All Questions
14 questions with no upvoted or accepted answers
15
votes
0
answers
376
views
Reducible polynomials of the shape $f(t^2)$, where $f$ is irreducible
Let $f(x) \in \mathbb{Z}[x]$ be a monic, irreducible polynomial. What are necessary and sufficient conditions for $g(t) = f(t^2)$ to be reducible over $\mathbb{Q}$?
For instance, if $f(x) = x-1$ then $...
- 26.9k
6
votes
0
answers
106
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. ...
- 96.4k
5
votes
0
answers
153
views
On factorization algorithms for $\mathcal{O}[x]$
We know that $\mathsf{LLL}$ algorithm provides factorization procedure that runs in poly time for polynomials in $\Bbb Z[x]$ that are primitive.
What other rings $\mathcal{O}$ can we use instead of $\...
user76479
4
votes
0
answers
171
views
Nilpotent elements of $K=(\mathbb{Z}/n \mathbb{Z})[x]/f(x)$
This is related to an open problem.
Let $n$ be integer and $f(x)$ polynomial with integer coefficients and set $K=(\mathbb{Z}/n \mathbb{Z})[x]/f(x)$.
Let $S$ be the set of degree 2 nilpotent elements ...
- 25.4k
4
votes
0
answers
236
views
Is this property of polynomials generic?
Let $n \geq 2$, and consider a polynomial $f$ in $n$ variables, say over a field $K$ of characteristic 0. Recall that $f$ is geometrically irreducible if $f$ is irreducible over the algebraic closure ...
- 26.9k
3
votes
0
answers
222
views
Mod p reduction of geometrically irreducible polynomials
Let $f\in \mathbb Z[t,x]$ be a polynomial of positive degree that is irreducible over $\overline{\mathbb Q}[t,x]$. Is it true that for all but finitely many primes $p$ the reduced polynomial $f_p\in \...
- 101
3
votes
0
answers
95
views
Sign of bivariate polynomial evaluated over two algebraic numbers
I would like to compute the sign of a bivariate polynomial $f$ evaluated over two algebraic numbers $a$, $b$. The numbers are in "isolating interval representation" meaning that each one is defined by ...
- 151
2
votes
0
answers
121
views
When $\gcd(P(x),Q(x))\bmod R(x)=\gcd(P(x) \bmod R(x),Q(x) \bmod R(x))$?
Let's $P,Q\in\mathbb K[x]$, with $\mathbb K$ a finite field.
On what necessary and sufficient condition on $R \in \mathbb K[x]$ is it :
$\gcd(P(x),Q(x))\bmod R(x)=\gcd(P(x) \bmod R(x),Q(x) \bmod R(x))...
- 4,074
2
votes
0
answers
93
views
The prime spectrume of integral-valued polynomial ring
Let $ D $ be an integral domain with quotiont field $K $ and let $Int (D) $be the set of all integral-valued polynomials on $D $, that is, $ Int (D):=\{f \in K[x]\mid f (D) \subseteq D\} $. The ...
- 21
1
vote
0
answers
57
views
Families of polynomials given by tuples of binary forms with finitely many reducible members
Let $G_1, \cdots, G_n \in \mathbb{Z}[x,y]$ be binary forms, and put $\mathbf{G} = (G_1, \cdots, G_n)$. Consider the family of monic polynomials
$$\displaystyle \mathcal{F}_\mathbf{G} = \{x^n + G_1(p,q)...
- 26.9k
1
vote
0
answers
90
views
Mod $N^2$ evaluation of a polynomial defined by first $N-1$ roots
Given a prime $N$ and integer $g$, where $g$ is able to generate the multiplicative subgroup $(\mathbb{Z}/N^2\mathbb{Z})^*$, I am interested in any results simplifying or evaluating $f\in (\mathbb{Z}/...
- 11
1
vote
0
answers
53
views
On a structural decomposition of polynomials based on integral roots
Given an irreducible polynomial of structure $$f(x,y)=\sum_{\substack{i,j\in\{0,1,2\}\\i+j\leq3}}a_{i, j}x^iy^j\in\mathbb Z[x,y]$$ with $a_{2,1}a_{1,2}a_{1,1}a_{1,0}a_{0,1}a_{0,0}\neq0$ if $f(m,n)=0$ ...
- 1,826
1
vote
0
answers
289
views
gcd of polynomial values
Suppose that $f$ and $g$ are two coprime polynomials in $\mathbb Z[x]$. I'm interested in any sort of upper bound on $gcd(f(a),g(a))$, in terms of the integer $a$. Are there any results of this type?...
user47305
0
votes
0
answers
126
views
Nilpotent elements of $(\mathbb{Z}/n \mathbb{Z})[x_1,...,x_m]/\langle f_1(x_1,...,x_m),f_2(x_1,...,x_m),...f_k(x_1,...,x_m)\rangle$
This is generalization of the univariate case
and also related to open problem.
Let $n,k,m,B>1$ be positive integers and $f_1(x_1,...,x_m),f_2(x_1,...,x_m),...f_k(x_1,...,x_m)$ be polynomials with ...
- 25.4k