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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2
votes
1answer
110 views

a second order difference equation related to a real polynomials which seems to have only real roots

I am seeking solutions to the following difference equation: $$2c_k-c_{k-1}-c_{k+1}=\ln(k+A)-\ln(k+B)$$ where $A>B>0$. This equation is related to a real polynomial (see here) which I want to ...
0
votes
0answers
65 views

unique positive real root fast computation

What is the fastest way to compute the value of the unique positive real root corresponding to the following polynom: :p(x) = a*x^5 + b*x^4 + c*x^3 + d*x^2 + e*x - f = 0 where a, b, c, d, e, f are ...
4
votes
1answer
181 views

A palindromic polynomial and its derivative have the same number of zeros outside the unit circle. Reference?

I am trying to find the original reference for a lemma attributed to Cohn (as in Schur-Cohn method): Let $A(z)$ be a palindromic or skew-palindromic polynomial, and denote its derivative by ...
0
votes
1answer
140 views

Irreducibility of a resultant of real and imaginary parts of a characteristic polynomial

The following question is motivated by the study of a stability border for a robust linear time-invariant control system. Let us we have an affine family of $n\times n$ matrices with indeterminate ...
6
votes
0answers
191 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: $$p_{n}(x)=\sum_{k=0}^{n}\binom{2n}{2k}b_k x^k$$ ...
1
vote
1answer
117 views

Truncated sums of symmetric polynomials; reference request for an algebraic derivation

EDIT: This is a case of being too wrapped up in a formulation ($e_j,p_i,$ and the like) to try something simple. It did not occur to me to pull exp to the outside in the weeks I have stared at this. ...
2
votes
1answer
198 views

Powers of linear functions span the space of polynomial functions?

Let $P_n$ be the space of polynomials in $n$ variables over a field of characteristic 0. I'm very sure that the space spanned by powers of linear functions is the whole space $P_n$. Anyone can come ...
0
votes
0answers
37 views

What is maximum number of m-complex solutions to a order n polynomial (say with real coefficients)?

I know the answer is n^2 for bicomplex numbers. Does anyone know if a general answer has been found for m-complex numbers ( http://en.wikipedia.org/wiki/Multicomplex_number)?
0
votes
1answer
110 views

Finding an interpolating polynomial on a set of points such that the polynomial only has extrema on those points

Suppose that we have an arbitrary set of points $S\subseteq\mathbb{C^2}$ and want to find a polynomial $P(X)$ such that $(x,y)\in S\Rightarrow y=P(x)$ and $P'(x)=0\Rightarrow (x, P(x))\in S$. In other ...
4
votes
2answers
297 views

Can a polynomial be almost always divisible by a member of a finite set of primes?

Special case of Bunyakovsky conjecture Let $f(x)$ be non-constant irreducible polynomial with integer coefficients, no fixed prime factor and positive leading coefficient. Let $S$ be a finite set of ...
8
votes
1answer
258 views

Irreducibility of a class of polynomials

This question is directly inspired by this question. Consider polynomials of the form $$p(x) = \prod_{i=1}^n(x-i)^2 - d.$$ For which values of $n$ and $d$ is $p(x)$ irreducible? There is a theorem of ...
2
votes
0answers
218 views

For which constant $d$ this polynomial is reducible over $\mathbb Q$? [closed]

Let $$f(x)=(x-1)^2(x-2)^2(x-3)^2\cdots(x-2013)^2+2014\tag{1}$$ Prove or disprove: $f(x)$ is reducible over $\mathbb Q$. See :http://www-irma.u-strasbg.fr/~bugeaud/travaux/PolyaType.pdf ...
3
votes
3answers
300 views

Can you efficiently solve a system of quadratic multivariate polynomials?

Given a system of 2nd-degree polynomials, $P=\{p_1,\dots,p_m\}$ where $p_i: \mathbb{R}^n \rightarrow \mathbb{R}$, can you efficiently find $x_1,\dots,x_n\in\mathbb{R}$? I know that in the general ...
3
votes
0answers
207 views

The Poisson-kernel in the plane and polynomials

Let \begin{align*} p(z) & = \ \prod_{j = 1}^{n} (z - z_j) \, ; \ |z_j| \ = \ 1 \\ & = \ \prod_{j = 1}^{l+1} (z - z_j)^{M_j} \end{align*} be a non-constant complex polynomial with ...
9
votes
3answers
505 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 ...
4
votes
2answers
393 views

Characterizing $\mathbb{Q}[X]$ via a property of its tensor powers

Let $\varphi: \mathbb{Q}[X] \longrightarrow R$ an inclusion of commutative rings. Suppose that the map $$- \circ \varphi: \operatorname{Hom}_{\mathbb{Q}\operatorname{-alg}}(R, R^{\otimes_{\mathbb{Q}} ...
10
votes
1answer
249 views

Reducibility of polynomials maps

Motivated by this question. Let $f \in \mathbb{Q}[x]$ or$f \in \mathbb{Z}[x]$ . Consider the sequence $f(x),f(f(x)), \ldots f^n(x)$. If some $f^k(x)$ is reducible, the rest iterates will be ...
7
votes
2answers
87 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 ...
1
vote
0answers
53 views

Irreducibility of reflexive compositions of polynomials

Let $m$ be a positive integer. For $k = 1, 2, 3, ... m$, fix $g_k(x_1, ..., x_{k + 1}) \in \mathbb{Z}[x_1, ..., x_{k + 1}]$. For any polynomial $p(x) \in \mathbb{Z}[x]$, let $P_0(x) = p(x), P_1(x) ...
3
votes
2answers
336 views

The holomorphic version of Galois theory

We identify the space of polynomials of degree n with $\mathbb{C}^{n+1}-\mathbb{C}^{n}$, that is an $n+1$ tuple $(a_{n},a_{n-1},\ldots,a_{0})$ with $a_{n} \neq 0$ is identified with $p(z)=a_{n}z^{n} ...
3
votes
2answers
266 views

Irreducible polynomials in $\mathbb{Q}_p((X))[Y]$

I'm looking for some criteria for the irreducibility of polynomials with coefficients in $\mathbb{Q}_p((X))$. In particular, is the polynomial $Y^2+1$ irreducible over $\mathbb{Q}_3((X))$? And how ...
4
votes
2answers
422 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 $\mathbb{Q}[X]$?
2
votes
1answer
231 views

Random algebraic numbers are linearly disjoint almost surely?

I already posted this question at MSE here, but since it received no answer or comment so far I cross-post it here. It is well-known that if one considers a “random” monic polynomial of fixed degree, ...
4
votes
1answer
119 views

If the sequence of degrees of the iterates of a self-map of $\mathbb{A}^2$ is bounded, is it eventually periodic?

Let $f : \mathbb{A}_k^2 \to \mathbb{A}_k^2$ be a regular self-map of the affine plane over a field $k$ of characteristic zero. Assume that the sequence $(\deg{f^n})_{n \in \mathbb{N}}$ is bounded. Is ...
6
votes
1answer
105 views

How “accidental” are equalities between parts of Ehrhart quasi-polynomials? When do they persist to Euler-Maclaurin?

Background What I think of Ehrhart theory (http://en.wikipedia.org/wiki/Ehrhart_polynomial) asserts that if we take a lattice polytope $P$, and count the number of lattice points in the $t$th ...
7
votes
0answers
221 views

Do the coefficients of these irreducible polynomials always become periodic?

Fix $n\in\mathbb N$ and a starting polynomial (or seed) $p_n=a_0+a_1x+\dots+a_nx^n$ with $a_k\in\mathbb Z\ \forall k$ and $a_0a_n\ne0$. Define $p_{n+1},p_{n+2},\dots$ recursively by $p_r = ...
6
votes
1answer
157 views

Knots indistinguishable by HOMFLY

Is there any list (incomplete of course) of knots, that have similar HOMFLY polynomials? I am mainly interested in torus knots.
6
votes
0answers
78 views

Irreducibility testing and factoring

It is a result of van Hoeij and Novicin (Algorithmica, 2012) that factoring polynomials of degree $d$ over the integers can be done in $O(d^6 + d^4 \log^2 A)$ time, where $A$ is the coefficient bound. ...
5
votes
1answer
185 views

Large gaps between consecutive irreducible polynomials with small heights

For a prime gap of length at least $n$, a trivial upper bound for its first occurrence is $N=n!$ or $N=lcm(2,\dots,n)$. A bit better is $N=p_1\cdots p_n$ where $p_k$ is the $k$th prime, as then ...
3
votes
1answer
202 views

Are there any efficient (polynomial time) algorithms for finding if a multivariate quadratic polynomial has a root?

I know that in general, polynomial satisfiability is NP; however, I'm curious to know what work has been done on special classes of polynomials, and in particular quadratic polynomials of multiple ...
1
vote
2answers
264 views

Bounds on the largest root of a polynomial

Consider the following polynomial: $p(x)=x^{3}-(k-1)x^{2}-(2k-1)x+(k-1)^{2}$, where $k \geq 5$ is a fixed parameter. I am trying to find a strong lower bound on the largest root $x_{\max}$ of the ...
0
votes
0answers
106 views

On the Brioschi-like quintic $v^5 - 5d v^3 + 10 d^2 v - d^2 =0 $

The general quintic can be transformed in radicals using a rational Tschirnhausen transformation to the one-parameter Brioschi quintic, $$u^5 - 10c u^3 + 45 c^2 u - c^2 = 0\tag{1}$$ which can be ...
4
votes
1answer
113 views

Do interpolation nodes have to be dense?

Let $f(x) = \exp(x)$ and $(\xi_i)_{i=0}^\infty, \, \xi_i \in (0,1)$ be a sequence of points from the unit interval. For $n \in \mathbb{N}$ let $P_n$ be a polynomial of degree $n$ that interpolates ...
1
vote
0answers
218 views

When $x^n - (x+a)^{n-k} (x+b)^k$ is irreducible?

Fix a nonzero integer $a$ and a positive integer $k$. I'm looking for some criterion to establish for which nonzero integers $b$ and $n \geq 2$ the polynomial $$f(x) := x^n - (x+a)^{n-k} (x+b)^k$$ is ...
2
votes
2answers
221 views

Irreducible Polynomials from a Reccurence

This question is inspired by a recent one : Let $c$ be a variable and define a sequence by $a_0=0$ $a_1=1$ and $a_{n+1}=a_{n}c-a_{n-1}$ . So $$\begin{align*} a_2 &= c \\ a_3 &={c}^{2}-1= ...
16
votes
5answers
742 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 ...
3
votes
2answers
314 views

noncommutative polynomials equality

Suppose $x$, $y$, $z$ are three variables satisfying $yz=zy$, $zx=xz$, $xy=yzx$. Could anyone give me two (non-commutative) polynomials $f$ and $g$ in the above three variables such that the ...
3
votes
0answers
143 views

What can we say about a semigroup that's acted on by an ideal of polynomials?

This got no response on MSE, so posting here. Let $S = \{ (a,b) \in \Bbb{Z}^2 : \gcd(a,b) \neq 1 \} \cup (1,1)$. Then $S$ forms a semigroup. The operation being componentwise multiplication. Let ...
10
votes
3answers
641 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 $f_2 = x^{m_2} + (1+x)^{n_2}$ ...
2
votes
2answers
158 views

Finding a simpler “local” lower bound for a rational function

I have obtained as the expression for some quantity the following gargantuan formula: $$ \frac{k^8 + 3k^7 + 8k^6 + 3k^5 - 16k^4 - 32k^3 + 63k^2 - 34k + 6}{k^6 + 3k^5 + 6k^4 - 24k^2 + 21k - 5}$$. ...
0
votes
0answers
82 views

Lower-Upper bounds on the cardinality of a set

Let $S$ be a finite set which is a subset of $\{(\alpha ,\beta ):\alpha , \beta \in \mathbb{Z}, \alpha\geq 0, \beta \geq 0\}$ and $ T(x,y)=\sum_{(\alpha ,\beta ) \in S} h_{\alpha, \beta} ...
1
vote
1answer
135 views

Roots of the derivative as symmetric functions of the roots of the polynomial

Let $p(t)=(t^2-a_1^2)\ldots(t^2-a_n^2)$ be an even polynomial with distinct real non-zero roots. Can the roots of its derivative $p'(t)$ be expressed nicely (e.g. as rational symmetric functions) in ...
6
votes
1answer
518 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 ...
1
vote
0answers
134 views

Finite fields: alternating sums of values of polynomials

Notation In what follows let $p$ be a (odd, if needed) prime, $e$ a positive integer, $q = p^e$; $\mathbb{F}_q$ will denote a finite field with $q$ elements whose prime subfield will be denoted as ...
0
votes
0answers
136 views

Image of critical points

Let $K$ be a field of characteristic $0$, $f:K^n\rightarrow K^n$ be an algebraic function, that is, $n$ polynomial functions in $n$ variables. Let $S$ be the set of critical points of $f$. If ...
14
votes
1answer
578 views

When is $f(x_1, \dots, x_n)+c$ an irreducible polynomial for almost all constants $c$?

Let $f\in \mathbb C[x_1, \dots, x_n]$, $n\ge 1$, be a non-constant polynomial. Consider the polynomial $f+t\in \mathbb C[t, x_1,\dots, x_n]$. This is an irreducible polynomial in $\mathbb C(t)[x_1, ...
3
votes
1answer
234 views

Menon’s identity

I also put this question in stackexchange, but remained unanswered. http://math.stackexchange.com/questions/506996/menons-identity Let $G$ be a group of order $n$. Consider an action of $U_n$, the ...
5
votes
1answer
196 views

Polynomiality of functions over residue rings

Suppose $\mathbb{Z}/m \mathbb{Z}$ is a residue ring for some $m \in \mathbb{N}$. If $m=p$ is a prime number then every function $f:\mathbb{Z}/p \mathbb{Z} \rightarrow \mathbb{Z}/p \mathbb{Z}$ is a ...
7
votes
0answers
449 views

Getting a bound via polynomial equations

When studying the existence problem of power residue deference sets, I came across the following system of polynomial equations over $\mathbb{C}$, \begin{cases} ...
2
votes
1answer
311 views

About the practice of Bernstein-Kushnirenko theorem

The following refers to common roots of bivariate polynomial equations and, in particular to the quim's and auniket's comments. The BKK theorem (cf. arXiv:0812.4688. Theorem 5.4) asserts that if we ...