# Tagged Questions

587 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 ...
210 views

### Uniform bound for the number primes $p$ s.t. a polynomial has a root modulo $p$

It is well known that for a given polynomial $f \in \mathbb{Z}[x]$ the number of primes $p$ s.t. $f$ has a root modulo $p$ is infinite. In fact, one can even write down a formula for the density of ...
183 views

### A question about the Sylvester determinant

I am writing an invited article related to the diophantine equations like the abc-conjecture by the way of the determinant of the Sylvester matrix. We know that Sylvester never really finalize some ...
252 views

### Counting solutions modulo primes

Let $P(x)$ be an irreducible polynomial in $\mathbb{Z}[x]$ of degree $n.$ By $N(k,x)$ we denote the number of primes up to $x,$ such that $P(x)$ has exactly $k$ solutions in $\mathbb{Z}_p.$ Is it ...
113 views

### Question related to the number of rational curves on a hypersurface

The following is from a PhD thesis "Algebraic Circle Method" by Thibaut Pugin, which I am currently reading. Let $k$ be a finite field of order $q$. Let $\mathfrak{X} \subseteq \mathbb{P}^n_k$ be ...
124 views

1k views

### An algebraic number is not a root of unity?

This problem is related to my study of the Burau representation of the braid group $B_3$: I was trying to show that certain "congruence subgroups" are of infinite index. There is an approach that ...
767 views

### A curious identity related to finite fields

To three elements $a_1$, $a_2$, $a_3$ in the finite field $\mathbb F_q$ of $q$ elements we associate the number $N(a_1,a_2,a_3)$ of elements $a_0\in \mathbb F_q$ such that the polynomial ...
281 views

### Can a polynomial be almost always divisible by a member of a finite set of primes?

Special case of Bunyakovsky conjecture Let $f(x)$ be non-constant irreducible polynomial with integer coefficients, no fixed prime factor and positive leading coefficient. Let $S$ be a finite set of ...
236 views

### Reducibility of polynomials maps

Motivated by this question. Let $f \in \mathbb{Q}[x]$ or$f \in \mathbb{Z}[x]$ . Consider the sequence $f(x),f(f(x)), \ldots f^n(x)$. If some $f^k(x)$ is reducible, the rest iterates will be ...
52 views

496 views

### Is OEIS A007018 really a subsequence of squarefree numbers?

A comment in A007018 a(n) = a(n-1)^2 + a(n-1), a(0)=1 claims Subsequence of squarefree numbers (A005117). - Reinhard Zumkeller, Nov 15 2004 Is it really so? As far as I know, it is open problem ...
272 views

### Polynomial version of the conjecture about Power free-values of polynomials

The conjecture about Power free-values of polynomials is: Let $F(X)$ be a polynomial with integer coefficients and no repeated roots. For any $\epsilon > 0$, there exists a constant ...
1k views

### Forbidden polynomial identities by the abc conjecture

The Masonâ€“Stothers theorem states Let $a(t), b(t)$, and $c(t)$ be relatively prime polynomials such that $a + b = c$, with coefficients that are either real numbers or complex numbers. Then ...
262 views

### Cyclotomic polynomial question [closed]

To avoid case distinction overload, I also call (say) $Z^3-Z^4$ cyclotomic. Just divide out the $Z=0$ solutions in the following if they offend. In the following, all exponents are assumed to be ...
162 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 ...
843 views

### Is there any theorem like implicit function theorem in $\mathbb{Q}$ ?

My qeustion is that, is there any theorem like implicit function theorem in $\mathbb{Q}$ ? More precisely, let $p(\bar{x},\bar{y})$ be in $\mathbb{Z}[\bar{x},\bar{y}]$ such that in $\mathbb{Q}$, for ...
469 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 ...
786 views

### Cyclotomic polynomials with coefficients $0,\pm1$

Let me begin with what looks like a joke. According to a Bourbaki member, the following conversation occurred during a meeting dedicated to polishing the but-last version of an Algebra Bourbaki ...