1
vote
2answers
90 views

Factorisation of a biquadratic polynomial

Let $u,v\in\mathbb{Z},$ and let $f=X^4+uX^2+v.$ Let $p$ be a prime number, and let $r\geq 1.$ In a paper I'm reading, one can find the following result. $\bf{Proposition. }$If $f$ is reducible ...
0
votes
0answers
51 views

Multivariable irreducible polynomials over finite fields [migrated]

It is not difficult to prove the following result, and it seems that it should be already proved. I would appreciate it if someone offer me some reference to it. For any $f(x_1,\dots, x_n)=\sum ...
0
votes
1answer
127 views

Irreducibility of a resultant of real and imaginary parts of a characteristic polynomial

The following question is motivated by the study of a stability border for a robust linear time-invariant control system. Let us we have an affine family of $n\times n$ matrices with indeterminate ...
0
votes
0answers
37 views

What is maximum number of m-complex solutions to a order n polynomial (say with real coefficients)?

I know the answer is n^2 for bicomplex numbers. Does anyone know if a general answer has been found for m-complex numbers ( http://en.wikipedia.org/wiki/Multicomplex_number)?
8
votes
1answer
229 views

Irreducibility of a class of polynomials

This question is directly inspired by this question. Consider polynomials of the form $$p(x) = \prod_{i=1}^n(x-i)^2 - d.$$ For which values of $n$ and $d$ is $p(x)$ irreducible? There is a theorem of ...
4
votes
2answers
391 views

Characterizing $\mathbb{Q}[X]$ via a property of its tensor powers

Let $\varphi: \mathbb{Q}[X] \longrightarrow R$ an inclusion of commutative rings. Suppose that the map $$- \circ \varphi: \operatorname{Hom}_{\mathbb{Q}\operatorname{-alg}}(R, R^{\otimes_{\mathbb{Q}} ...
6
votes
0answers
77 views

Irreducibility testing and factoring

It is a result of van Hoeij and Novicin (Algorithmica, 2012) that factoring polynomials of degree $d$ over the integers can be done in $O(d^6 + d^4 \log^2 A)$ time, where $A$ is the coefficient bound. ...
0
votes
0answers
127 views

Image of critical points

Let $K$ be a field of characteristic $0$, $f:K^n\rightarrow K^n$ be an algebraic function, that is, $n$ polynomial functions in $n$ variables. Let $S$ be the set of critical points of $f$. If ...
15
votes
1answer
547 views

When is $f(x_1, \dots, x_n)+c$ an irreducible polynomial for almost all constants $c$?

Let $f\in \mathbb C[x_1, \dots, x_n]$, $n\ge 1$, be a non-constant polynomial. Consider the polynomial $f+t\in \mathbb C[t, x_1,\dots, x_n]$. This is an irreducible polynomial in $\mathbb C(t)[x_1, ...
8
votes
0answers
440 views

Getting a bound via polynomial equations

When studying the existence problem of power residue deference sets, I came across the following system of polynomial equations over $\mathbb{C}$, \begin{cases} ...
2
votes
1answer
286 views

About the practice of Bernstein-Kushnirenko theorem

The following refers to common roots of bivariate polynomial equations and, in particular to the quim's and auniket's comments. The BKK theorem (cf. arXiv:0812.4688. Theorem 5.4) asserts that if we ...
7
votes
2answers
407 views

Dimension of a homogeneous polynomial system

Let $m\geq4$ be an even integer, $V\subset\mathbb{C}^{m-1}$ be the solution set of the following polynomial equations: \begin{cases} ...
5
votes
1answer
268 views

Is an ideal generated by multilinear, irreducible, homogeneous polynomials of different degrees always radical?

I asked this question on math.se and someone even put a bounty on it, yet there was no answer. Hence, I am asking here. Assume $\Bbbk$ to be a field of characteristic zero. Definition. A ...
4
votes
1answer
295 views

On composition of polynomials

Given two irreducible polynomials $f_{u}(x),f_{r}(x) \in \Bbb Q[x]$, can one find two polynomials or rational functions $h_{u}(x),h_{r}(x) \in \Bbb Q[x]$ or $\Bbb Q(x)$ respectively such ...
6
votes
3answers
391 views

Idempotent polynomials

Let $R$ be a commutative ring with identity and let $f \in R[x]$. There are well known characterizations for $f$ to be a nilpotent element of $R[x]$ or to have a multiplicative inverse in $R[x]$. Is ...
27
votes
3answers
1k views

Are surjectivity and injectivity of polynomial functions from $\mathbb{Q}^n$ to $\mathbb{Q}$ algorithmically decidable?

Is there an algorithm which, given a polynomial $f \in \mathbb{Q}[x_1, \dots, x_n]$, decides whether the mapping $f: \mathbb{Q}^n \rightarrow \mathbb{Q}$ is surjective, respectively, injective? -- And ...
2
votes
1answer
96 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 ...
2
votes
1answer
238 views

Is there an irreducible integral polynomial in two variables which is reducible for every value of one of the variables?

Is there a polynomial $f(x,y)$ in two variables, with integer coefficients, such that $f$ is irreducible over the complex numbers (i.e., in $\mathbb{C}[x,y]$), but for every integer $n$, the ...
6
votes
1answer
399 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 $k$ if and only if ...
2
votes
1answer
447 views

common roots of bivariate polynomial equations

Let $f(x,y)=0$ and $g(x,y)=0$ be bivariate polynomial equations where the polynomials have the same degree, say, $N\geq 3$. Furthermore, both of them have the same terms but different coefficients. ...
3
votes
1answer
267 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 $k$, ie ...
1
vote
2answers
172 views

Resultant of system with 3 polynomials and 3 variables

Let us say I have a system of 3 polynomials, f1(x,y,z), f2(x,y,z), f3(x,y,z). How to find the resultant of these 3 polynomials? What I mean is: is there any special method to do this? Does the ...
0
votes
1answer
112 views

Ideal membership (concerning polynomial invariants of orthogonal groups)

Let $\mathbb F _q$ be finite field of odd characteristic and consider the polynomials $$ \xi_i = x_1^{q^i+1} - x_2^{q^i+1} + x_3^{q^i+1} - x_4^{q^i+1} \in \mathbb F_q[x_1,x_2,x_3,x_4].$$ I'm ...
1
vote
0answers
661 views

Prime ideals in polynomial rings over integers

Im trying to find a characterization of the prime ideals in the polynomial ring $R = \mathbb Z[X,Y]$ in two variables over the integers. Actually I need to find the maximal ideals in quotient rings ...
2
votes
1answer
339 views

Is there an algorithm to decide if an ideal contains monomials?

Let $I\subset k[x_1,\dots,x_n]$ be an ideal in a polynomial ring in commuting variables. Is there a procedure to decide if $I$ contains a monomial and possibly to find one? Gröbner bases come to ...
1
vote
0answers
200 views

Algebraic Independence of Polynomials in n Variables with Real Coefficients

I am considering the problem of determining the algebraic independence of $n$ polynomials in $m$ variables with real coefficients, where $m \geq n$. The variables will be denoted by $a_{1}, a_{2}, ... ...
1
vote
2answers
179 views

Transformation of a bivariate polynomial into a homogeneous one

For a given a bivariate polynomial $P(x,y)$ with rational coefficients: Q1. How compute such (invertible) substitutions of its variables that would transform the polynomial into a homogeneous one? In ...
0
votes
2answers
451 views

On reducible polynomials with positive coefficients, $1$ as constant coefficient and certain bounds on coefficients

Given $a \in \mathbb{Z}$ with $a > 1$. Let $g(x) \in \mathbb{Z}[x]$ be a polynomial with $g(a)=\pm 1$. Let $h(x) \in \mathbb{Z}[x]$ be a polynomial with $h(a)= p$, a prime. Let $g(x)$ and $h(x)$ ...
0
votes
1answer
160 views

minimal spans of polynomial companions of co-prime polynomials.

Is there an algorithm to determine for given $P,Q$ in $\mathbb Z[x,x^{-1}]$ with $gcd(P,Q)=1$, the value of $min\lbrace Span(A)+Span(B): A,B\in \mathbb Z[x,x^{-1}],\ A\cdot P+B\cdot Q=1\rbrace$, where ...
7
votes
1answer
343 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
1answer
425 views

Multivariate Hensel's Lemma, but with only one polynomial

One version of Hensel's Lemma is the following statement: Let $R$ be a commutative ring with a unit. Given a polynomial $Q\in R[X]$ and a root $\alpha$ of $Q$ modulo some ideal $I$ (i.e. $Q(\alpha) ...
1
vote
1answer
181 views

Special case of testing integer polynomials for irreducibility

How much easier is testing polynomials of the form x^n + ax + b for irreducibility (in Z[x]) than testing polynomials in general? I am especially interested in the case where n is prime, which may be ...
10
votes
2answers
744 views

Symmetric group action on squarefree polynomials

The following dynamical system on polynomials comes mostly from idle curiosity, but I hope it is of some interest. Background Fix some natural number $n$. Let $P$ be the quotient of the polynomial ...
3
votes
3answers
405 views

Are there any neat algorithm to factor a homogenous polynomial, given we know this polynomial factors into linear forms?

Are there any neat algorithm to factor a multivariable (more than 2 variables) homogenous polynomial, given we know this polynomial factors into linear forms?
0
votes
1answer
96 views

Uniqueness of Hensel factors of a polynomial (invariant to change of “basepoint”)?

An important component of algorithms for factoring multivariate polynomials over a commutative ring $R$ is Hensel lifting. Here's a brief, concrete example to set the stage for my question: Let $f ...
6
votes
1answer
1k views

Maximal ideals in a polynomial ring over the real numbers.

Let $\mathbf{R}$ be the field of real numbers. What are the generators of the maximal ideals of the polynomial ring $\mathbf{R}[x_1, ... , x_n]$? If instead of $\mathbf{R}$ one considers the field ...
11
votes
1answer
485 views

The word problem in the ring of polynomials

This question must be well known but I cannot find it in the literature. Question: What is the computational complexity of the word problem in a subring of the ring of polynomials in $n\ge 1$ ...
5
votes
1answer
285 views

Are plethories a theory of basis-free polynomials?

This question is a follow-up to a question about the theory of polynomials. It should be quite clear by now that matrix theory and linear algebra are quite different topics. As the various answers ...
7
votes
1answer
996 views

Is there an alternative formula for solving cubic equations?

It is known that Cardano's formula for solving cubic equations is not good in the case of positive discriminants. In this case it expresses the solution through cubic roots of complex numbers. ...
2
votes
2answers
371 views

Criteria for system of parameters in polynomial rings

Let $k$ be a field and let $R=k[x_1,...,x_n]$ be a polynomial ring over $k$. A subset $\lbrace y_1,...,y_n\rbrace$ of $R$ is called a homogenous system of parameters (hsop) of $R$, if the $y_i$ are ...
0
votes
1answer
188 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
503 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 $B$, $M$ ...
2
votes
1answer
260 views

Polynomials mapping the twisted cubic power series ring to itself

If $f(X) \in \mathbb{F}_2((T))[X]$ and $f(\mathbb{F}_2[[T^2,T^3]]) \subseteq \mathbb{F}_2[[T^2,T^3]]$, then does it follow that $f'(0) \in \frac{1}{T}\mathbb{F}_2[[T]]$? This might seem like an ...
1
vote
4answers
1k views

A proof for a statement about polynomial automorphism

I already got a proof for the fact that if a polynomial map is surjective then it is also injective. However, I used the invariant dimension of a ring and I want a simpler proof. Bravo for any try. ...
1
vote
3answers
637 views

Detecting if a polynomial is a Pfaffian

Given an explicit polynomial, is there any kind of trick/algorithm to check whether it is a pfaffian of a matrix with linear entries?
5
votes
0answers
165 views

“Unknot” algebraic set defined by two mutually dependent set of variables

Let $n$ be an integer $\geq 4$, and let $V \subseteq {\mathbb C}^{2n-1}$ be the set of all $(a_1,a_2, \ldots ,a_n,b_1,b_2, \ldots ,b_{n-1}) \in {\mathbb C}^{2n-1}$ such that the derivative of the ...
6
votes
0answers
346 views

“Consecutive” irreducible polynomials

If $P\in {\mathbb Z}[X]$ is a polynomial of degree $2$, then it is easy to see that for any integer $m$, at least one of the polynomials $P-(m+1),P-(m+2),P-(m+3),P-(m+4)$ is irreducible in ${\mathbb ...
6
votes
3answers
505 views

A question about an application of Molien's formula to find the generators and relations of an invariant ring

In the very beginning of the book "Introduction to Invariants and Moduli" Shigeru Mukai proves Molien's formula for the Hilbert series of the invariant ring of a finite group action on $\mathbb C^n$. ...
8
votes
1answer
770 views

When are complex polynomial maps almost surjective?

Consider a complex polynomial map $f: \mathbb{C}^n \rightarrow \mathbb{C}^n$. For $n = 1$, the fundamental theorem of algebra says that, for any $y \in \mathbb{C}$ there exists $x \in \mathbb{C}$ ...
2
votes
1answer
620 views

Count the number of homogeneous polynomials

Is there a general way of counting the number of homogeneous polynomials of certain degree in a complex projective space or a weighted complex projective space, mod the ideal generated by some ...