**4**

votes

**1**answer

357 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

185 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

173 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 ...

**2**

votes

**1**answer

245 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

234 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

189 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

273 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

240 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

612 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

413 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

113 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

742 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

259 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

356 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

773 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 ...

**3**

votes

**1**answer

385 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 $a_k$ ...

**19**

votes

**1**answer

937 views

### Have we ever proved any non-solvable case of reciprocity without the Langlands program ?

The reciprocity of the title is the following not completely well-posed problem:
Fix $P(X)$ a monic irreducible polynomial of degree $n$, with coefficients in $\mathbb Z$. "Describe"
(in some sense) ...

**2**

votes

**0**answers

39 views

### Put positive polynomial in finite intersection of half-spaces

This is a cross-posting of a MSE question (which did not attract any attention there so far).
Denote by $V={\mathcal P}_{n,d}$ the space of polynomials in $n$ variables with degree at most $d$, ...

**0**

votes

**0**answers

90 views

### Two Different Representations of Multivariate Bernstein Polynomials

In the literature the multivariate Bernstein polynomial of a function $f:[0,1]^m\rightarrow\mathbb{R}$ is often defined as the following:
$$B_{f,n}(x_1,\dots,x_m)=\sum_{\mathbf{k}\in ...

**1**

vote

**0**answers

230 views

### Algebraic Independence of Polynomials in n Variables with Real Coefficients

I am considering the problem of determining the algebraic independence of $n$ polynomials in $m$ variables with real coefficients, where $m \geq n$. The variables will be denoted by $a_{1}, a_{2}, ... ...

**7**

votes

**1**answer

273 views

### Is the evaluation of polynomial functors appropriately continuous?

I'd like a nice proof of the following fact.
Let $C$ and $D$ be categories, and let $\mathbf{Cat}/(C\times D)$ be the usual (1-categorical) slice category whose objects are triples $(X,F\colon X\to ...

**1**

vote

**0**answers

129 views

### Polynomials satisfying a three-term recurrence

Let ${p_n}(x) = x{p_{n - 1}}(x) - {a_{n - 2}}{p_{n - 2}}(x)$ for some numbers ${a_n}$ with initial values ${p_{ - 1}}(x) = 0$ and ${p_0}(x) = 1.$
By Favard’s theorem about orthogonal polynomials ...

**7**

votes

**0**answers

220 views

### When adding a constant makes a multivariate polynomial reducible?

Given a multivariate polynomial $f(x_1,\dots,x_n)$ with integer coefficients, how to find an integer $m$ (if it exists) such that $f(x_1,\dots,x_n) + m$ factors into polynomials of smaller degrees?
...

**9**

votes

**2**answers

594 views

### When polynomial f(x^2) can be factored as g(x)·g(-x) ?

In relation to my question Expression for the sum of square roots of zeros of a polynomial
How to characterize polynomials $f(x)$ with rational coefficients such that $f(x^2)=g(x)\cdot g(-x)$ where ...

**1**

vote

**2**answers

190 views

### Transformation of a bivariate polynomial into a homogeneous one

For a given a bivariate polynomial $P(x,y)$ with rational coefficients:
Q1. How compute such (invertible) substitutions of its variables that would transform the polynomial into a homogeneous one? In ...

**6**

votes

**2**answers

433 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 ...

**1**

vote

**1**answer

230 views

### Homogenous polynomials as sum or differences of squares and symmetric polynomials

I seem to recall that a general homogenous real polynomial $P$ of even degree in $n$ variables, $n\geq 3,$
cannot always be expressed as $P(x_1,\dotsc,x_n)=\sum_j a_j Q_j^2(x_1,\dotsc,x_n)$ where $a_j ...

**8**

votes

**1**answer

499 views

### Polynomial with all zeros on a circle and many real coefficients

On a circle (or a line) $\Omega$ in the complex plane that is not symmetric w.r.t. the real axis, choose $n\ge5$ distinct points $z_1,...,z_n$ and consider the polynomial ...

**1**

vote

**2**answers

215 views

### Reducing system of polynomials with symbolic factors

Getting nowhere with maple using its triangularize and groebner decompositions for even moderate size systems with any symbolic factors. Any suggestions on how better to approach this would be ...

**11**

votes

**0**answers

156 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, ...

**4**

votes

**1**answer

343 views

### Inequality on Trigonometric polynomials

My question comes from trying to understand a technical step in this paper by Bourgain.
Let $R,L$ be positive integers and let $f(x)=\sum_{|n|\leq RL}a_ne^{2\pi inx}$ be a trigonometric polynomial. ...

**5**

votes

**1**answer

224 views

### A “known” Pythagorean identity in algebra?

Some will recognize this as similar to a question I asked before, but
I want to ask it without the trigonometry.
Let $e_k$ be the $k$th-degree elementary symmetric polynomial in
$x_1,x_2,x_3,\ldots$. ...

**0**

votes

**2**answers

469 views

### On reducible polynomials with positive coefficients, $1$ as constant coefficient and certain bounds on coefficients

Given $a \in \mathbb{Z}$ with $a > 1$. Let $g(x) \in \mathbb{Z}[x]$ be a polynomial with $g(a)=\pm 1$. Let $h(x) \in \mathbb{Z}[x]$ be a polynomial with $h(a)= p$, a prime. Let $g(x)$ and $h(x)$ ...

**2**

votes

**3**answers

502 views

### Decomposing irreducible polynomials with a prescribed condition - Existence

Let $f(x)$ be an irreducible polynomial in $\mathbb{Z}[x]$ or $\mathbb{F}_{q}[x]$ with $deg(f(x)) \ge 2$ (assume constant coefficient is $1$).
Let $a \in \mathbb{Z}$ or $\mathbb{F}_{q}$. Let $f(a)$ ...

**6**

votes

**0**answers

355 views

### How many values a polynomial map misses?

Let $F$ be a field. For a uni-variate polynomial $f(x)$ over $F$,let $M_f(F)$ denote the number of values that $f$ misses, that is, the cardinality of the subset $F - f(F)$ in $F$. Assume that $f$ is ...

**2**

votes

**3**answers

2k views

### Intersection of Cones in Three Space

In several branches of applied mathematics the problem arises to describe the intersection of two cones in three space.
I have searched and found a few references that discuss the problem for cones ...

**3**

votes

**1**answer

128 views

### Decision problem wrt Pairs of Polynomials with Integer Coefficients

Given two arbitrary polynomials $G(x)$ and $H(y)$, with integer coefficents, are there any circumstances in which it is possible to decide whether or not $G(x) = H(y)$ has solutions with $x, y \in ...

**0**

votes

**1**answer

93 views

### Probability of summing products of irreducible polynomials in a finite field to zero

Let $f(x), g(x), h(x)$ be randomly chosen irreducible polynomials over the finite field $GF(2^n)$.
What would the probability be for $\sum_{(i,j,k:i,j,k\in\mathbb{N},i+j+k=C)} f^i(x)g^j(x)h^k(x)=0$, ...

**1**

vote

**0**answers

338 views

### Properties of a rational function of multiple variables

Suppose you are given a multivariable rational function f(x0,x1,x2,x3,..,xn), so the only four operation are +,-,*,/.
Assume that all constants and exponents are integers within certain range.
I ...

**11**

votes

**0**answers

518 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 it really so?
As far as I know, it is open problem ...

**0**

votes

**1**answer

161 views

### minimal spans of polynomial companions of co-prime polynomials.

Is there an algorithm to determine for given $P,Q$ in $\mathbb Z[x,x^{-1}]$ with $gcd(P,Q)=1$, the value of $min\lbrace Span(A)+Span(B): A,B\in \mathbb Z[x,x^{-1}],\ A\cdot P+B\cdot Q=1\rbrace$, where ...

**7**

votes

**1**answer

354 views

### Constructive proof of “Projective implies proper”

For every ring $A$, the structural morphism of schemes $\pi_A : {\bf P}^n_{A} \to {\rm Spec}{A}$ is a closed map. The usual proof of this fact is not constructive : given equations of a closed subset ...

**1**

vote

**0**answers

346 views

### What is $f(x)$ if $f^{-1}(x)=\frac{\ln(x+1)}{\ln x}$? [closed]

I asked this on MSE but no one could get it. http://math.stackexchange.com/questions/257455/inverse-function-of-y-frac-lnx1-ln-x It's been bothering me for a really long time. I think it's equivalent ...

**17**

votes

**0**answers

433 views

### What is the best probabilistic estimate from below for a random polynomial on an arc?

I'm currently involved in a small (but quite time consuming) project where we are trying to get some decent bound for the number $N(P)$ of real zeroes of a random polynomial $P(x)=\sum_{k=0}^n\xi_k ...

**0**

votes

**1**answer

274 views

### Polynomial version of the conjecture about Power free-values of polynomials

The conjecture about Power free-values of polynomials
is:
Let $F(X)$ be a polynomial with integer coefficients and no repeated roots. For any $\epsilon > 0$, there exists a constant ...

**1**

vote

**1**answer

282 views

### Combinatorics: Product Rules.

I couldn't find a way to figure this out, though it is a somewhat basic question that came up when studying the stationary phase expansion of an integral. The abstract version is the following:
I ...

**0**

votes

**1**answer

334 views

### transform a polynomial into another one upto a constant

I have a polynomial $p(x)=a_Nx^N+a_{N-1}x^{N-1}+\dots+a_0$. I want to convert this into another polynomial of same order, say $b_Ny^N+b_{N-1}y^{N-1}+\dots+b_0$. Is it possible to find a transformation ...

**4**

votes

**5**answers

413 views

### Spectrum of transition matrix for symmetric random walk

I asked this question previously on math.stackexchange.com, where it had little traction.
Consider the symmetric random walk on $\{0,1,…,n\}$ with transition probabilities $P(j→j±1)=1/2$ for $0 ...

**3**

votes

**2**answers

223 views

### Hausdorff dimension of Julia sets of quadratics not in the Mandelbrot set.

What are the bounds on the possible values of the Hausdorff dimension of the Julia sets of quadratics not in the Mandelbrot set? In particular, assume we have a quadratic $q_c: z \mapsto z^2 + c$ on ...

**8**

votes

**3**answers

443 views

### Relating a Polynomial equation to the characteristic equation of a Hermitian matrix

This question arose out of mere curiosity. Given a polynomial equation and I happen to know that its roots are real (but not the roots itself). Does it mean it is the characteristic equation of a ...