power series

Suppose we have a power series $f(x) = \sum_{i=0}^\infty a_nx^n$ that converges for all reals where $f(\pm1) \neq 0$. We want to find the number of real roots. In order to do that we find the roots ...
56 views

Proving Vizing's and Brooks' theorem using the polynomial approach

It is known that the graph polynomial defined by $\prod_{i<j}(x_i-x_j)$ where the vertices $x_k\ \ , \ \ k=\{1,2\ldots,n\}$ are ordered with respect to some order; can be used to verify the proper ...
23 views

Product of edge monomials and chromatic number

As a continuation of this question and as stated in the comments, I wish to ask if the minimum value of $rad$ for all monomials equals the chromatic number when the number of variables in a monomial ...
127 views

If two monic polynomials of $\mathbb{Z}_p[X]$ (p-adic integer coefficient) are relatively prime modulo p, then do they generate $\mathbb{Z}_p[X]$?

I'm currently reading a paper of Rene Schoof, and I got stuck in a line. And I'm trying to check the above sentence. Although that seems to be elementary, I hope someone can give me a counterexample ...
145 views

Probability that a Random Monic Polynomial Has Few Real Zeros

In the paper https://arxiv.org/pdf/math/0006113.pdf, it is shown that the probability that a random polynomial $a_0 + a_1x + \cdots + a_n x^n$ has $o(\log n / \log\log n)$ real zeros is $n^{-b + o(1)}$...
169 views

Chromatic number and graph polynomial

If $\prod_{i=1}^t x_i^{e_i}$ is a monomial, define $$rad\biggl(\prod_{i=1}^t x_i^{e_i}\biggr)$$ to be the number of distinct (nonzero) values of $e_i$. Now let $G$ be a simple graph with vertices ...
187 views

Polynomial defined recursively by a resultant

Cross posting from MSE. Definition: For any natural number $n\ge 3$, define the polynomial $P_{n}\left(x_1,x_2,...,x_{n-1},x_{n} \right)$, with indeterminates $x_{i}$, where $i\in\{1,2,...,n-1,n\}$, ...
106 views

I cam across with this conjecture (http://mathworld.wolfram.com/SchinzelsHypothesis.html) My question is: Does these conditions imply the existence of some integers $x$ such that all the polynomials ...
113 views

Example of applying real quantifier elimination algorithm for polynomials

Sorry if any of this is unclear, or doesn't make much sense, I'm still trying to figure it out, a practical example such as this would likely help me understand better than anything else. I have read ...
303 views

Existence of algebraic integers with certain properties

Is the following statement true? ($\star$) Given integers $n > k > 0$, there exists a monic polynomial of degree $n$ with integer coefficients and constant term $\pm 1$, irreducible over \$\...

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