**3**

votes

**1**answer

169 views

### Irreducibility of trinomials

I wonder if the following is known or, not very difficult to see:
Let $K$ be a number field and $A, B \in \mathcal{O}_K$ be nonzero integers of $K$. Does there necessarily exist a positive integer $n ...

**1**

vote

**1**answer

147 views

### If $F(a_1,\ldots,a_k)=0$ whenever $a_1,\ldots,a_k$ are integers such that $f(x)=x^k-a_1x^{k-1}-\cdots-a_k$ is irreducible, then $F\equiv0$

I'm trying to understand a proof of the following theorem (from section II of Hall's paper An Isomorphism Between Linear Recurring Sequences and Algebraic Rings):
If $F(a_1, \ldots, a_k)$ is a ...

**0**

votes

**1**answer

169 views

### Integral points on genus 0 curves related to polynomial identities

Consider the identity
$$(x+1)^5+(y+1)^5 = (2x + 6y) ( -x +y )^4 + f(x,y)$$
where $f(x,y)=-x^5 + 2*x^4*y + 12*x^3*y^2 - 28*x^2*y^3 + 22*x*y^4 - 5*y^5 + 5*x^4 + 5*y^4 + 10*x^3 + 10*y^3 + 10*x^2 + ...

**6**

votes

**3**answers

441 views

### Idempotent polynomials

Let $R$ be a commutative ring with identity and let $f \in R[x]$. There are well known characterizations for $f$ to be a nilpotent element of $R[x]$ or to have a multiplicative inverse in $R[x]$. Is ...

**9**

votes

**3**answers

725 views

### Why is the gcd so large in an identity related to the $abc$ conjecture?

Consider the identity
$$ (x+z)^5+(y-z)^5 = (-3 x + 4 y)^2 (x + y)^3 + (x+y) f(x,y,z) $$
Where $f(x,y,z)=(-8*x^4 + 5*x^3*y + 24*x^2*y^2 - 9*x*y^3 - 15*y^4 + 5*x^3*z - 5*x^2*y*z + 5*x*y^2*z - 5*y^3*z + ...

**2**

votes

**0**answers

79 views

### Usefulness of polynomial ideals for graphs

Given a graph $G=(V,E)$ on $n$ vertices, one can associate to it the polynomial $f \in k[x_1,\dots,x_n]$ given by $f = \prod_{\{i,j\}\in E}(x_i - x_j)$. Then one can map various graph properties into ...

**28**

votes

**3**answers

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 ...

**5**

votes

**2**answers

150 views

### Collision polynomials

Consider $P_n(x)$ polynomials defined through the recurrence relations
$$P_n(x)=2(1-x)P_{n-1}(x)-(1+x)^2P_{n-2}(x),$$ with $P_0(x)=1$ and $P_1(x)=1-3x$.
In fact, the explicit solution of these ...

**7**

votes

**2**answers

191 views

### Finding a low-degree polynomial vanishing on half the zeroes of a polynomial system

Let $f(x)$ be a real polynomial of degree $2d$ without real roots. Let the complex roots be $z_1$, $\bar{z_1}$, $z_2$, $\bar{z_2}$, ..., $z_d$, $\bar{z_d}$ with $z_i$ in the upper half plane. Let ...

**4**

votes

**0**answers

164 views

### Pair of two-variable polynomial equations of high order

I have the following pair of equations to be solved for two variables $\rho$ and $D$ resulting from a certain Maximum Likelihood Estimation for a time series $X_n > 0$, $n=0, \ldots, N+1$ with $N ...

**2**

votes

**1**answer

101 views

### Degree principles for non-symmetric polynomials

A theorem of Timofte says that a symmetric polynomial inequality of degree $d$ holds on $\mathbb{R}^{n}_{+}$ if and only if it holds for all vectors in $\mathbb{R}^{n}_{+}$ with at most $\max\{\lfloor ...

**8**

votes

**1**answer

326 views

### certain trigonometric homeomorphisms

Are there any simple characterizations of rational functions $f(x,y)$ with real coefficients such that $\theta\mapsto f(\cos\theta,\sin\theta)$ is a homeomorphism from $\mathbb R\bmod 2\pi$ to ...

**4**

votes

**4**answers

557 views

### The Icosahedron Equation

$$1728 V^5 + F^3 = E^2 \;.$$
Can anyone point me to a concise, modern derivation and explanation of
the significance of the icosahedron equation, more modern and
concise than Klein's description in ...

**1**

vote

**1**answer

96 views

### “Symmetric” Polynomial 4-cocycles

It is an old theorem of Heaton's (based on work of Eilenberg and MacLane), that a polynomial 3-cocycle $f(x,y,z)$ which is "symmetric," in the sense that $f(x,y,z)-f(x,z,y)+f(z,x,y)=0$, is always a ...

**10**

votes

**1**answer

393 views

### Least prime $p$ such that an irreducible polynomial of degree $n$ has no root modulo $p$?

This question is inspired by an old question of Greg Kuperberg, about how small is the first prime $p$ which makes a given monic polynomial $P$ with integral coefficient have a (simple) root modulo ...

**33**

votes

**2**answers

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 ...

**3**

votes

**2**answers

334 views

### Counting polynomials with same coefficient sum and a given value at a point

Call an univariate polynomial $f(x) = \sum_{i=0}^{n}a_{i}x^{i} \in \Bbb{Z}[x]$ symmetric if $a_{i} = a_{n-i}$ and $a_{0} = a_{n} > 0$.
For a given $\sum_{i=0}^{n}a_{i}$ and $a_{i} \geq 0$, how ...

**2**

votes

**1**answer

184 views

### Algorithm for representing a polynomial as a composition of lower degree polynomials

Let $q$ be a large prime and $e$ an integer such that $GCD(e,q-1)=1$. Let $p(x)$ be a polynomial of degree $e^n$ with coefficients in $\mathbb Z_q$ such that there exists a progression of polynomials ...

**2**

votes

**1**answer

275 views

### Is there an irreducible integral polynomial in two variables which is reducible for every value of one of the variables?

Is there a polynomial $f(x,y)$ in two variables, with integer coefficients, such that $f$ is irreducible over the complex numbers (i.e., in $\mathbb{C}[x,y]$), but for every integer $n$, the ...

**28**

votes

**1**answer

1k 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
...

**1**

vote

**0**answers

60 views

### Jack symmetric functions and their inner products

I have some questions regarding Jack polynomials. I use the notation of of I.G. Macdonald's book "Symmetric Functions and Hall polynomials".
Let $\Lambda$ be the ring of symmetric functions over ...

**1**

vote

**2**answers

144 views

### Polynomials giving Lower Degree Elements in an Algebraic Number Field

My earlier related question
Lower Degree Elements in an Algebraic Number Field
has been given a clean answer for the first part. My present question is below:
Take a number field ...

**9**

votes

**1**answer

346 views

### Irreducibility of the trinomial over Q

I'm trying to find an algebraic proof of irreducibility of the polynomial $x^n-x-1$ over rational numbers (or integers, which the same). I've read the Selmer's paper "On the irreducibility of certain ...

**1**

vote

**2**answers

324 views

### Existence of non-trivial solution to non linear polynomial system

I need to find conditions for the existence of non-trivial solutions to a multivariable polynomial system in two cases:
The first case:
$f1: a_1x^2+a_2xy+a_3y^2+a_4z^2=0$
$f2: ...

**20**

votes

**4**answers

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 ...

**1**

vote

**1**answer

115 views

### Special values of continuous q - Hermite polynomials

The continuous $q-$Hermite polynomials are defined by
$${H_{n + 1}}(x|q) = 2x{H_n}(x|q) +( {q^n}-1){H_{n - 1}}(x|q)$$
with initial values ${H_{ - 1}}(x|q) = 0$ and ${H_0}(x|q) = 1.$
Cf. e.g. ...

**2**

votes

**0**answers

175 views

### Reducing a System of Polynomial Equations

I am currently writing a program in SAGE which computes Nilpotent Orbit Varieties for an Algebraic Geometry research project and I have reduced my problem to the following:
Consider a system of ...

**4**

votes

**1**answer

295 views

### Bounding Roots of a Polynomial by Coefficients

I'm using Samuelson's result and a chapter from Marden's monograph "The Geometry of Polynomials". These are sophisticated results. Are these independent from the Jury-Cohn test to show that a ...

**18**

votes

**5**answers

660 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:
...

**7**

votes

**4**answers

528 views

### Minimal representation of a polynomial as a linear combination of squares

Given a polynomial of degree $2n$ over $\mathbb{Q}$, how to represent it as a linear combination (with rational coefficients) of squares of polynomials of degree at most $n$ over $\mathbb{Q}$ such ...

**6**

votes

**1**answer

416 views

### Are roots of transcendental elements transcendental?

This looks extremely easy, but then again it's late at night...
Let $k$ be a commutative ring with unity. An element $a$ of a $k$-algebra $A$ is said to be transcendental over $k$ if and only if ...

**6**

votes

**0**answers

291 views

### Polynomials and divided differences

I would greatly appreciate any hint for proving the following.
Question: Let $f:[0, 1] \to {\bf R}$. Can it be proved that if $[0, 1/(N+m),\dots, (N+m)/(N+m) ; f ]=0$ for all $m=1,2, 3,\dots$, then ...

**1**

vote

**1**answer

273 views

### Help me on proof of an equation.

I wanna prove following equation
$ \sum_{i=1}^n \prod_{k=1,k\neq i}^n \prod_{j=1,j\neq k}^{n+1}(x_j - x_k) = -\prod_{i=1}^n \prod_{j=1,j\neq i}^n (x_j - x_i) $
I have verified several numbers such as ...

**1**

vote

**1**answer

563 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. ...

**3**

votes

**0**answers

158 views

### identity for number of monomials

Fix a positive integer $d \geq 2$, and let $n,k$ be natural numbers with $k \leq n$.
Let $b(n,k)$ denote the number of monomials of degree $kd-(n+1)$ in $n+1$ variables $x_0,\ldots,x_n$ with all ...

**4**

votes

**1**answer

366 views

### Algebraic closure of a polynomial ring

What could be conditions on $k\in\mathbb{C}[x,y,z]$ that would ensure that any polynomial $f\in\mathbb{C}[x,y,z]$ that is algebraically dependent of $k$ is indeed a polynomial in $k$, ie ...

**1**

vote

**2**answers

199 views

### Resultant of system with 3 polynomials and 3 variables

Let us say I have a system of 3 polynomials, f1(x,y,z), f2(x,y,z), f3(x,y,z). How to find the resultant of these 3 polynomials? What I mean is: is there any special method to do this? Does the ...

**0**

votes

**1**answer

185 views

### Existence of nontrivial roots of a homogeneous polynomial over a finite field in extension fields

Consider a homogeneous polynomial, f, of total degree n in n variables, with coefficients in a prime order finite field, GF(p).
Are there any general rules regarding the existence of nontrivial roots ...

**3**

votes

**1**answer

265 views

### Multivariate polynomial approximation of smooth functions

Let $f$ be a function defined on $[-1,1]^d$. Assume that all partial derivatives of $f$ up to order $r$ are continuous; and the $\infty$-norm of these partial derivatives are uniformly upper bounded ...

**1**

vote

**1**answer

247 views

### How to show an ideal is Zero-dimensional [closed]

I have the following past exam paper question, a similar sort of question seems to come up every year.. And i'm completely lost with it...
Let $J$ denote the ideal in $\mathbb{Q}[x,y,z]$ generated by ...

**0**

votes

**1**answer

192 views

### Zeros of compositions of polynomials and derivatives

Suppose we have polynomials $f$ and $g$ over a field $F$ and for some $a\in F$ we know $(x-a)^m$ divides $f\circ g$, i.e. $f\circ g$ has a zero of order $m$ at $a$. Then we have that $(x-a)^{m-n}$ ...

**1**

vote

**0**answers

277 views

### When integer polynomials take integer values, does their GCD also take integer values?

Suppose you have two polynomials $P$ and $Q$ with integer coefficients. Let their GCD (more specifically, the smallest integer multiple of the GCD in $\mathbb Q$ such that all the coefficients are ...

**2**

votes

**1**answer

258 views

### Why can an infinite-order polynomial be written as the ratio of two finite-order polynomials?

I am studying GARCH processes in Time Series Analysis by Hamilton.
Something that has regularly been used in the book is the assumption that an infinite-order polynomial can be written as the ratio ...

**3**

votes

**3**answers

624 views

### A basis of the symmetric power consisting of powers

I have asked this question on math.se, but did not get an answer - I was quite surprised because I thought that lots of people must have though about this before:
Let $V$ be a complex vector space ...

**6**

votes

**3**answers

420 views

### What is the correspondence between combinatorial problems and the location of the zeroes of polynomials called?

(From MSE)
In the wikipedia article on the Italian-born American mathematician and philosopher Gian-Carlo Rota, it is stated that the one combinatorial idea he would like to be remembered for
...

**0**

votes

**1**answer

114 views

### Ideal membership (concerning polynomial invariants of orthogonal groups)

Let $\mathbb F _q$ be finite field of odd characteristic and consider the polynomials
$$ \xi_i = x_1^{q^i+1} - x_2^{q^i+1} + x_3^{q^i+1} - x_4^{q^i+1} \in \mathbb F_q[x_1,x_2,x_3,x_4].$$
I'm ...

**1**

vote

**0**answers

787 views

### Prime ideals in polynomial rings over integers

Im trying to find a characterization of the prime ideals in the polynomial ring $R = \mathbb Z[X,Y]$ in two variables over the integers.
Actually I need to find the maximal ideals in quotient rings ...

**4**

votes

**0**answers

262 views

### $a^5+b^5=c^5+d^5$ and polynomial identities

No nontrivial integer solutions to $$ a^5+b^5=c^5+d^5 \qquad (1)$$ are known.
(1) has infinitely many solutions in an extension of $\mathbb{Z}$
(root of $9-15x+37x^2 $ ) resulting
from genus 0 curve ...

**2**

votes

**1**answer

362 views

### Is there an algorithm to decide if an ideal contains monomials?

Let $I\subset k[x_1,\dots,x_n]$ be an ideal in a polynomial ring in commuting variables. Is there a procedure to decide if $I$ contains a monomial and possibly to find one?
Gröbner bases come to ...

**26**

votes

**1**answer

796 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 ...