5
votes
0answers
124 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
2answers
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
2answers
315 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
0answers
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
1answer
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
4answers
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
5answers
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
3answers
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 $f(a)$ ...
8
votes
1answer
356 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
2answers
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
1answer
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
4answers
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
1answer
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
1answer
294 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
1answer
407 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
0answers
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
1answer
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
1answer
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
3answers
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
2answers
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
1answer
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
1answer
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
2answers
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
4answers
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
4answers
899 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
6answers
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
1answer
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
2answers
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
3answers
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
1answer
531 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
11answers
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
4answers
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 ...