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Tagged Questions

0
votes
2answers
157 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: b …
18
votes
5answers
504 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 fo …
12
votes
2answers
695 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 ch …
7
votes
4answers
332 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 $ …
1
vote
1answer
130 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 …
2
votes
0answers
115 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: Con …
5
votes
1answer
343 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 $k$ …
1
vote
0answers
42 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) = …
-2
votes
0answers
89 views

Shift a polynomial function

Let $f$ be a polynomial function on [0,1]. Let $g$ the function obtained by making a periodic shift of $f$ : $$g(s)=1_{\alpha \leq s \leq 1}(s)f(s-\alpha)+1_{0 \leq s < \alpha} …
1
vote
1answer
255 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 sev …
3
votes
1answer
220 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 $ …
0
votes
1answer
152 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 th …
3
votes
0answers
141 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,\ldo …
3
votes
2answers
408 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 com …
2
votes
1answer
84 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 unifo …

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