**1**

vote

**1**answer

218 views

### Composition of rational functions

Given a rational function $R\in\Bbb R(x_1,\dots,x_n)$ with multilinear numerator and denominator, is there always a rational function $G\in\Bbb R(x)$ such that $G\circ R\in\Bbb R[x_1,\dots,x_n]$, ...

**2**

votes

**0**answers

108 views

### What are the general zonal spherical functions for ${\rm SO}(n)/{\rm SO}(n-1)$?

The zonal spherical functions [1] on the sphere $(G={\rm SO}(n)$, $K={\rm SO}(n-1))$ are the Gegenbauer or ultraspherical polynomials if one considers the irreducible representations of ${\rm SO}(n)$ ...

**5**

votes

**0**answers

163 views

### Degree of Chebyshev polynomial necessary

In general, given $0<a<1$, I want to find a polynomial $f(x)\in\Bbb R[x]$ such that $f(x)\in[1-\frac{a}2,1+\frac{a}2]$ at every $x\in[1-a,1+a]$ and $f(0)=0$. What is minimum degree that is ...

**0**

votes

**1**answer

289 views

### cup-length of the first Chern class of complex grassmannian

Let $G_2(\mathbb{C}^{n+1})$ be the complex grassmannian.
Then the cohomology ring $H^*(G_2(\mathbb{C}^{n+1});\mathbb{C})=\mathbb{C}[c_1,c_2]/(f_n,f_{n+1})$, where ...

**8**

votes

**3**answers

474 views

### Degree necessary of a polynomial?

Given $-1<a<b<0$, I want to find a polynomial $f(x)\in\Bbb R[x]$ such that $f(x)\in[a,b]$ at every $x\in[b^2,a^2]$ and $f(0)=0$. What is minimum degree that is needed and maximum degree that ...

**1**

vote

**1**answer

168 views

### a problem about ideals of polynomial rings

Let $\{f_n\}_{n=1}^\infty\in \mathbb{C}[x,y]$ be a sequence of polynomials given by the following expressions
$$
f_n(x,y)=\sum_{i=0}^{[\dfrac{n}{2}]}(-1)^{n-i}{{n-i}\choose i}x^{n-2i}y^i.
$$
Let ...

**13**

votes

**1**answer

386 views

### Positive roots of a polynomial

Let $a_i>0$, $i=1,\dots,n$, and put $\overline{a}:=\frac{1}{n}\sum_{i=1}^n a_i$. Assuming not all $a_i$'s are equal, take
$$
p(x):=\sum_{i=1}^n a_i (a_i-\overline{a})\prod_{k=1,\dots,n\;k\neq i} ...

**9**

votes

**3**answers

352 views

### Computing minimal polynomials using LLL

I would like to use the Lenstra–Lenstra–Lovász lattice basis reduction algorithm (LLL algorithm) to compute the minimal polynomial of a (real) algebraic number $\alpha$ from a decimal approximation ...

**0**

votes

**2**answers

117 views

### Irreducibles in polynomial rings

Let R be a reduced ring with characteristic zero which is not an integral domain. Is "x" necessarily non irreducible in R[x]?

**2**

votes

**1**answer

181 views

### Smallest degree of approximating polynomial

Let $\{0,1\}^n=S_0\cup S_1$ withh $S_0\cap S_1=\emptyset$.
Let $\epsilon\in[\frac{1}2,1)$.
Let $f:\Bbb R^n\rightarrow\Bbb R$ be a polynomial such that $$f(S_0)=0,\mbox{ ...

**0**

votes

**0**answers

75 views

### Factoring a polynomial in a specific manner

Let $f(x,y,z) = ax^d + b y^k z^{d-k} + c$, where $a,b,c \in \mathbb{C}$ and $abc \ne 0$, and $d \geq 3$. Let $d'$ denote the smallest non-negative integer $l$ such that there does not exist a positive ...

**1**

vote

**0**answers

88 views

### Generalization of Schur polynomials

I am making a list of generalizations of Schur polynomials and other closely related polynomials that appear in representation theory. My motivation is to eventually make a nice poster of ...

**3**

votes

**3**answers

211 views

### ideals of polynomial ring with complex number coefficients

Let $\mathbb{C}[x,y]$ be the polynomial ring with variables $x,y$ and coefficient in $\mathbb{C}$.
Let $f,g\in \mathbb{C}[x,y]$.
Let $(f,g)$ be the ideal of $\mathbb{C}[x,y]$ generated by $f,g$.
...

**13**

votes

**4**answers

776 views

### Random Diophantine polynomials: Percent solvable?

Suppose one generates a random polynomial
of degree $d$ with integer coefficients
uniformly distributed within $[-c_\max,c_\max]$.
For example, for
$d=8$, $|c_\max|=100$, here is one random ...

**4**

votes

**1**answer

174 views

### Multivariable function analysis

Sorry if the terms I'm going to use is not professional enough:) This is about the complexity analysis of an algorithm.
Let $\alpha$ be the greatest real root of the polynomial ...

**1**

vote

**1**answer

66 views

### ideals of polynomial ring of two variables generated by two elements

Let $f,g$ be two polynomials in $\mathbb{Z}[x,y]$, given by
$$
f(x,y)=x^4-3xy+y^2,$$
$$
g(x,y)=x^5-4xy+3xy^2.$$
Let $I=(f,g)$ be the ideal in $\mathbb{Z}[x,y]$ generated by $f$ and $g$.
Is ...

**5**

votes

**2**answers

189 views

### Common roots of polynomial and its derivative

Suppose $f$ is a uni-variate polynomial of degree at most $2k-1$ for some integer $k\geq1$. Let $f^{(m)}$ denote the $m$-th derivative of $f$. If $f$ and $f^{(m)}$ have $k$ distinct common roots ...

**1**

vote

**1**answer

49 views

### local bernstein type inequality for multivariate polynomials

Let's say $p(x_1,...,x_n)$ is an n-variate degree d homogenous polynomial. Assume $U \subset S^{n-1}$ and $ vol(U) > 0 $ is there any Bernstein type inequality saying
$$ \max_{x \in U , y \in ...

**1**

vote

**2**answers

250 views

### Ideals generated by two elements in the polynomial ring of two variables over a field

Let $k$ be a field. For example, $k=\mathbb{Q}$ or $\mathbb{Z}/p$, $p$ prime.
Let $k[x,y]$ be the polynomial ring.
Let $f,g\in k[x,y]$.
Let the ideal $I=(f,g)$ be the ideal of $k[x,y]$ ...

**3**

votes

**4**answers

329 views

### Examples of polynomials $x_1(t), x_2(t)$ such that $(x_1(t))^4 + (x_2(t))^4$ has a double root

Are there examples of polynomials $x_1(t), x_2(t) \in \mathbb{Q}[t]$ of equal degree at least one, with $\gcd(x_1(t), x_2(t)) = 1$, such that the sum $(x_1(t))^4 + (x_2(t))^4$ is divisible by the ...

**9**

votes

**3**answers

634 views

### $S_k(x+y)-S_k(x)-S_k(y)$ where $S_k$ is symmetric polynomial

Let $S_k$ be the $k$-th elementary symmetric polynomial of $n$ variables. How can I rewrite $$S_k(x+y)-S_k(x)-S_k(y)$$ by just using $x,y,S_1,S_2,\cdots S_{k-1}$ where $x=(x_1,x_2,\cdots,x_n)$ and ...

**3**

votes

**0**answers

170 views

### How to handle a polynomial whose roots exhibit obvious symmetry

I've got a sequence of polynomials, and for each of them the roots obviously follow a definite pattern. Here are the roots of the 34th one
All others have their roots arranged in a similar ...

**1**

vote

**1**answer

125 views

### Zeros of Polynomial with decreasing coefficients [closed]

If $n_1<n_2<n_3\cdots<n_m$ are positive integers. Does the polynomial $a_0+a_1z^{n_1}+a_2z^{n_2}+\cdots+a_mz^{n_m}$ satisfying
$$ 0<a_0\leq a_1\leq \cdots\leq a_m $$ has all its zeros ...

**3**

votes

**2**answers

239 views

### Proof that image of a polynomial map is a cone

Consider the nonlinear mapping $\phi: \mathbb R^{2 \times 2} \to \mathbb R^3$ given by $X \mapsto \begin{pmatrix} x_{11} x_{21} \\ x_{11} x_{22} + x_{21} x_{12} \\ x_{12}x_{22} \end{pmatrix}$.
I ...

**2**

votes

**2**answers

127 views

### Integrating a barycentric monomial over a simplex

Are there standard formulas for the integral over a simplex of a monomial in the barycentric coordinates? Can someone supply a reference? I think I have seen such formulas, but I am unable to find ...

**1**

vote

**0**answers

268 views

### Generalization of proposition of Granville related to abc conjecture

Related to this question.
For polynomial $f$, let $rad(f)$ denote the radical of $f$,
the product of irreducible factors.
Suppose that $G(x,y) \in \mathbb{C}[x,y]$ is homogeneous
without any ...

**6**

votes

**2**answers

613 views

### Counterexample to Proposition of Granville related to abc conjecture

Looks like there is counterexample to Proposition related to
abc conjecture. Confusion is likely.
From RATIONAL AND INTEGRAL POINTS ON QUADRATIC TWISTS OF A GIVEN HYPERELLIPTIC CURVE, Andrew ...

**3**

votes

**0**answers

99 views

### Small polynomial roots modulo prime powers

Let $f(x) \in \mathbb{Z}[x]$ have no rational roots,
$n>1,q=p^n$ for prime $p$.
Let $r$ be the smallest root of $f(x)=0$ modulo $q$.
Smallest mean smallest absolute value when lifted
to the ...

**0**

votes

**0**answers

91 views

### Polynomial generated with primitive element modulo p

This question is equivalent to the question "Normal basis in cyclotomic number fields" that I asked recently. I am posing this question because maybe in this format somebody can have an answer:
Let ...

**1**

vote

**2**answers

195 views

### Algorithm for fast factorization of polynomial over $\mathbb Z$ or over $\mathbb F_p$

I want to fast decompose polynomial over ring of integers (original polynomial has integer coefficients and all of factors have integer coefficients) and also over ring of integers modulo prime ...

**4**

votes

**2**answers

288 views

### Find all possible rational values of a parametric quartic such that it is reducible

Description: Given the following parametric quartic polynomial
$y^4 - 28 z y^3 - 14 (656 - 328 z + 83 z^2) y^2 +
4 z (-20464 + 10232 z + 3409 z^2) y +
91 (62208 - 62208 z + 41504 z^2 - 12976 z^3 ...

**2**

votes

**2**answers

94 views

### Moment problem for discrete distributions

Let $x_1, \dots, x_N \in \mathbb R$ and consider the discrete distribution $\mu := \frac{1}{N} \sum_{i=1}^N \delta_{x_i}$, where $\delta_x$ denotes the Dirac measure, i.e. for any measurable set $B ...

**4**

votes

**1**answer

187 views

### Represent matrix immanants using Schur functions

For each irreducible character $\chi^\lambda$ of the symmetric group $S_n$, the immanant of an $n\times n$ square matrix $A$ is defined as
\begin{equation*}
d_\lambda(A) := \sum_{\sigma \in S_n} ...

**5**

votes

**0**answers

83 views

### Real Zeros - tail estimate

Given a random polynomial with Gaussian coefficients, the Kac-Rice formula tells us what the expected number of real zeros is (for more on this, see the excellent paper of Edelman and Kostlan in the ...

**2**

votes

**1**answer

116 views

### Prove or disprove an inequality concerning zeros of a polynomial

If a polynomial $p(z)$ of degree $n$ with zeros $z_1,z_2,\cdots,z_n$ assumes maximum at $w$ on $|z|=1.$ Prove or disprove that the Harmonic mean of $|z_k-w|,$ $k=1,2,\cdots,n$ is greater or equal to ...

**2**

votes

**1**answer

210 views

### If a polynomial $p(z)$ omits a value, then $p(z)-\dfrac{(1-e^{i\psi})}{n}zp^{\prime}(z)$ also omits that value

Suppose that a polynomial $p(z)$ of degree $n$ does not assume the value $w$ for $|z|<1$, that is $p(z)\neq w$ for $|z|<1.$ Show that $p(z)-\dfrac{(1-e^{i\psi})}{n}zp^{\prime}(z)\neq w$ for ...

**8**

votes

**2**answers

255 views

### Root criterion for polynomial over number fields

It's well known that if $\alpha $ is a rational root to an integer coefficient polynomial, then its denominator divides the leading coefficient and its numerator divides the constant term. I'm asking ...

**0**

votes

**0**answers

132 views

### About distinct eigenvalues of a graph

if a graph with adjacency matrix $A$ and Laplacian $L$ has $k$ distinct eigenvalues then does this fact naturally help define or prove existence of a polynomial $p$ of degree $k-1$ such that ...

**2**

votes

**1**answer

62 views

### Reference for multivariate orthogonal polynomials

I want to learn about multivariate orthogonal polynomials. Is there a good textbook/survey that you could suggest? I need to see common examples like Jack's polynomials etc .. and also general ...

**0**

votes

**0**answers

42 views

### t-linked extension

Let $A\subseteq B$ be an extension of commutative integral domains. the extension is t-linked if it satisfies the following property:
If P is a finitely generated ideal of A such that $P^{-1}=A$ than ...

**1**

vote

**1**answer

139 views

### polynomials with similar maxima-minima

Assume $p$ and $q$ are n-variate degree d, homogeneous polynomials. Define $ |p|_{\infty}= \max_{x \in S^{n-1}} |p(x)|$
$D(p)=\{ x \in S^{n-1} : p(x)= |p|_{\infty} \}$
$E(p)=\{x \in S^{n-1} : p(x)= ...

**0**

votes

**1**answer

219 views

### Find all possible rational values of the parameter of a parametric cubic such that it is reducible

Description: Given the following parametric cubic polynomials ${E}^{3}
- 15\, {\beta}_{\pm}\, {E}^{2} - 3 \left({71\, {\beta}_{\pm}^{2} + 352\, {\beta}_{\mp}}\right) E
+ 135\, {\beta}_{\pm} ...

**4**

votes

**1**answer

242 views

### Permutation polynomials mod $p$ of the form $(x+1)^n-x^n$

Among some permutation polynomials I've been studying, $f(x)=(x+1)^n-x^n$ is one of the polynomials that I cannot grasp.
Question : For a given odd prime $p$, how can we find every positive ...

**8**

votes

**2**answers

591 views

### Sum of consecutive cubes

I'm investigating when the sum of $n$ consecutive cubes equals a cube, i.e., for which $n$ does
$$\sum_{i=0}^{n-1} (k+i)^3 = k^3 + (k+1)^3 + \cdots + (k+n-1)^3 = Y^3 $$
have nontrivial solutions ...

**3**

votes

**1**answer

163 views

### Periodic points in C^2

I came up with a problem which is similar to the following quesitons:
Consider a map: $f(x,y)=(y^2-2,xy-2)$. It is seems that the number of periodic points of given period is bounded.
I want to ...

**6**

votes

**0**answers

88 views

### Is there a ``Ladner's Theorem" for the PH-vs-PSPACE scenario?

Like a statement of the kind, ``If the Polynomial Hierarchy (PH) $\neq$ PSPACE then there exists $L \in PSPACE \backslash PH$ which is not PSPACE-complete"?
Or is there something else that states ...

**0**

votes

**1**answer

69 views

### On a reciprocal of Ostrowski theorem on Newton polytopes and factorization

$\newcommand\KK{\mathbb{K}}$Let $\KK$ be any field and $f\in\KK[x_1,\dotsc,x_n]$ be a polynomial. Its support $S_f$ is the set $\{(e_1,\dotsc, e_n) : x_1^{e_1}\dotsb x_n^{e_n}$ has a nonzero ...

**1**

vote

**1**answer

53 views

### About convex combinations of real-stable multivariable complex polynomials

Say $f: \mathbb{C}^{n+1} \rightarrow \mathbb{C}$ is a real stable multivariable polynomial on the variables $(z,w_1,w_2,...,w_n)$. (a "real-stable" polynomial is one which has no zeroes in the open ...

**6**

votes

**0**answers

405 views

### Find a polynomial not in any ideal generated by polynomials of total degree $o(n)$

Is there an explicit nontrivial (= not a constant) polynomial $p \in \mathbb{C}[x_1, \ldots, x_n]$ such that, for any ideal $I \not= \mathbb{C}[x_1, \ldots, x_n]$ generated by $f_1, f_2, \ldots, f_m$ ...

**1**

vote

**1**answer

81 views

### About preserving real-rootedness of multivariable polynomials

Say $f_i(z_1,z_2,..,z_m)$ are polynomials real rooted in the $z$s for a bunch of polynomials indexed by $i$. When can one say that $\sum_{i} p_i f_i(z_1,z_2,..,z_m)$ is also real rooted?
If ...