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.

Filter by
Sorted by
Tagged with
30 votes
4 answers
3k 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 ...
Joseph O'Rourke's user avatar
28 votes
2 answers
1k 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 ...
Wolfgang's user avatar
  • 13.2k
25 votes
4 answers
6k 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, ...
Daniel Miller's user avatar
25 votes
5 answers
4k views

Monic polynomial with integer coefficients with roots on unit circle, not roots of unity?

There are certainly non-monic polynomials of degree 4 with all roots on the unit circle, but no roots are roots of unity; $5 - 6 x^2 + 5 x^4$ for example. Now, for a monic polynomial of degree $n$, ...
Per Alexandersson's user avatar
23 votes
1 answer
1k 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, ...
Ofir Gorodetsky's user avatar
20 votes
1 answer
3k views

Density of polynomials in $C^k(\overline\Omega)$

Let $\Omega$ be an open and bounded subset of $\mathbb{R}^2$ and let $C^k(\Omega)$, $1\leq k<\infty$, be the space of functions $f$ with continuous derivatives of order $\leq k$ in $\Omega$, ...
user111's user avatar
  • 3,761
19 votes
1 answer
1k views

Is OEIS A007018 really a subsequence of squarefree numbers?

A comment in A007018 a(n) = a(n-1)^2 + a(n-1), a(0)=1 claims Subsequence of squarefree numbers (A005117). - Reinhard Zumkeller, Nov 15 2004 Is that really so? As far as I know, it is an open ...
joro's user avatar
  • 24.2k
18 votes
3 answers
858 views

$\prod_k(x\pm k)$ in binomial basis?

Let $x$ be an indeterminate and $n$ a non-negative integer. Question. The following seems to be true. Is it? $$x\prod_{k=1}^n(k^2-x^2)=\frac1{4^n}\sum_{m=0}^n\binom{n-x}m\binom{n+x}{n-m}(x+2m-n)^...
T. Amdeberhan's user avatar
18 votes
4 answers
7k 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 ...
Peter Erskin's user avatar
17 votes
3 answers
2k views

About the prime divisors of values of polynomials

Let $P$ be a polynomial having integer coefficients (and degree $\geq 3$), and let $\mathscr P_P$ be the set of prime numbers dividing some value $P(n)$ with $n \in \mathbb Z$. Is it true that $\...
Konstantinos Gaitanas's user avatar
17 votes
1 answer
3k 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.
Pablo's user avatar
  • 11.2k
17 votes
6 answers
3k 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 ...
jlk's user avatar
  • 3,234
16 votes
1 answer
552 views

Cyclotomic polynomials 2

My naive question may actually lead to something interesting. Let $\Phi_m(x)$, $\Phi_n(x)$ be cyclotomic polynomials, $m<n$. These polynomials are relatively prime and so there are polynomials $...
user avatar
16 votes
1 answer
2k views

Reduced resultants and Bezout's identity

Introduction: Given two univariate and coprime integer polynomials $f(x), g(x)$, we can always write \begin{equation} u(x)f(x)+v(x)g(x)=c \end{equation} for a unique pair of integer polynomials $u(...
Konstantinos Kanakoglou's user avatar
15 votes
1 answer
639 views

Is a wild automorphism of $k[x_1,\ldots,x_n]$, $n \geq 3$, necessarily of infinite order?

Let $k[x_1,\ldots,x_n]$ be a polynomial ring over a field $k$ of characteristic zero. When $n=2$, it is known that every automorphism of $k[x_1,x_2]$ is tame, namely, a finite product of elementary ...
user237522's user avatar
  • 2,783
13 votes
3 answers
833 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 ${\mathbb{R}}^n$?
user avatar
13 votes
2 answers
2k views

Galois group of a product of polynomials

How can I compute the Galois group of the polynomial $fg\in K[x]$ assuming that I know the Galois groups of $f\in K[x]$ and $g\in K[x]$? Let's suppose for simplicity that the field $K$ is perfect.
roger123's user avatar
  • 2,712
13 votes
2 answers
2k views

Irreducible polynomial over number field with roots in every completion?

Let K/Q be a field, probably not a finite extension. Is it possible for a polynomial to be irreducible over K but have a root in every completion of K? What about all but finitely many completions? ...
Dror Speiser's user avatar
  • 4,563
11 votes
1 answer
330 views

Analogue of conic sections for the permutohedra, associahedra, and noncrossing partitions

Slicing cones in various ways with a plane generates conic sections identified geometrically as hyperbolas, parabolas, or ellipses and algebraically, when suitably rotated, as certain rescaled ...
Tom Copeland's user avatar
  • 9,937
9 votes
3 answers
861 views

Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in Z[t]$ be unbounded as n varies?

We've recently seen this question: Can the number of solutions $ab(a+b+1)=n$ for $a,b,n \in \mathbb{Z}$ be unbounded as $n$ varies? It appears initially plausible that the answer is yes, but evidently ...
Aaron Meyerowitz's user avatar
9 votes
0 answers
673 views

Van der Pol's identity for the sum of divisors and a quartic polynomial equation for odd perfect numbers

In Touchard (1953) it is mentioned that the sum of divisors $\sigma(n)$ satisfies the following recurrence relation ($n>1$): $$n^2(n-1) = \frac{6}{\sigma(n)} \sum_{k=1}^{n-1}(3n^2-10k^2)\sigma(k)\...
Perfect Number's user avatar
8 votes
0 answers
1k views

Is the Collatz conjecture known to be true for interesting unbounded classes of numbers?

The Collatz or the $3n+1$ conjecture is open. Is there a specific polynomial $f(x)\in\mathbb Z[x]$ whose range is unbounded for which every integer of form $|f(m)|$ at $m\in\mathbb Z$ satisfies $3n+1$...
Turbo's user avatar
  • 13.7k
8 votes
2 answers
1k 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 ...
Minkov's user avatar
  • 1,117
8 votes
3 answers
597 views

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)| &...
DUO Labs's user avatar
  • 265
8 votes
2 answers
304 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 ...
Edward's user avatar
  • 81
8 votes
7 answers
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 $$ \...
Luis H Gallardo's user avatar
8 votes
2 answers
1k 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 ...
Max Alekseyev's user avatar
8 votes
1 answer
2k views

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 ...
mostafa's user avatar
  • 367
8 votes
2 answers
430 views

Inductive definition of Bernstein polynomials

For $n\in \mathbb{N}$ let $B_n$ be the linear operator taking a function $f$ on the unit interval $I=[0,1]$ to its $n$-th Bernstein polynomial $B_nf$, $$ B_nf(x):=\sum_{k=0}^n\binom{n}{k} f\Big(\...
Pietro Majer's user avatar
  • 56.5k
7 votes
2 answers
778 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 ...
მამუკა ჯიბლაძე's user avatar
7 votes
1 answer
1k 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 $a_k$ ...
Wolfgang's user avatar
  • 13.2k
6 votes
1 answer
915 views

Irreducible polynomial $p_{n}(x)=\sum_{k=0}^{n}\frac{x^k}{k!}$ for all positive integers $n$

Let $n$ be a positive integer greater than $1$, and define the polynomial $$p_{n}(x)=\sum_{k=0}^{n}\dfrac{x^k}{k!}$$ Is $p_{n}(x)$ irreducible in $\mathbf{Q}[x]$? I can show it when $n$ is a ...
math110's user avatar
  • 4,230
6 votes
1 answer
697 views

Find the maximum of $|a_{p}|$, if $a_0+a_1x+\dots+a_nx^n:[-1,1]\mapsto [-1,1]$

Let $n$ be a given positive integer, and let $f(x)=\displaystyle\sum_{k=0}^{n}a_{k}x^k$, where $a_{i}\in \mathbb{R}$, $0 \le i \le n$. If $$|f(x)|\le 1,\qquad \text{for } ~|x|\le 1,$$ what is the ...
math110's user avatar
  • 4,230
6 votes
2 answers
784 views

Is every square root of an integer a linear combination of cosines of $\pi$-rational angles?

For example, $\sqrt 2 = 2 \cos (\pi/4)$, $\sqrt 3 = 2 \cos(\pi/6)$, and $\sqrt 5 = 4 \cos(\pi/5) + 1$. Is it true that any integer's square root can be expressed as a (rational) linear combinations of ...
pavpanchekha's user avatar
  • 1,461
6 votes
2 answers
923 views

Counterexample to Proposition of Granville related to abc conjecture

Looks like there is counterexample to Proposition related to abc conjecture. Confusion is likely. From RATIONAL AND INTEGRAL POINTS ON QUADRATIC TWISTS OF A GIVEN HYPERELLIPTIC CURVE, Andrew ...
joro's user avatar
  • 24.2k
6 votes
1 answer
746 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 ...
Ali Taghavi's user avatar
6 votes
1 answer
515 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 irreducible ...
Pablo's user avatar
  • 11.2k
5 votes
0 answers
152 views

Nullstellensatz with nilpotents and $I=J(V(I))$

Let $R$ be the ring $$\mathbb{R}[t_1,t_2\ldots]/(t_1^2,t_2^2,\ldots)$$ Let $p_1,\ldots p_\ell$ be polynomials in $R[x_1,\ldots,x_n]$ whose constant terms are 0. Let $f$ be a polynomial which is zero ...
user avatar
5 votes
1 answer
348 views

(Variation of an old question) Are these functions polynomials?

This is a followup to the question here: How to show that the following function isn't a polynomial over Q?. As before, let $b_1,b_2,\dots$ be an enumeration of $\mathbb Q$. The question might ...
Asvin's user avatar
  • 7,648
5 votes
0 answers
515 views

Log-concave polynomial is a log-concave function?

A polynomial $\sum\limits_{k=0}^n a_kx^k$ is log-concave if $a_0,\ldots,a_n$ constitute a log-concave sequence. I wonder whether the log-concave polynomial is also a log-concave function with respect ...
Johnny Yin's user avatar
5 votes
1 answer
1k views

Ideal generated by two univariate, coprime, integer polynomials

Let $f(x)$, $g(x)$ be two univariate, coprime, integer polynomials and let $I=\big(f(x),g(x)\big)$ the ideal of $\mathbb{Z}[x]$ generated by $f, g$. Let $I \cap \mathbb{Z}$, that is, the elements of $\...
Konstantinos Kanakoglou's user avatar
4 votes
2 answers
332 views

algorithm for finding radical expressions of all conjugates of an arbitrary algebraic number expressed in radicals

By an algebraic number expressed in radicals, I mean one that is an element of a set $S$ characterized as follows: $\mathbb{Z}\subset S$. For any $a,b\in S$, $a+b,a·b\in S$. For $a,b\in S$ with $b\...
Alex Kindel's user avatar
4 votes
1 answer
239 views

A discrete operator begets even/odd polynomials

Given a function $f(x)$ define the forward shift operator by $Ef(x)=f(x+1)$ and the discrete derivative $\delta f(x)=(E-1)f(x)=f(x+1)-f(x)$. Given a partition $\lambda=(\lambda_1,\lambda_2,\dots,\...
T. Amdeberhan's user avatar
3 votes
0 answers
261 views

Guises of the refined Eulerian numbers, generated by tangent vectors (OEIS A145271)

The Eulerian numbers (OEIS A008292, not to be confused with the Euler numbers) pop up in numerous scenarios in combinatorics and advanced analysis, one as the components of the h-vectors of the ...
Tom Copeland's user avatar
  • 9,937
3 votes
1 answer
249 views

Is the restriction of a graded automorphism of a polynomial ring to a polynomial subring linearizeable?

Let $k$ be a field and let $A=k[x_1,\dots,x_n]$ be a polynomial algebra over $k$, and let $B\subset A$ be a graded subalgebra that is itself a polynomial ring, i.e. $B=k[f_1,\dots,f_m]$ for some ...
benblumsmith's user avatar
  • 2,831
3 votes
0 answers
199 views

On an exact expression for the squares of the distances of the critical points to a given zero of a polynomial

Let $p(z) = \prod_{j=1}^{l+1} (z - z_j)^{M_j}$ be a complex polynomial of degree $n$, where the $z_j$ are distinct for $1, \ldots, l+1$. The first $l$ entries in the list $\{z'_1, \ldots, z'_{n-1} \}$ ...
thomashennecke's user avatar
2 votes
1 answer
593 views

Half spaces free of roots of a given polynomial

I thank Loic Teyssier and Emil Jerabek who helped me to revise the two previous version This question is motivated by the following fact in complex variable:(I learned this fact from the book of ...
Ali Taghavi's user avatar
2 votes
0 answers
185 views

Infinite products from the fake Laver tables-Now with no set theory

We say that a sequence of algebras $(\{1,\dots,2^{n}\},*_{n})_{n\in\omega}$ is an inverse system of fake Laver tables if for $x\in\{1,\dots,2^{n}\}$, we have $2^{n}*_{n}x=x$, $x*_{n}1=x+1\mod 2^{n}$,...
Joseph Van Name's user avatar
2 votes
0 answers
94 views

Cryptography signature scheme based on hardness of finding points on varieties?

Related to this question Complexity of finding solutions of trapdoored polynomial. I am trying to build signature scheme based on hardness of finding points on varieties. Let $K$ be field and $M=K[x_1,...
joro's user avatar
  • 24.2k
2 votes
2 answers
2k views

A bounded polynomial having bounded coefficients: several variables

Consider the multivariate case for the question "Approximation theory reference for a bounded polynomial having bounded coefficients" (Approximation theory reference for a bounded polynomial having ...
Felix Y.'s user avatar
  • 123

1
2
3 4 5
8