**2**

votes

**2**answers

103 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.
$\bf{Proposition. }$If $f$ is reducible ...

**16**

votes

**1**answer

801 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**

vote

**1**answer

456 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

**0**answers

68 views

### How to show non-existence of elements in the intersection of two ideals?

Given l, k any two natural numbers, define
$I_1 =\langle y^{l+2k}, (x+y)^{3l+2k} \rangle: x^{l+2k} + \langle y^{2l+3k}, (x+y)^{3l} \rangle: x^{l+2k}; $
$I_2 = \langle x^{l+k} \rangle.$
I want to ...

**64**

votes

**11**answers

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 ...

**1**

vote

**0**answers

46 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 ...

**4**

votes

**0**answers

91 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

**0**answers

35 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 ...

**1**

vote

**0**answers

155 views

### Integer roots of a polynomial [closed]

I am interested in the following problem due to its connection to the variance of two random variables. Do there exist integers $p\ge1$ and $j\in \mathbb Z$ and real numbers ...

**7**

votes

**3**answers

213 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 ...

**15**

votes

**2**answers

395 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 ...

**9**

votes

**1**answer

636 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 ...

**1**

vote

**0**answers

185 views

### A question about the Sylvester determinant

I am writing an invited article related to the diophantine equations like the abc-conjecture by the way of the determinant of the Sylvester matrix. We know that Sylvester never really finalize some ...

**3**

votes

**1**answer

79 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

**1**answer

254 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

**0**answers

74 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

**0**answers

32 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

**1**answer

61 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

**2**answers

111 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 ...

**2**

votes

**1**answer

497 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 ...

**17**

votes

**3**answers

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 ...

**8**

votes

**1**answer

130 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 $O(1/\sqrt{n})$. ...

**5**

votes

**1**answer

825 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 ...

**3**

votes

**0**answers

86 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

**1**answer

56 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
...

**3**

votes

**0**answers

196 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 ...

**1**

vote

**2**answers

349 views

### Finding regions where multi-variate polynomials are positive

Given a constant $k \in \mathbb N$, and a set of $p$ multi-variate polynomials {$P_j:\mathbb N^n\to \mathbb Z$}$\_{j=1...p}$, with $P_j \not\equiv 0$.
Is the following true:
There exists $n$ sets ...

**10**

votes

**0**answers

151 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 1893, ...

**3**

votes

**2**answers

317 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

**0**answers

113 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 ...

**6**

votes

**1**answer

127 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

**0**answers

124 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 ...

**1**

vote

**1**answer

145 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 $\left( \frac{\partial f}{\partial z_1}, \ldots, \frac{\partial f}{\partial z_n}\right)$ ). ...

**2**

votes

**3**answers

131 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 ...

**5**

votes

**2**answers

537 views

### Would such polynomial identity exist? (related to sum of four squares)

Let $f_1,f_2,f_3,f_4,f_5 \in \mathbb{Q}[x]$ be linear and
coprime and not all constant.
Is it possible $ f_1^2+f_2^2+f_3^2+f_4^2=f_5^2$?
I suppose the answer is negative.
If this is possible, ...

**4**

votes

**2**answers

157 views

### Finding roots of a recursively given polynomial sequence / eigenvalues of an almost Toeplitz matix

I would like to find the roots of the polynomial sequence given by a recurrence relation as follows:
$V_0(x) = 1-a^2$
$V_1(x) = 1-a^2 - x$
$V_{k \geq 2}(x) = (1+a^2 - x)V_{k-1}(x) - a^2V_{k-2}(x)$
...

**18**

votes

**3**answers

936 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 ...

**2**

votes

**0**answers

110 views

### Must the coordinates of a polynomial iteration have about the same size?

Original post. The following statements seem plausible (not to say intuitively obvious), but I do not see how to prove them.
Let us say that a polynomial mapping of $\mathbb{C}^2$ is reducible if ...

**1**

vote

**0**answers

75 views

### Irreducibility of reflexive compositions of polynomials (II)

Given an unknown polynomial $p \in \mathbb{Z}[x]$, write on the blackboard a finite expression that defines a polynomial $q \in \mathbb{Z}[x]$ by using only constants $1, x, p$ of the ring ...

**5**

votes

**0**answers

47 views

### Bounding volume of cell in complement of zero set

I am given an integer polynomial $f \in \mathbb{Z}[X_1, \ldots, X_n]$ of bounded degree and bounded coefficient size. The polynomial's zero set partitions $\mathbb{R}^n$ into cells. What I am looking ...

**9**

votes

**1**answer

478 views

### Real-rootedness, interlacing, root-bounds of a sequence of polynomials

Problem: the number $a(n,k)$ is defined by the following recurrence
\begin{equation}
a(n,k)=(k+1)(k+2)\, a(n-1, k)+\frac{(k+1)(k+2)(k+3)}{k} \,a(n-1, k-1),
\end{equation}
with $a(1,1)=1$ and ...

**30**

votes

**1**answer

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 ...

**6**

votes

**1**answer

499 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 ...

**7**

votes

**1**answer

102 views

### Positivity of coefficients of a power series

How does one check for the positivity of coefficients of a rational function,say, for example $\frac{p_1(x,t)}{(1-xt)(1-x^2t)(1-x^3t)}$ where $p_1(x,t) = 1 + tx + 2t^2x^2 - 3x^3t^2 -x^5t^3 - ...

**34**

votes

**2**answers

768 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
...

**2**

votes

**0**answers

147 views

### square-free parts of values of polynomials

Given a polynomial $f(x) \in \mathbb{Z}[x]$ of degree $d$, consider the following three sets:
$$N_1(x) = \#\{k \leq x: f(k) \text{ is square-free}\}$$
$$N_2(x) = \#\{n \leq x: n = f(k) \text{ is ...

**10**

votes

**1**answer

248 views

### Minimize norm of a polynomial around a circle (count the solutions)

I already posted this question at MSE here, but as it received no significative feedback for a while I cross-post it here.
I also noticed a related question here on MO (which does not answer my ...

**9**

votes

**3**answers

1k views

### An algebraic number is not a root of unity?

This problem is related to my study of the Burau representation of the braid group $B_3$: I was trying to show that certain "congruence subgroups" are of infinite index.
There is an approach that ...

**1**

vote

**1**answer

68 views

### Polynomials are dense in $A_{B(0,1)}$

Let $D(0,1)$ be the disk of center 0 and radius 1 and call $A_{D(0,1)}= \{ f:\overline{D(0,1)} \rightarrow \mathbb{C} : f \text{ is continuous and } f|_{D(0,1)} \text{ is holomorphic} \}$.
Can ...

**12**

votes

**1**answer

343 views

### Probability that random cubic polynomials meet in a square

Let $p_1(x)$ and $p_2(x)$ be cubic polynomials with
random coefficients in $[-1,1]$.
I wanted to compute the probability that $p_1$ and $p_2$
share at least one point within
the square $[-1,1]^2$.
Of ...