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, ...

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273
votes
15answers
38k views

Why do roots of polynomials tend to have absolute value close to 1?

While playing around with Mathematica I noticed that most polynomials with real coefficients seem to have most complex zeroes very near the unit circle. For instance, if we plot all the roots of a ...
9
votes
2answers
1k views

Riemann zeta function at positive integers and an Appell sequence of polynomials related to fractional calculus

I was exploring some raising and lowering operators related to an infinitesimal generator for fractional integro-derivatives and found an Appell sequence of polynomials, i.e., an infinite sequence of ...
39
votes
9answers
11k views

Irreducibility of polynomials in two variables

Let $k$ be a field. I am interested in sufficient criteria for $f \in k[x,y]$ to be irreducible. An example is Theorem A of this paper. Does anyone know of similar results in the same vein? How about ...
47
votes
6answers
4k views

How to recognise that the polynomial method might work

A couple of days ago I was at a nice seminar given by Christian Reiher, during which he told us about a short proof of the following special case of a theorem of Olson. Theorem. Let ...
34
votes
2answers
2k views

A question on maps from $\mathbb{Z}/p\mathbb{Z}$ to itself

Let $p\geq 3$ be a prime number, and let $u:\mathbb{Z}/p\mathbb{Z}\to \mathbb{Z}/p\mathbb{Z}$ be a map such that, for all $l\in \mathbb{Z}/p\mathbb{Z}$,$l\neq 0$, the map $k\mapsto u(k+l)-u(k)$ is a ...
7
votes
1answer
1k views

Cycling through the Zeta Garden: Zeta functions for graphs, cycle index polynomials, and determinants

Zeta functions abound in mathematics. Audrey Terras describes in Zeta Functions and Chaos three zeta functions--the zeta fct. of a projective non-singular algebraic variety; the Artin-Mazur zeta ...
2
votes
2answers
351 views

Can one estimate the distribution of eigenvalues of a matrix by its Cauchy/Stieltje transform?

Given a real symmetric $n$ dimensional matrix $A$, with eigenvalues $\lambda_i$ I am defining its Cauchy transform as the function, $f_A(z) = \sum_i \frac{1}{z-\lambda_i}\,$ Is there any information ...
14
votes
4answers
680 views

Finite interpolation by a nondecreasing polynomial

Let $x_1 < x_2 < \ldots < x_n$ and $y_1 < y_2 < \ldots < y_n$ be two sequences of $n$ real numbers. It is well known that there are polynomials that "interpolate" in that ...
63
votes
7answers
4k views

Roots of truncations of $ e^x - 1$

During a talk I was at today, the speaker mentioned that if you truncate the Taylor series for $e^x - 1$, you'll get lots of roots with nonzero real part, even though the full Taylor series only has ...
15
votes
5answers
10k views

Criteria to determine whether a real-coefficient polynomial has real root?

Given a polynomial equation $x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0=0$, where $n$ is even and all the coefficients $a_i$ are real, what is the best way to determine whether it has a real root or not? I ...
33
votes
5answers
2k views

The resultant and the ideal generated by two polynomials in $\mathbb{Z}[x]$

I was asked the following question by a colleague and was embarrassed not to know the answer. Let $f(x), g(x) \in \mathbb{Z}[x]$ with no root in common. Let $I = (f(x),g(x))\cap \mathbb{Z}$, that is, ...
22
votes
4answers
2k views

Distribution of roots of complex polynomials

I generated random quadratic and cubic polynomials with coefficients in $\mathbb{C}$ uniformly distributed in the unit disk $|z| \le 1$. The distribution of the roots of 10000 of these polynomials are ...
10
votes
2answers
3k views

Conditions for smooth dependence of the eigenvalues and eigenvectors of a matrix on a set of parameters

Let $A\in\mathcal M_n$ be an $n\times n$ real [symmetric] matrix which depends smoothly on a [finite] set of parameters, $A=A(\xi_1,\ldots,\xi_k)$. We can view it as a smooth function $A:\mathbb ...
23
votes
1answer
531 views

Hilbert's Theorem on $L_2$ norm of polynomials in $\mathbb{Z}[X]$ - Explicit construction and a converse?

Consider the set of polynomials with real coefficients as a vector space with the following inner-product: $\langle f, g \rangle = \int_{a}^{b} f(x)g(x) dx$. Hilbert showed, in a paper from 1894, ...
12
votes
2answers
965 views

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 ...
15
votes
3answers
774 views

Cyclotomic polynomials: $\Phi_n(p)$ is like $p^{\phi(n)}$ for big enough $p$, right?

Apologies in advance if this turns out to be simple. So far I haven't found a proof or a reference. Although I like $p$ to be a prime, I can ask the following for positive integers $n$ and $p$, ...
1
vote
4answers
1k views

A proof for a statement about polynomial automorphism

I already got a proof for the fact that if a polynomial map is surjective then it is also injective. However, I used the invariant dimension of a ring and I want a simpler proof. Bravo for any try. ...
7
votes
7answers
1k views

Old question of Serre on discriminants of a sequence of polynomials

Let $P_n(t)$ be polynomials with integer coefficients with $d_n = \deg(P_n(t))$ going to infinity when $n$ goes to infinity and with nonzero discriminants $disc(P_n(t)) \neq 0$. Question: Is $$ ...
6
votes
6answers
673 views

Getting a bound on the coefficients of the factor polynomial

Suppose $f(x):=a_0+a_1x+\cdots+a_nx^n$ is a polynomial in $\mathbb{Z}[x]$ and $|a_i|\leq M$ for each $i=0,\ldots ,n.$ Now suppose $g(x)$ is a factor of $f(x)$ in $\mathbb{Z}[x]$, then is it possible ...
4
votes
1answer
729 views

Why are all these families of polynomials finally log-concave?

This started when I was examining certain families of unimodal polynomials, i.e. $\sum_{k=0}^n a_kx^k$ where $a_0\le a_1\le\cdots \le a_k\ge\cdots \ge a_n$. (Notation: in the following, the $a_k$ ...
23
votes
2answers
572 views

Are there irreducible polynomials with all zeros on two concentric circles?

This is somewhat similar to this recent question, but extending in a different direction. Let $f(x)$ be an irreducible polynomial of degree $n$ with integer coefficients. Call such $f$ a bicycle ...
6
votes
1answer
503 views

An equivalence relation on the space of polynomials in one complex variable

Let $P(z)$ be a polynomial with complex variable $z$. We consider the following distribution for the roots of $P(z)=0$: the distribution is a triple $(n_{1},n_{2},n_{3})$ where these integers are ...
6
votes
2answers
1k views

When does a rational function have infinitely many integer values for integer inputs?

Consider rational functions $F(x)=P(x)/Q(x)$ with $P(x),Q(x) \in \mathbb{Z}[x]$. I'd like to know when I can expect $F(k) \in \mathbb{Z}$ for infinitely many positive integers $k$. Of course this ...
6
votes
3answers
2k views

Rational exponential expressions

Consider the following extension of polynomials. The rational exponential expressions (REXes) are given by: The leaves 1 and $x$ for $x$ drawn from a class of variables; and Closed under the binary ...
5
votes
1answer
420 views

Unexpectedly prime rich cubic polynomial

We got a cubic polynomial which is unexpectedly prime rich. Let $f(x)=29160 x^3 + 30132 x^2 + 8046 x + 643$ and $\pi_f(n)$ the number of primes values of $f(x)$ for $x \in [1,n]$. Let ...
6
votes
1answer
413 views

Is every polynomial a factor of a trinomial?

We say that $T(X) \in \mathbb{Q}[X]$ is a trinomial if there exist $A,B,C \in \mathbb{Q}$ such that $T(X) = AX^n + BX^m + C$ for some $n \geq m \in \mathbb{N}$. Is it true that for each ...
5
votes
2answers
131 views

Moment matching: construction of a mixture of Gaussian distribution with lower moments identical to Gaussian

This is a question related to the statistical model behind independent component analysis (ICA). We assume that $Z \sim N(0,1)$. Our goal is to construct a random variable $X$ that follows a ...
2
votes
0answers
63 views

Relating face polytopes of permutohedra to integer partitions

The OEIS entries A019538, A049019, and A133314, relate a refinement of the face polynomials of the permutohedra (A049019) to partition polynomials (A133314) defined by multiplicative inversion of an ...
1
vote
1answer
122 views

Is there a limit definition for the roots of a polynomial with arbitrary degree? [closed]

I know there's no general formula for all the roots of a polynomial with a degree greater than 4, but is there some sort of limit (or other) definition to calculate the roots (particularly the largest ...
1
vote
0answers
334 views

When is a cubic polynomial a cube? [closed]

I've been researching cubes and I'm trying to solve this Diophantine equation over the integers. $$ax^3 + bx^2 + cx + d = y^3$$where a, b, c, d are parameters for a given $n$. For example, for $n = ...
69
votes
11answers
4k views

Can a non-surjective polynomial map from an infinite field to itself miss only finitely many points?

Is there an infinite field $k$ together with a polynomial $f \in k[x]$ such that the associated map $f \colon k \to k$ is not surjective but misses only finitely many elements in $k$ (i.e. only ...
44
votes
4answers
4k views

Degree of sum of algebraic numbers

This is an elementary question (coming from an undergraduate student) about algebraic numbers, to which I don't have a complete answer. Let $a$ and $b$ be algebraic numbers, with respective degrees ...
25
votes
7answers
2k views

When is a monic integer polynomial the characteristic polynomial of a non-negative integer matrix?

Suppose $P(x)$ is a monic integer polynomial with roots $r_1, ... r_n$ such that $p_k = r_1^k + ... + r_n^k$ is a non-negative integer for all positive integers $k$. Is $P(x)$ necessarily the ...
28
votes
12answers
4k views

What Are Some Naturally-Occurring High-Degree Polynomials?

To construct J. H. Conway's look-and-say sequence, begin by putting down a 1 as the first entry. The other entries are found by saying the previous entry aloud, and writing what you hear. ...
31
votes
1answer
3k views

Integers not represented by $ 2 x^2 + x y + 3 y^2 + z^3 - z $

EDIT, 9 March 2014: when I asked this in 2010, I did not have the courage of my convictions, and so did not ask for an if and only if proof, as Kevin Buzzard quite properly pointed out. Such problems ...
36
votes
4answers
2k views

The sum of squared logarithms conjecture

I am searching for the first proof of (or counterexample to) the following conjecture. (The sum of squared logarithms conjecture) For all natural numbers $n$ and positive numbers $x_1,x_2, \ldots , ...
30
votes
3answers
1k views

Polynomials with the same values set on the unit circle

Assume that $P(z)$, $Q(z)$ are complex polynomials such that $P(S)=Q(S)$, where $S=\{z\colon |z|=1\}$ (equality is understood in the sense of sets, but I do not know the answer even for multisets). ...
29
votes
2answers
971 views

Polynomial $g:\mathbb R^n \rightarrow\mathbb R^n$ with no critical point may have no root

Version 1 (solved): If $g$ : $\mathbb R^n \rightarrow \mathbb R^n$ is a polynomial, $Dg(x)$ is non-degenerate for every $x$, then there exists $x$, such that $g(x)=0$. Version 2: If $f$ : $\mathbb R^n ...
28
votes
1answer
2k views

Zeros of polynomials with real positive coefficients

The following problem arose in collaborative work with Subhro Ghosh: Question: To any polynomial $P_n(z)=\sum_{i=0}^n a_i z^i =a_n \prod_{i=1}^n(z-z_i)$, attach the empirical measure of zeros ...
25
votes
0answers
527 views

Finding a path through real rooted polynomials

This is a lemma I wanted in order to solve Patrizio Neff's conjecture. It turned out to be the wrong way to think about it, but I am still curious if it is true. Let $z^n+a_{n-1} z^{n-1} + \cdots + ...
14
votes
2answers
1k views

Using higher-order Bring radicals to solve arbitrary polynomials

It is well known that there is no general formula for the solution of the quintic. Of course, what this really means is that there is no general formula that only involves addition, subtraction, ...
7
votes
2answers
657 views

Upper density of the set of $n$'s such that $p(n)$ is prime, where $p$ is polynomial

The starting point for this question is the following (false) statement $\forall n\in \mathbb{N} (n^2 + n + 41 \text{ is prime}).$ Given a polynomial function $p:\mathbb{N} \to ...
15
votes
1answer
1k views

Is $x^{n}-x-1$ irreducible?

Is it true that for every $n \in \mathbb{N}$, $x^{n}-x-1$ is irreducible in $\mathbb{Z}[x]$? The standard irreducibility criteria seem to fail.
22
votes
5answers
830 views

Bass' stable range of $\mathbf Z[X]$

Let $n$ be a positive integer and $A$ be a commutative ring. The ring $A$ is said to be of Bass stable range $\mathrm{sr}(A)\leq n$ if for $a, a_1, \dots, a_n \in A$ one has the following implication: ...
17
votes
2answers
408 views

Non-negative polynomials on $[0,1]$ with small integral

Let $P_n$ be the set of degree $n$ polynomials that pass through $(0,1)$ and $(1,1)$ and are non-negative on the interval $[0,1]$ (but may be negative elsewhere). Let $a_n = \min_{p\in P_n} \int_0^1 ...
12
votes
3answers
907 views

Probability of coprime polynomials

Given positive integer $N$, we choose $m_1, m_2, n_1, n_2$ independently and with equal probabilities from $\{0,1,\ldots,N\}$, and let $f_1 = x^{m_1} + (1+x)^{n_1}$ and $f_2 = x^{m_2} + (1+x)^{n_2}$ ...
11
votes
4answers
689 views

Relating the roots of polynomials to the solution sets of certain functional equations

Consider a functional equation of the following form: $$\sum_{k=0}^n a_k\,\underbrace{f(f(\cdots f}_{k}(x)\cdots )=0\quad \big(f:\,\mathbb{R}\to\mathbb{R},\;a_i\in ...
10
votes
4answers
3k views

Expressing power sum symmetric polynomials in terms of lower degree power sums

Is there an explicit formula expressing the power sum symmetric polynomials $$p_k(x_1,\ldots,x_N)=\sum\nolimits_{i=1}^N x_i^k = x_1^k+\cdots+x_N^k$$ of degree $k$ in $N < k$ variables entirely ...
6
votes
5answers
2k views

Application of polynomials with non-negative coefficients

Question 1: Are there any deeper applications (in any field of mathematics) of polynomials (with possibly more than one variable) over the real numbers whose coefficients are non-negative? So far I ...
23
votes
6answers
1k views

Relations between sums of powers

This question is so naive that it could have been asked before on this site. If so, I'll delete it. Among beautiful formula, I like a lot this one: $$\left(\sum_{n=1}^Nn\right)^2=\sum_{n=1}^Nn^3.$$ ...