4
votes
0answers
215 views

Polynomial dynamical systems

The question is somewhat related to the theory of permutation polynomials. Let $\mathbb{F}_p$ be a finite field of $p$ elements ($p$ is prime) and $\mathcal{V} = \mathbb{F}_p^2 = \{ (t_1,t_2)\::\: ...
17
votes
1answer
988 views

How to prove that every polynomial in an infinite family is irreducible over Q?

Consider the bivariate polynomial $$p(X,Y) = X^5 - (2 Y + 1) X^3 - (Y^2 + 2) X^2 + Y (Y-1) X + Y^3.$$ For every integer $y \ge 4$, I conjecture that $p(X,y)$ is irreducible in $\mathbb{Q}[X]$. How can ...
4
votes
0answers
139 views

The Alexander-Conway polynomial: from knots to braids?

The Alexander-Conway polynomial was the first knot invariant to be discovered, as far back as 1923 according to this link. Given that knots can be expressed in terms of quasi-toric braid closures, it ...
5
votes
2answers
412 views

Which polynomials are Fricke polynomials ?

Let me recall the definition which seems the most standard of Fricke polynomials. Let $G$ be the free group with two generators $u,v$. It is not very hard to prove that there exists a unique ...
1
vote
2answers
264 views

Completing unimodular vectors with $3$ entries in $F_2[t]$ to a $3$ by $3$ matrix with determinant equal to $1.$

Given $a_1,b_1,c_1$ in $F_2[t]$ with $\gcd(a_1,b_1,c_1)=1$ it is known that there exists an element $g$ of $SL(3, F_2[t])$ (by explicit construction) such that $g$ has first line $$ [a_1^2, b_1,c_1], ...
3
votes
2answers
335 views

Polynomial group Laws on $\mathbb{R}^2$

When students are first learning about groups, a classic example of a group that is not defined as a set of functions is the group whose underlying set is $\mathbb{R}\setminus-1$, and whose operation ...
6
votes
0answers
200 views

Involution on sextic polynomials?

The strangeness of $Aut(S_{6})$ suggests the following question: Let $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}$ be the zeros of a 6th degree polynomial $P$ over a field $F$. Let ...
4
votes
2answers
1k views

Algebraic integers on the unit circle

Consider a set of algebraic integers which lie on the unit circle, they will generate a multiplicative subgroup of $\mathbb S^1$. Do these objects have a name? I would guess they contain useful ...
1
vote
3answers
447 views

A question regarding polynomials whose roots satisfy certain algebraic relation

Suppose I know the following information about a function : 1) Its a polynomial (not an explicit equation, neither the roots nor the degree is known) 2) I have managed to find an algebraic relation ...
9
votes
2answers
1k views

How to show the galois group of a polynomial is not an alternating group?

I am trying to prove that a certain class of polynomials have symmetric galois group. Using the Newton polygon, I have shown that the galois groups of these polynomials are transitive on $k$-sets for ...