**13**

votes

**1**answer

230 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} ...

**2**

votes

**0**answers

101 views

### A multivariable polynomial degree question

Given $\mathsf{F,G}\in\Bbb R[x_1,\dots,x_n]$, minimum multivariate polynomials of least total degree $\mathsf{degF}$, $\mathsf{degG}$ such that, given unequal $a,b\in\Bbb R$,
$$\mathsf{F(p)}=a, ...

**-2**

votes

**0**answers

58 views

### Standard Deviation of an equation [on hold]

is it possible to derive an equation of standard deviation out of a given function? Is there such method? Let say Log n?
I read the work of Seokkun Chang about the standard deviation the expected ...

**5**

votes

**1**answer

130 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

115 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

145 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

42 views

### compute standard basis in local rings

Let$>'$ be the order in $k[t,x_1,\cdots,x_n]$ as follows:
Each semigroup order > on monomial in the $x_i$ extends to a semigroup order >' on monomial in $t,x_1,\cdots,x_n$ in the following way. We ...

**0**

votes

**0**answers

67 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

73 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 ...

**-1**

votes

**0**answers

15 views

### Compute Faber polynomials in Matlab [closed]

I want to compute some faber polynomials associated to an ellipse centered at a point (u,v) (in the complex plane: u+iv) in Matlab.
Say the ellipse has minor axis a along the x coordinates (real part) ...

**2**

votes

**3**answers

163 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 $f,g$.
...

**12**

votes

**4**answers

684 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

163 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

46 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 ...

**0**

votes

**0**answers

71 views

### Unbounded polynomials [migrated]

Let $p(x)$ be a polynomial of degree $d$ on $R^n$, and let $\tilde{p}(x)$ be the homogeneous components with degree $d$, then how do we prove that:
if $\tilde{p}(x)$ is unbounded below, then $p(x)$ ...

**5**

votes

**2**answers

151 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

43 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 ...

**0**

votes

**0**answers

94 views

### Polynomial approximation on affine varieties [migrated]

Let $V,W \subseteq \mathbb{A}^n$ be two affine varieties over an algebraically closed field $k$ of characteristic zero and let $a,b\in k$.
Q: Can we find a polynomial $f \in k[X_1,...,X_n]$ such ...

**1**

vote

**0**answers

244 views

### Total degree of a polynomial

Let $\mathsf{F,G}\in\Bbb R[x_1,\dots,x_n]$ be minimum multivariate polynomials of least total degree $\mathsf{degF}$, $\mathsf{degG}$ such that, given unequal $a,b\in\Bbb R$,
$$\mathsf{F(p)}=a, ...

**1**

vote

**2**answers

220 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

312 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 ...

**3**

votes

**1**answer

211 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

162 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

110 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

222 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

107 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

258 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

571 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

89 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

68 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

161 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

268 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

79 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

154 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

72 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

109 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

199 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

242 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

102 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

51 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

38 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

137 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

211 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

219 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

578 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

152 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

82 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

64 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

39 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

372 views

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

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 $f_1, f_2, \ldots, f_m$ ...