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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0
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0answers
28 views

Closed-for expression for Newton-Girard symmetric polynomials with 0/1 variables

There are $n$ Bernoulli $s_i\in\left\{0,1\right\}$, $i=1,...,n$ with equal marginals $\Pr(s_i=1)=\theta$ $\forall i$ so that E$(s_i)=\theta$. Their standardized mean deviations are \begin{equation*} ...
5
votes
0answers
152 views

The equation $f(x)=f(a^n)^k$ always has a solution in $\mathbb{Q}$ [closed]

Fix two positive integers $a,k\ge2$, and let $f\in\mathbb{Z}[x]$ be a nonconstant polynomial. Suppose that for all sufficiently large positive integers $n$, there exists a rational number $x$ ...
-1
votes
1answer
86 views

derivatives and uniformly convergence [closed]

Let $f$ be a function of a real variable expandable in power series on $\mathbb R$: there exists a sequence $(a_n)_{n\in\mathbb N}$ of reals such that for all $x\in\mathbb R$, one has ...
4
votes
0answers
110 views

The closures in $C^0(\mathbb C,\mathbb C)$ of the set of integer valued polynomials

This question is closely related to the thread The closures in $C^0(\mathbb R,\mathbb R)$ of the set of integer valued polynomials, resp, of polynomials with integer coefficients. (Recall that a ...
3
votes
1answer
192 views

Polynomial without roots in a ring

Let $A$ be a commutative ring which is not an integral domain. I try to find a polynomial $P$ of $A[X]$ such that $d°P = 1$ and $P$ admits no root in any ring $B$ such that $A$ is a subring of $B$.
5
votes
1answer
216 views

Literature about metapolynomials

We say that a function $f:\mathbb{R}^k \rightarrow \mathbb{R}$ is a metapolynomial if, for some positive integer $m$ and $n$, it can be represented in the form $$f(x_1,\cdots , x_k ...
3
votes
0answers
73 views

Unimodular triangulation and Ehrhart polynomials

Let $P$ be a convex lattice polytope. Then it has a polynomial Ehrhart function. I am interested in what can be said about the Ehrhart polynomial when $P$ has any of the properties is integrally ...
1
vote
0answers
101 views

polynomial with rational coefficients [closed]

Do you think that the following statement is true? Do you have any idea about the proof? Let $\; f(x) \in \mathbb{C}[x]$ be a polynomial. If $f(n) \in \mathbb{Z}$ for an infinite number of $n \in ...
3
votes
0answers
93 views

Discriminant polynomial generalizing the usual discriminant

I wonder if anybody has seen the following natural polynomial. Given a monic univariate polynomial $P(z)$ of degree $N$, denote its roots by $z_1,..., z_N$. Now form a new polynomial $Q(z)$ of ...
2
votes
1answer
98 views

Roots of modified polynomials

Consider the following two polynomials: $$ g=x^3 - x^2 - (c + 2)x + c $$ and $$ h=x^3 - x^2 - cx + c $$ The roots of $h$ are $1$ and $\pm \sqrt{c}$. I am interested in obtaining the roots of $g$, ...
14
votes
2answers
522 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 $n$ ...
5
votes
7answers
488 views

Source for roots of matrix polynomials?

A matrix polynomial is a polynomial whose variables are square $n \times n$ matrices, let's say with entries in $\mathbb{C}$, and with coefficients in $\mathbb{C}$. I am seeking a source of results on ...
4
votes
1answer
172 views

Stronger versions of Schwartz-Zippel for random linear subspaces

This is a (self-contained) followup question to http://math.stackexchange.com/questions/380672/analogue-of-the-schwartz-zippel-lemma-for-subspaces. Let $f : \mathbb{R}^n \to \mathbb{R}$ be a nonzero ...
2
votes
1answer
72 views

Lagrange Interpolation and integer polynomials

Suppose that there is a polynomial $P$ with integer coefficients such that $P(x_i)=y_i$ for $i=1,\ldots,n$. Is it true that the result of Lagrange interpolation through the data $(x_i,y_i)$ is a ...
1
vote
0answers
53 views

Generalization of the Hermite-Beihler-Kakeya Theorem (2)

This is a follow up to the questions posed in Generalization of the Hermite-Bielher-Kakeya Theorem. Here is an interesting follow up to those comments. Firstly we remark that: $f(x)+g(x)\cdot w$ is ...
1
vote
0answers
135 views

The gcd of coprime polynomials evaluated at integers

Let $p(x),q(x)$ be coprime squarefree polynomials with integer coefficients. For integer $n$ is $\gcd(p(n),q(n))$ bounded by an absolute constant? In case the answer is negative what is the fastest ...
1
vote
0answers
34 views

Find a common expression (parameterized by y = 1/2, 1 and 2) uniting three (rational) polynomials in k [closed]

I am seeking a common "simple" expression (preferably/presumably a sum of products of linear factors, or sum of products of low-degree factors) uniting the three polynomials (parameterized by ...
2
votes
2answers
138 views

Sextic resolvent has no rational root

An irreducible quintic $f(x)\in\mathbb{Q}[x]$, is solvable by radicals if and only if its sextic resolvent $\theta_f (y)=(y^3+py^2+qy+r)^2-2^{10}\Delta(f)y$ has a rational root ($\Delta$ is the ...
0
votes
0answers
26 views

number of bipolar orientations to acyclic ones

Given a $k$-regular graph $G$, is there an upper bound on the number of bipolar orientations that $G$ has? I am trying to show that the number of bipolar orientations is much much lower than the ...
5
votes
0answers
148 views

Are the roots of chromatic polynomials plus a fixed constant dense in $\mathbb{C}$?

Alan Sokal proved that chromatic roots are dense in the whole complex plane. I.e., if $P(G;z)$ denotes the chromatic polynomial of a finite simple graph $G$ evaluated at $z \in \mathbb{C}$, then ...
3
votes
2answers
325 views

Degrees of factors of polynomial $f(x)=x^q-(ax^2+bx+c)\in \mathbb{F}_q[x]$

I’m facing the problem of factoring polynomials of type $f(x)=x^q-(ax^2+bx+c)\in \mathbb{F}_q[x]$ and the degrees of factors seem to be quite special. For example, according to my experimental results ...
10
votes
5answers
798 views

Is $x^p-x+1$ always irreducible in $F_p[x]$?

It seems that for any prime number $p$ and for any non-zero element $a$ in the finite field $\mathbb F_p$, the polynomial $x^p-x+a$ is irreducible over $\mathbb F_p$. (It is of course obvious that ...
3
votes
1answer
131 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 ...
1
vote
0answers
75 views

Estimating decay of certain trigonometric polynomials

For $p=0,1,2,\dots$ and $n=0,1,2,\dots,$, let $f_{n,p}(z)=\sum_{k=0}^n k^p z^k$ be a sequence of polynomials. Restricted to the unit circle, the functions $g_{n,p}(t):=f_{n,p}(e^{it})$ are ...
1
vote
1answer
141 views

Polynomial convex coefficients

Assume we have an arbitrary high order polynomial $$f(L)=1-L\theta_1-L^2\theta_2-L^3\theta_3-...-L^N\theta_N$$ and we know all roots of this polynomial site outside the unit circle. It is obvious that ...
2
votes
1answer
116 views

Nilradical and Newton's identities

Let $R$ be a commutative ring with unity such that $n!$ is not a zero-divisor. Let $s_1=\sigma_1,s_2,s_3\cdots$ and $\sigma_1,\sigma_2\cdots$, (convention: if $k>n$, then $\sigma_k=0$) be elements ...
3
votes
2answers
164 views

“Degree 3 fields”

I was wondering what was known about fields $k$ having the property that any polynomial over $k$ of degree $3$ has at least one root in $k$. Does such a field have a special name ? Is there some kind ...
17
votes
1answer
990 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 ...
1
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0answers
36 views

Identification of model involving convex polynomials

I want to solve a nonlinear least squares problem on the following form \begin{equation} \begin{array}{l} \min_{\theta,\phi} J(\theta,\phi) &=& \min_{\theta,\phi} \sum_{i=1}^k ...
4
votes
0answers
139 views

The Alexander-Conway polynomial: from knots to braids?

The Alexander-Conway polynomial was the first knot invariant to be discovered, as far back as 1923 according to this link. Given that knots can be expressed in terms of quasi-toric braid closures, it ...
1
vote
0answers
52 views

Whether r.v. with p.g.f. $\exp [\sum\limits_{i = 1}^\infty {{q_i}({z^i}} - 1)]$ is overdispersion?

When discrete r.v. $X$ is not Poisson distributed and ${\rm{Var}}X,EX < \infty $, I want to know whether r.v. $X$ with p.g.f. $\exp [\sum\limits_{i = 1}^\infty {{q_i}({z^i}} - 1)],({q_i} \in ...
10
votes
3answers
252 views

Uniform bound for the number primes $p$ s.t. a polynomial has a root modulo $p$

It is well known that for a given polynomial $f \in \mathbb{Z}[x]$ the number of primes $p$ s.t. $f$ has a root modulo $p$ is infinite. In fact, one can even write down a formula for the density of ...
9
votes
1answer
672 views

How small can a totally positive integer be?

Consider a large, fixed $M>2$. For each $n$, let $\alpha_n$ denote the smallest algebraic integer of degree at most $n$, all of whose Galois conjugates lie in the real interval $(0,M)$. Is there ...
3
votes
1answer
101 views

Convergence of the Double Integral of a Polynomial Reciprocal

Let $f \in \mathbb{R}[x,y]$ be a polynomial satisfying the following conditions: (i) $f(\mathbb{R}^2) \subset [a,\infty)$ where $a>0$; (ii) $f$ is non-degenerate, in the sense that there isn't a ...
3
votes
1answer
285 views

Counting solutions modulo primes

Let $P(x)$ be an irreducible polynomial in $\mathbb{Z}[x]$ of degree $n.$ By $N(k,x)$ we denote the number of primes up to $x,$ such that $P(x)$ has exactly $k$ solutions in $\mathbb{Z}_p.$ Is it ...
0
votes
0answers
77 views

Complexity of turning a d-degree polynomial to 2-degree polynomial

For a very simple example, $(1+x)^4=x^4+4x^3+6x^2+4x+1$ is a 4 degree polynomial, and I want to change it to a 2-degree polynomial by add more variables, for this example, we can simply let $y=x^2$, ...
0
votes
1answer
125 views

The Largest Root of Associated Laguerre Polynomial

The Laguerre polynomial $L_n(x)$ is the solution to the Laguerre differential equation \begin{equation*} x\,y'' + (1 - x)\,y' + n\,y = 0. \end{equation*} The associated Laguerre polynomial ...
0
votes
1answer
97 views

k-th largest root in common interlacing polynomials

In their proof of the celebrated Kadison-Singer conjecture, Marcus, Spielman and Srivastava exploited so-called interlacing families which are originally defined for their work on Ramanujan graphs. ...
1
vote
2answers
142 views

Under what conditions does the Mittag-Leffler function ${E_{\alpha ,1}}(z),(0 < \alpha < 1)$ has no real zero?

I want to know under what conditions does the Mittag-Leffler function ${E_{\alpha ,1}}(z),(0 < \alpha < 1)$ has no real zero, where ${E_{\alpha ,1}}(z) = \sum\limits_{k = 0}^\infty ...
8
votes
1answer
179 views

is there any such result about Bernstein polynomials?

It is well known that for any lipschitz function $f:[0,1]\rightarrow [0,1]$, we can approximate it by $\sum_{i=1}^n f(i/n) {n\choose i} x^i (1-x)^{n-i}$, and the $L_\infty$ error is $O(1/\sqrt{n})$. ...
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 ...
3
votes
0answers
125 views

Correspondence between numerical semigroups and polynomials?

A numerical semigroup $A$ is defined as a subsemigroup of the semigroup $(\mathbb{N},+)$ of the positive integers such that the set $\mathbb{N}\setminus A$ is finite. Equivalently (for a subsemigroup) ...
3
votes
1answer
68 views

Counting extrema on a simplex

Let $p(x_{1},x_{2},\ldots,x_{n})=\sum_{i,j=1}^{n}{a_{ij}x_{i}x_{j}}$ be a homogenous multivariate polynomial of degree $2$. I would like to know how many extrema $p$ has on the standard simplex ...
2
votes
2answers
208 views

Factorisation of a biquadratic polynomial

Let $u,v\in\mathbb{Z},$ and let $f=X^4+uX^2+v.$ Let $p$ be a prime number, and let $r\geq 1.$ In a paper I'm reading, one can find the following result. Proposition. If $f$ is reducible modulo ...
3
votes
0answers
117 views

Question related to the number of rational curves on a hypersurface

The following is from a PhD thesis "Algebraic Circle Method" by Thibaut Pugin, which I am currently reading. Let $k$ be a finite field of order $q$. Let $\mathfrak{X} \subseteq \mathbb{P}^n_k$ be ...
7
votes
1answer
158 views

Inequality for Laguerre polynomials

Let $L_n$ be the $n$-th Laguerre polynomial defined by $\quad L_n (x)=\frac{e^x}{n!}\frac{d^n}{dx^n}(x^n e^{-x}).\quad $ I want to prove that $$ \forall n\in \mathbb N,\forall x\ge 0,\quad \sum_{0\le ...
2
votes
1answer
522 views

Absolute convergence of logarithm of polynomial with positive coefficient ($\ln G(z) = \sum\limits_{i = 0}^\infty {{q_i}{z^i}} $)

Special problem: Let $G(z)$ be a probability generating function(pgf, the $z$ can be seen as real number or complex number), that is $$G(z) = \sum\limits_{i = 0}^\infty {{p_i}{z^i}} ,(\left| z ...
1
vote
1answer
153 views

Zeroes of a complex polynomial on a sphere as a manifold

Let $ f \in \mathbb{C}[z_1, \ldots, z_n]$ be a polynomial such that $f'(z) \neq 0$ if $z \neq 0$ ($f'$ means $\left( \frac{\partial f}{\partial z_1}, \ldots, \frac{\partial f}{\partial z_n}\right)$ ). ...
3
votes
0answers
127 views

Number of rational curves on varieties over finite fields

Let $k$ be a finite field. Let $P_r$ be set of degree $r$ binary forms over $k$. We define $$ \mathcal{M}_r = \{ (x_0, ..., x_n) \in P_r^{n+1} : x_0^d + ... + x_n^d = 0 \text{ and the }x_i \text{'s ...
2
votes
3answers
178 views

Multivariate Hermite Polynomials

Let $h_0, h_1, \dots$ be the classical univariate Hermite polynomials, renormalized to have constant norm. Is $$x\mapsto\prod_{j=1}^n h_{l_j}(x_j), \quad l_j\in \mathbb N$$ a complete orthogonal ...