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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6
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 ...
213
votes
15answers
30k 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 ...
62
votes
6answers
3k 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 ...
33
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 ...
21
votes
4answers
1k 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 ...
6
votes
1answer
980 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 ...
9
votes
1answer
696 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 ...
7
votes
7answers
950 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 $$ ...
5
votes
6answers
448 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
459 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 $a_k$ ...
10
votes
4answers
542 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 ...
6
votes
2answers
883 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 ...
1
vote
0answers
293 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 = ...
36
votes
9answers
8k 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 ...
65
votes
11answers
3k 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 ...
26
votes
12answers
3k 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 ...
28
votes
1answer
1k 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 ...
12
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, ...
6
votes
5answers
1k 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 ...
21
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: $$\left(\sum_{n=1}^Nn\right)^2=\sum_{n=1}^Nn^3.$$ ...
9
votes
4answers
2k 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 ...
7
votes
2answers
2k 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 ...
21
votes
4answers
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, ...
20
votes
4answers
1k views

Freeness of a Z[x]-module

Definition: Call a mapping $f: \mathbb{Z} \rightarrow \mathbb{Z}$ a generalized polynomial if for any distinct integers $m$ and $n$ we have $(m - n)|(f(m)-f(n))$. It is easy to check that polynomial ...
17
votes
2answers
455 views

Stability of real polynomials with positive coefficients

Say that a polynomial in an indeterminate $x$ with real coefficients of degree $d$ has positive coefficients if each of the coefficients of $x^d,\ldots,x^1,x^0$ is (strictly) positive. For $f$ a ...
10
votes
3answers
662 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 $f_2 = x^{m_2} + (1+x)^{n_2}$ ...
10
votes
6answers
2k views

What's an example of a transcendental power series?

Let $k$ be a field. What is an explicit power series $f \in k[[t]]$ that is transcendental over $k[t]$? I am looking for elementary example (so there should be a proof of transcendence that does ...
9
votes
1answer
654 views

The closures in $C^0(\mathbb{R}, \mathbb{R})$ of the set of integer valued polynomials, resp, of polynomials with integer coefficients

This is a follow up of an interesting recent question on the topic. The answer given there by fedia shows that the matter is rich and complicated, and I can't resist to submit here a further question. ...
6
votes
0answers
360 views

“Consecutive” irreducible polynomials

If $P\in {\mathbb Z}[X]$ is a polynomial of degree $2$, then it is easy to see that for any integer $m$, at least one of the polynomials $P-(m+1),P-(m+2),P-(m+3),P-(m+4)$ is irreducible in ${\mathbb ...
3
votes
4answers
1k views

Minimizing the modulus of a polynomial around a circle

I'm probably missing something elementary here, but I guess the only way to be sure is to ask here. Now, I have encountered a situation where given an nth-degree polynomial $p_n(z)$ with complex ...
10
votes
2answers
1k views

Giant Rat of Sumatra singularity

I would be grateful for explanations of the issues raised in any of these three questions, or pointers to the relevant literature (now updated with answers): How did a particular singularity come ...
7
votes
1answer
612 views

Can roots of any polynomial be expressed using Eulerian function?

I encountered an interesting function which is called "Eulerian" by the Wolfram's MathWorld: $$\phi(q)=\prod_{k=1}^{\infty} (1-q^{k})$$ It is interesting because it is claimed that roots of any ...
4
votes
2answers
427 views

Is there an integer a such that f(X)+a is irreducible in Z[X]?

Let $f\in\mathbb{Z}[X]$ be an irreducible polynomial. Is there an integer $a\neq 0$ such that $f(X)+a$ is also irreducible in $\mathbb{Z}[X]$? Can this be also extended to $\mathbb{Q}[X]$?
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. ...
10
votes
3answers
446 views

Effective algorithm to test positivity

Let $f(x_1,\ldots, x_n)$ be a real polynomial in several variables. Is there an effective algorithm to test whether $f$ is positive (or nonnegative) on the whole of ${\mathbb{R}}^n$?
7
votes
2answers
90 views

Bounds on coefficients of factors of a multivariate polynomial

Given a multivariate polynomial $F(x, y, ..)$ what is the smallest bound B that can be quickly found such that $|G|_{\infty} \le B$ for all factors $G$ of $F$. (I'm using $|G|_{\infty}$ to denote the ...
7
votes
2answers
451 views

Dimension of a homogeneous polynomial system

Let $m\geq4$ be an even integer, $V\subset\mathbb{C}^{m-1}$ be the solution set of the following polynomial equations: \begin{cases} ...
6
votes
2answers
597 views

“MultiCatalan numbers”

Could anyone provide a reference for the following (sort of) generalization of Catalan numbers: the multinomial coefficient $$ \binom{2k_1+3k_2+4k_3+...}{k_1+2k_2+3k_3+...,k_1,k_2,k_3,...} $$ is ...
5
votes
1answer
525 views

The resultant of an arbitrary polynomial and a cyclotomic polynomial

This is a natural generalization of this question. Let $f$ be a monic irreducible polynomial over $\mathbb Z$. Let $S_f$ be the set of natural numbers $n$ such that one of the three equivalent ...
5
votes
2answers
1k views

Polynomial reducible modulo every integer

Hi, let $f\in\mathbb{Z}[X]$ be a monic polynomial. Assume that the reduction of $f$ modulo $m $ is reducible for all integers $m\geq 2$. Q1: Is $f$ reducible in $\mathbb{Z}[X]$ ? I've thought ...
1
vote
1answer
574 views

common roots of bivariate polynomial equations

Let $f(x,y)=0$ and $g(x,y)=0$ be bivariate polynomial equations where the polynomials have the same degree, say, $N\geq 3$. Furthermore, both of them have the same terms but different coefficients. ...
1
vote
1answer
1k views

Roots of bivariate polynomials

A bivariate polynomial of degree $m+n$ is, $ p(x,y) = \sum_{k=1}^n\sum_{j=1}^m a_{jk}x^ky^j$ where $a_{mn}\neq0$ and $a_{jk}\in\mathbb{R}$ for $1\leq j\leq m$, $1\leq k\leq n$. I would like to ...
28
votes
3answers
2k views

Are surjectivity and injectivity of polynomial functions from $\mathbb{Q}^n$ to $\mathbb{Q}$ algorithmically decidable?

Is there an algorithm which, given a polynomial $f \in \mathbb{Q}[x_1, \dots, x_n]$, decides whether the mapping $f: \mathbb{Q}^n \rightarrow \mathbb{Q}$ is surjective, respectively, injective? -- And ...
6
votes
0answers
204 views

Zeros of polynomials related to Jensen polynomial associated with Riemann xi function $\xi(x)$

We encountered polynomials defined by the recursive relations for the coefficients $b_k>0$ as defined below: $$p_{n}(x)=\sum_{k=0}^{n}\binom{2n}{2k}b_k x^k$$ ...
6
votes
2answers
473 views

Expression for the sum of square roots of zeros of a polynomial

Let $f(x)$ be a polynomial of degree $n$ with rational coefficients whose zeroes are nonnegative real numbers: $x_1, \dots, x_n\geq 0$. General question. Does there exist a simple expression for the ...
6
votes
2answers
372 views

bound for zeros of a polynomial with bounded integer coefficients

Let $f$ be a monic polynomial with bounded integer coefficients and such that all zeros are (in absolute value) greater than $1$. How close can the zeros of $f$ reach $1$ (in absolute value)? ...
6
votes
3answers
1k 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
355 views

degree of polynomials in nullstellensatz

$(A)$ If $K$ is algebraically closed field, then consider $m$ polynomials $f_i$ for $i=\{1,\dots,m\}$ in $K[x_1,\dots,x_n]$ of degree $d_i$ each with no common root. We know there exists $g_i$ for ...
3
votes
1answer
147 views

Generalization of the Hermite-Bielher-Kakeya Theorem

A stable polynomial is one with zeros in the upper half plane or lower half plane. Interlacing polynomials are polynomials with only real zeros, where between every two zeros of one polynomial lies a ...