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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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0answers
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### 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 ...
1answer
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 ...
1answer
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 ...
1answer
130 views

0answers
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 ...
0answers
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 ...
0answers
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 ...
1answer
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 ...
0answers
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 $n^{-b + o(1)}$...
1answer
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 ...
0answers
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 $i\in\{1,2,...,n-1,n\}$, ...
0answers
106 views

### About the Schinzel's hypothesis

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 ...
0answers
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 ...
2answers
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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