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.
Questions tagged [polynomials]
2548 questions
24
votes
1 answer
$f(x)$ is irreducible but $f(x^n)$ is reducible
Does there exist an irreducible polynomial $f(x)\in \mathbb{Z}[x]$ with degree greater than one such that for each $n>1$, $f(x^n)$ is reducible (over $\mathbb{Z}[x]$)?
Hesam
- 615
22
votes
2 answers
Number of zeros of a polynomial in the unit disk
Suppose $m$ and $n$ are two nonnegative integers. What is the number of zeros of the polynomial $(1+z)^{m+n}-z^n$ in the unit ball $|z|<1$?
Some calculations for small values of $m$ and $n$ suggests the following…
Mostafa
- 4,454
18
votes
2 answers
Polynomials with many zeros of absolute value 1
Let $S$ be a finite subset of the positive integers. Define $N_S(x) =
1-(1-x)\sum_{j\in S}x^j$. Assume that $N_S(x)$ is symmetric, i.e.,
$x^dN_S(1/x)=N_S(x)$, where $d=\deg N_S(x)$. It seems that $N_S(x)$
tends to have many zeros of absolute value…
Richard Stanley
- 49,238
15
votes
2 answers
sum of derivatives in roots of a polynomial of odd degree
Given odd positive integer $n$ and a monic polynomial $f(x)=(x-x_1)\dots (x-x_n)$ with $n$ distinct real roots. Is it always true that $\sum f'(x_i) > 0$? I may prove it for $n=3$ and $n=5$ and it looks plausible.
Fedor Petrov
- 102,548
14
votes
4 answers
Is a polynomial with 1 very large coefficient irreducible?
I am asking for some sort of generalization to Perron's criterion which is not dependent on the index of the "large" coefficient. (the criterion says that for a polynomial $x^n+\sum_{k=0}^{n-1} a_kx^k\in \mathbb{Z}[x]$ if the condition…
Gjergji Zaimi
- 85,056
14
votes
3 answers
Sets that can be mapped onto R^n by a polynomial
The question was edited several times. Most recent version, suggested by Fedja:
Does there exist an open set $U\subset \mathbb R^n$ (n>1) that contains balls of arbitrarily large radius and such that no polynomial mapping $p\colon \mathbb…
002
- 3
13
votes
2 answers
About irreducible trinomials
This question was inspired by this one. For every $n>m>0$ consider the polynomial $p_{m,n}=x^n-x^m-1$.
For which $m,n$ is $p_{m,n}$ irreducible over $\mathbb Q$?
In particular, if $m$ is odd, is it always irreducible?
user6976
8
votes
3 answers
Can irreducibility of polynomials be figured out in polynomial time?
I remember seeing somewhere "primarity test (of numbers) is harder than irreducibility test (of polynomials)", now as primarity test in polynomial time is known, can irreducibility test of polynomials over the integers be done in a fast way?
(I'm…
Yuhao Huang
- 4,982
8
votes
1 answer
polarization formula for homogeneous polynomials
given a homogeneous polynomial p of dgree n on $R^d$, there is a unique symmetric n-linear functional $B$ on $(R^d)^n$ such that $p(x)=B(x,..,x)$. The question is: Can we get $B$ by means of a polarization formula as in the case $n=2$ for quadratic…
mostafa
- 367
8
votes
3 answers
Is it possible to create a polynomial $p(x)$ with this relation between $p(0)$ and $p(c)$?
Given $b$ and $c$ with $b,c>1$, is it possible to construct a polynomial $p(x)$, whose degree is $n$ for all $c$ and $b$, such that:
$|p|$ is strictly increasing on $[1,c]$
and $|b \cdot p(c)| < |p(0)|$?
This might be satisfied by an…
DUO Labs
- 265
8
votes
1 answer
Patterns in roots of integer-coefficient polynomials
Below are shown two displays of all the roots of polynomials
$$c_n x^n + c_{n-1} x^{n-1} + \cdots + c_1 x + c_0 \;=\; 0$$
with each coefficient $c_i$ an integer $|c_i| \le M$
(including $c_i=0$).
No doubt this is all well-known, but
I would be…
Joseph O'Rourke
- 149,182
- 34
- 342
- 933
8
votes
0 answers
How prove this polynomial inequality from a book
Question:
Let $$f(X)=(X-x_{1})(X-x_{2})\cdots (X-x_{n})$$ be an irreducible polynomial over the field of rational numbers,with integer coefficients and real zeros.
Prove that
$$\prod_{1\le i
math110
- 4,230
7
votes
3 answers
Roots of this sextic
I'm looking for the roots of the sextic equation in $x$
$$
x^6 - (3 m) x^5 + 5 m^2 x^4 - (5 m^3) x^3 + 3 m^4 x^2 - m^5 x + L = 0.
$$
I know that at most two of the roots of this are real when $m$ and $L$ are positive integers. Also mathematica finds…
Benjamin L. Warren
- 885
- 2
- 11
7
votes
0 answers
Irreducibility of a polynomial in two variables
Is the polynomial
$$P_n(x,y)=\displaystyle\sum_{a+b\leq n}x^ay^b$$
irreducible in $\mathbb Z[x,y]$?
For all $n\leq 500$ this is true (checked using Mathematica), so it is reasonable to presume that it is true for all $n$.
This question is related…
MarkoR
- 101
