# Tagged Questions

**5**

votes

**0**answers

123 views

### Operator connected with Hermite polynomials

For $n \geq 1$, define the following operator $M_n$ on the ring of all polynomials with real coefficients.
$$M_n P(x) = nP(x)^2 - x \int_0^x (P'(t))^2 \, \mathrm{d}t$$
Monomials $x^k$ are mapped to $n ...

**3**

votes

**2**answers

323 views

### Degrees of factors of polynomial $f(x)=x^q-(ax^2+bx+c)\in \mathbb{F}_q[x]$

I’m facing the problem of factoring polynomials of type $f(x)=x^q-(ax^2+bx+c)\in \mathbb{F}_q[x]$ and the degrees of factors seem to be quite special. For example, according to my experimental results ...

**3**

votes

**2**answers

314 views

### noncommutative polynomials equality

Suppose $x$, $y$, $z$ are three variables satisfying $yz=zy$, $zx=xz$, $xy=yzx$.
Could anyone give me two (non-commutative) polynomials $f$ and $g$ in the above three variables such that the ...

**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 ...

**2**

votes

**1**answer

99 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 ...

**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 ...

**18**

votes

**5**answers

651 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:
...

**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 $f(a)$ ...

**7**

votes

**1**answer

355 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 ...

**0**

votes

**2**answers

163 views

### power sums are enough for rationality? [closed]

If I have k algebraic integers like a_1, ..., a_k such that the sum of their n-power are integer for n=1, ...m
can we deduce that a_1, ..., a_k are integers? how large m should be? (how many power ...

**3**

votes

**1**answer

480 views

### When does a polynomial split over Q?

If P(x) is a polynomial in Q[X], is there any iff theorem that states all the roots of P(x) are rational based on the coefficients?!
In another words, what could you impose on the coefficients to ...

**12**

votes

**4**answers

770 views

### Showing that a family of polynomials has positive and real roots.

Hi everybody, for my research I am dealing with the following function:
$$\alpha_n(x):=\left.\frac{\partial^{2n+1}}{\partial z^{2n+1}}\frac{\sinh(z)}{\cosh(z)-1+x}\right|_{z=0},\quad n\in ...

**3**

votes

**1**answer

239 views

### Special polynomials over finite fields

My field of research is coding theory and I am working on cyclic codes. During my research, I tackled an algebraic problem. After some simple definitions, I asked my question. I will appreciate any ...

**0**

votes

**1**answer

292 views

### Do similar properties hold for fractional polynomials?

It is well-known that the number of zeros of a polynomial $P_n(z)$ of degree $n$ is precisely $n$ and $P_n(z)$ can be represented in the form $$
P_n(z)=a_n\prod_{i=1}^n(z-z_i),
$$
where $a_n$ is the ...

**12**

votes

**1**answer

406 views

### Is the ring of quaternionic polynomials factorial?

Denote by $\mathbb{H}[x_1,\dots,x_n]$ the ring of polynomials in $n$ variables with quaternionic coefficients, where the variables commute with each other and with the coefficients. Two polynomials ...

**0**

votes

**0**answers

123 views

### Expressing a polynomal as products of shifted Riemann zeta functions and reciprocals.

Suppose I have a polynomial $f$ in $p,t,$ say $f=1 \pm p^{a_1}t^{b_1} \pm \cdots \pm p^{a_s}t^{b_s}$ with $a_i,b_i \in \mathbb{N}\cup0$ and $b_i$ is non-zero. Let $X:=${$ \ (1-p^it^j) | i,j \in ...

**0**

votes

**1**answer

190 views

### How to consider a module over the ring Q[t,t^(-1)] to be a module over the polynomial ring Q[t]? [closed]

Can we view a module over the ring $\mathbb{Q}[t,t^{-1}]$ to be a module over the polynomial ring $\mathbb{Q}[t]$?
where $\mathbb{Q}$ denote any rational number coefficients.

**0**

votes

**1**answer

511 views

### Question about modules, quotient rings, and polynomial rings? [closed]

Consider an integer polynomial ring, $A = \mathbb{Z}[t]$, and a ring of fractions, $B = \mathbb{Z}[t, t^{-1}]$; obviously, $A$ is a subring of $B$.
Now we consider two modules over $A$ and $B$, $M$ ...

**24**

votes

**3**answers

2k views

### when is the power of a nonnegative polynomial a sum of squares?

There are polynomials that are not sum of squares. For example Motzkin gave the example $x^4y^2+x^2y^4+z^6-3x^2y^2z^2$ in 1967.
Is there a real polynomial $f\in{\mathbb{R}}[x_1,\ldots,x_n]$ in ...

**3**

votes

**2**answers

244 views

### $n$-forms representing zero (versus division rings)

Let me start with two observations.
In the classification of quadratic forms with rational coefficients, one has the following statement: a quadratic form in five indeterminates represents $0$ over ...

**7**

votes

**1**answer

320 views

### Is this Negativstellensatz with uniform denominators known?

A theorem of Reznick states that if $f>0$ is a real homogeneous polynomial in several polynomials that is positive away from the origin of ${\mathbb{R}}^n$, then for large $N$, the form $(\sum ...

**2**

votes

**1**answer

412 views

### Is every polynomial a limit of polynomials in quadratic variables?

Let $d>0$ be even. Consider ${\mathbb{R}}[x_1,\ldots, x_n]_d$, i.e. polynomials of degree $d$.
Call a homogeneous polynomial $f$ of degree $d$ a polynomial in quadratic variables if it is of the ...

**2**

votes

**2**answers

374 views

### on existence of matrices X, Y s.t. XAY is diagonal over non-commutative ring

Given $A\in Mat_{n\times n}(R)$ where $R$ is a non-commutative associative ring are there exist any (non-zero) matrices $X, Y\in Mat_{n\times n}(R)$ such that $XAY=diag(a_1, \ldots , a_n)$ for some ...

**17**

votes

**4**answers

913 views

### Why are polynomials easier to handle with than integers?

This may seems to be an elementary question, but I found no answers on MO nor google.
I have always heard "polynomials are easier to handle with than integers". For example:
When $n$ is quite ...

**4**

votes

**4**answers

896 views

### Are all parametrizations via polynomials algebraic varieties?

Suppose that we have a parametrization via polynomials as follows:
$$t\longrightarrow (f_1(t),\ldots,f_n(t)),$$
where $t$ is a vector in $\mathbb{C}^r$ and $f_i$ are polynomials of arbitrary degree.
...

**9**

votes

**6**answers

1k views

### What's an example of a transcendental power series?

Let $k$ be a field. What is an explicit power series $f \in k[[t]]$ that is transcendental over $k[t]$?
I am looking for elementary example (so there should be a proof of transcendence that does ...

**1**

vote

**1**answer

2k views

### When are the units of R[x] exactly the units of R?

I (Anton) have edited this question to be the question Pete and Zeb discuss in the first few comments.
What conditions on a ring $R$ imply that the units of $R[x]$ are exactly the units of $R$?

**9**

votes

**2**answers

1k views

### Galois group of a product of polynomials

How can I compute the Galois group of the polynomial $fg\in K[x]$ assuming that I know the Galois groups of $f\in K[x]$ and $g\in K[x]$? Let's suppose for simplicity that the field $K$ is perfect.

**6**

votes

**3**answers

1k views

### Rational exponential expressions

Consider the following extension of polynomials. The rational exponential expressions (REXes) are given by:
The leaves 1 and $x$ for $x$ drawn from a class of variables; and
Closed under the binary ...

**8**

votes

**1**answer

524 views

### what was Hilbert's geometric construction in his 17th problem?

Hilbert's 17th problem asked if a nonnegative real polynomial is the sum of squares of rational functions. It was answered affirmative by Artin in around 1920. However, in his speech, he also asked if ...

**64**

votes

**11**answers

3k views

### Can a non-surjective polynomial map from an infinite field to itself miss only finitely many points?

Is there an infinite field $k$ together with a polynomial $f \in k[x]$ such that the associated map $f \colon k \to k$ is not surjective but misses only finitely many elements in $k$ (i.e. only ...

**4**

votes

**4**answers

2k views

### Are quotients of polynomial rings almost UFDs?

If K is a field then the polynomial ring K[x1,..,xn] is a UFD. On the other hand, quotients of such a polynomial ring usually don't enjoy unique factorization: consider, for instance, R[x,y] modulo ...