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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242
votes
15answers
33k 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 ...
64
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 ...
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 ...
48
votes
2answers
1k views

Does one real radical root imply they all are?

Is there an example of an irreducible polynomial $f(x) \in \mathbb{Q}[x]$ with a real root expressible in terms of real radicals and another real root not expressible in terms of real radicals?
43
votes
6answers
3k 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 ...
43
votes
5answers
2k views

Bizarre operation on polynomials

There I was, innocently doing some category theory, when up popped a totally outlandish operation on polynomials. It seems outlandish to me, anyway. I'd like to know if anyone has seen this ...
42
votes
2answers
4k views

Polynomials having a common root with their derivatives

Here is a question someone asked me a couple of years ago. I remember having spent a day or two thinking about it but did not manage to solve it. This may be an open problem, in which case I'd be ...
41
votes
4answers
3k 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 ...
36
votes
9answers
9k 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 ...
36
votes
4answers
3k views

Polynomial roots and convexity

A couple of years ago, I came up with the following question, to which I have no answer to this day. I have asked a few people about this, most of my teachers and some friends, but noone had ever ...
36
votes
0answers
2k views

How to prove this polynomial always has integer values at all integers?

Let $m$ be any positive integer. $$ P_m(x)=\sum_{i=0}^{m}\sum_{j=0}^{m}{x+j\choose j}{x-1\choose j}{j\choose i}{m\choose i}{i\choose m-j}\frac{3}{(2i-1)(2j+1)(2m-2i-1)}. $$ Question: $P_m(x)$ always ...
35
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 , ...
35
votes
2answers
984 views

A curious identity related to finite fields

To three elements $a_1$, $a_2$, $a_3$ in the finite field $\mathbb F_q$ of $q$ elements we associate the number $N(a_1,a_2,a_3)$ of elements $a_0\in \mathbb F_q$ such that the polynomial ...
34
votes
2answers
3k views

Polynomial with the primes as coefficients irreducible?

If $p_n$ is the $n$'th prime, let $A_n(x) = x^n + p_1x^{n-1}+\cdots + p_{n-1}x+p_n$. Is $A_n$ then irreducible in $\mathbb{Z}[x]$ for any natural number $n$? I checked the first couple of hundred ...
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 ...
34
votes
4answers
3k views

The maximum of a polynomial on the unit circle

Encouraged by the progress made in a recently posted MO problem, here is a "conceptually related" problem originating from a 2003 joint paper of Sergei Konyagin and myself. Suppose we are given $n$ ...
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 ...
29
votes
4answers
980 views

Generalizing the notion of Farey neighbors to the algebraic numbers

"The Beauty of Roots" is about plots of roots of polynomials—specifically, those with degree less than a given number and height less than another given number. As you can see, these plots are ...
28
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, ...
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 ...
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 ...
27
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. ...
27
votes
1answer
949 views

How many polynomial Morse functions on the sphere?

Let $f$ be a homogeneous polynomial of degree $d$ in $n$ variables. Restricted to the unit sphere $S^{n-1}$, it might or might not be a Morse function. If $f$ is a Morse function of degree $1$, you ...
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 ...
24
votes
3answers
2k views

when is the power of a nonnegative polynomial a sum of squares?

There are polynomials that are not sum of squares. For example Motzkin gave the example $x^4y^2+x^2y^4+z^6-3x^2y^2z^2$ in 1967. Is there a real polynomial $f\in{\mathbb{R}}[x_1,\ldots,x_n]$ in ...
23
votes
1answer
462 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, ...
22
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: $$\left(\sum_{n=1}^Nn\right)^2=\sum_{n=1}^Nn^3.$$ ...
22
votes
2answers
1k views

Have we ever proved any non-solvable case of reciprocity without the Langlands program ?

The reciprocity of the title is the following not completely well-posed problem: Fix $P(X)$ a monic irreducible polynomial of degree $n$, with coefficients in $\mathbb Z$. "Describe" (in some sense) ...
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 ...
21
votes
0answers
427 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 + ...
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 ...
20
votes
1answer
696 views

$f(x)$ 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 $\mathbb{Z}[x]$)?
20
votes
5answers
749 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: ...
20
votes
1answer
2k views

Polynomials with rational coefficients

Long time ago there was a question on whether a polynomial bijection $\mathbb Q^2\to\mathbb Q$ exists. Only one attempt of answering it has been given, highly downvoted by the way. But this answer ...
19
votes
11answers
3k views

When does 'positive' imply 'sum of squares'?

Does anyone have examples of when an object is positive, then it has (or does not have) a square root? Or more generally, can be written as a sum of squares? Example. A positive integer does not ...
19
votes
2answers
1k views

Cyclotomic polynomials with coefficients $0,\pm1$

Let me begin with what looks like a joke. According to a Bourbaki member, the following conversation occurred during a meeting dedicated to polishing the but-last version of an Algebra Bourbaki ...
18
votes
5answers
901 views

Cayley-Hamilton revisited

Let $(A_i)_i$ be $n\times n$ matrices with entries in a field $K$ with characteristic $0$. We consider the equation (1) $f(X)=A_kX^k+\cdots+A_1X+A_0=0_n$ where $X\in\mathcal{M}_n(K)$ is unknown. Let ...
17
votes
6answers
2k views

What, if anything, makes homogeneous polynomials so great?

It should be obvious from the question that I am not any kind of algebraic geometer, so if there are definitions of hom-polys as comonoidal dyadic functors or whatnot, let's leave that to one side for ...
17
votes
4answers
924 views

A mixing property for finite fields of characteristic $2$

In connection with this MO post, here is a question somewhat implicitly contained in a joint paper of S. Kopparty, S. Saraf, M. Sudan, and myself. Let ${\mathbb F}$ be a finite field, and suppose ...
17
votes
1answer
467 views

Minimum value of $|p(1)|^2+|p(2)|^2 +…+ |p(n+3)|^2$ over all monic polynomials $p$

Let $n$ be a positive integer. Determine the smallest possible value of $|p(1)|^2+|p(2)|^2 +...+ |p(n+3)|^2$ over all monic polynomials $p$ of degree $n$. This question was proposed (problem ...
17
votes
2answers
597 views

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 $n$ ...
17
votes
2answers
541 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 ...
17
votes
1answer
1k views

How to prove that every polynomial in an infinite family is irreducible over Q?

Consider the bivariate polynomial $$p(X,Y) = X^5 - (2 Y + 1) X^3 - (Y^2 + 2) X^2 + Y (Y-1) X + Y^3.$$ For every integer $y \ge 4$, I conjecture that $p(X,y)$ is irreducible in $\mathbb{Q}[X]$. How can ...
17
votes
3answers
2k views

Where in mathematics do these polynomials appear?

Does anyone recognize the following sequence of polynomials? $f_0(x) = x-1$ $f_1(x) = x^2-x$ $f_2(x) = x^4-2x^2+x$ $f_3(x) = x^8-3x^4+3x^2-x$ $f_4(x) = x^{16}-4x^8+6x^4-4x^2+x$ $\vdots$ The ...
17
votes
4answers
921 views

Why are polynomials easier to handle with than integers?

This may seems to be an elementary question, but I found no answers on MO nor google. I have always heard "polynomials are easier to handle with than integers". For example: When $n$ is quite ...
17
votes
0answers
502 views

What is the best probabilistic estimate from below for a random polynomial on an arc?

I'm currently involved in a small (but quite time consuming) project where we are trying to get some decent bound for the number $N(P)$ of real zeroes of a random polynomial $P(x)=\sum_{k=0}^n\xi_k ...
16
votes
5answers
2k views

Given a polynomial f, can there be more than one constant c such that every root of f(x)-c is repeated?

The question Let $f$ be a nonconstant polynomial over $\mathbb{C}$. Let's say that a point $c \in \mathbb{C}$ is unusual for $f$ if every root $x$ of $f(x) - c$ is repeated. Can $f$ have more than ...
16
votes
1answer
721 views

Symmetric polynomial from graphs

Let $g$ be a directed, connected multigraph, on $n$ vertices, without loops. Define $$P_g(x_1,\dots,x_n) := Sym\left[ \prod_{(i,j) \in g} (x_i-x_j) \right]$$ where $(i,j)$ is the directed edge ...
16
votes
2answers
2k views

When is P(x)-Q(y) irreducible?

Let $k$ be an algebraically closed field (in my application, it is characteristic zero, but this probably doesn't matter so much), and let $P: k \to k$, $Q: k \to k$ be polynomials of one variable. ...
16
votes
2answers
451 views

Minimum number of variables on which a multivariate polynomial depends?

Let $p:F_2^n\rightarrow F_2$ be a multivariate polynomial, let's say of degree 3. (Both the degree and the order of the field could probably be replaced by other constants without affecting this ...