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Results tagged with polynomials
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user 26522
Questions in which polynomials (single or several variables) play a key role. It is typically important that this tag is combined with other tags; polynomials appear in very different contexts. Please, use at least one of the top-level tags, such as nt.number-theory, co.combinatorics, ac.commutative-algebra, in addition to it. Also, note the more specific tags for some special types of polynomials, e.g., orthogonal-polynomials, symmetric-polynomials.
13
votes
1
answer
526
views
Are the logarithms of the integer polynomials discrete in $L^1$ of the unit circle?
Tautologically, the integer polynomials form a discrete set in $L^1$ of the unit circle. On the other hand, a set of logarithms ordered by norm becomes generally rather denser than the original set. …
- 13.8k
10
votes
1
answer
581
views
Are the polynomials in $\{1/t\}$ dense in $L^2(0,1)$?
.
$$
Here $B_m(x)$ are the Bernoulli polynomials; so this (RH conditional) $L^2$ spanning set consists of certain polynomials in $\{1/t\}$ and $t$. …
- 13.8k
3
votes
0
answers
100
views
Independence of number fields generated by roots of Littlewood polynomials
Let $\mathcal{R}_d \subset \bar{\mathbb{Q}}$ be the set of all roots of degree $d$ polynomials with $\{-1,0,1\}$ coefficients and
$$
c(d) := \min_{\substack{ \alpha, \beta \in \mathcal{R}_d \\ \alpha^k … Thus, in particular, it follows easily that there are only finitely many irreducible and non-cyclotomic $\{-1,0,1\}$-polynomials $f(X)$ for which the number field $\mathbb{Q}[X] / (f(X))$ is Galois over …
- 13.8k
9
votes
0
answers
333
views
Is this a possible strengthening of the Lehmer conjecture?
Here is another possible refinement of the Lehmer conjecture.
For $\alpha \in \overline{\mathbb{Q}}^{\times}$, let $C_{\alpha} \subseteq \mathbb{Q}(\alpha)$ be the maximal cyclotomic field contained …
- 13.8k
8
votes
0
answers
220
views
Is there an approximate formula for the discriminant of a sparse polynomial?
Consider integer polynomials $P \in \mathbb{Z}[X] \setminus \{0\}$ of a degree $D \geq 1$ and without multiple complex roots. … An outstanding open problem is to prove (or disprove) that $d(P) \to \infty$ under any sequence of polynomials with degrees $D \to \infty$. …
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6
votes
0
answers
218
views
Extremal polynomial majorants of $\log{|f|}$: a multivariate extension of a theorem of Carne...
(The first two of those extremal polynomials are $P_0(z) = \log{2}$ and $P_1(z) = \frac{1}{2}( \log{2} - (z + z^{-1})/2 )$.) … arbitrary $f$ gets straightforwardly reduced to the $f(z) = 1 - z$ case, but this way of phrasing the result suggests an immediate generalization:
I would like to raise the following for multivariate polynomials …
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6
votes
Coefficients of shifted Bernoulli polynomials
Let me address (1). First, you need a correction: I suppose you intended $\frac{1}{n}B_n(x+k)$ in the empirical calculations, and not $\frac{1}{n-1}B_n(x+k)$. For instance, the $x^2$ coefficient of $B …
- 13.8k
12
votes
About irreducible trinomials
manifestly additive ($m(PQ) = m(P) + m(Q)$), the proof that the trinomial has not more than a single non-cyclotomic factor is an almost immediate consequence of two general, if not easy, facts about polynomials …
- 13.8k
8
votes
1
answer
326
views
Angular distribution of zero sets of sparse polynomials
Consider a sequence of complex polynomials $f \in \mathbb{C}[z]$, $f(0) \neq 0$, that are composed of a negligible fraction $o(\deg{f})$ of monomials. … Are the zeros of such polynomials necessarily equidistributed in angle, for the uniform measure $d\theta/2\pi$ on $S^1 = \mathbb{C}^{\times} / \mathbb{R}^{> 0}$? …
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11
votes
About the prime divisors of values of polynomials
Assume without loss of generality that $P$ is irreducible, and denote by $S_P(X)$ the set of primes $p < X$ that divide some value $P(n)$. Let $G$ be the Galois group of $P$ and $n_1 > 0$ the number o …
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13
votes
Accepted
Is the set of certain polynomials finite or infinite?
For the proof of this, write $p = \prod_{i=1}^d (x - 2\cos(2\pi \, t_i))$ and observe that the sequence of degree $d$ monic polynomials $p_n := \prod_{i=1}^d(x - 2\cos(2\pi n t_i))$ is also in $\mathbb … Combining these two statements, you can see that your set consists precisely of the products of pairwise different minimal polynomials of those $2\cos(2\pi / d)$ that do not exceed $1.99$. …
- 13.8k
39
votes
$f(x)$ is irreducible but $f(x^n)$ is reducible
There is no such polynomial.
It is clear that $f$ cannot be a cyclotomic polynomial (your condition $\deg{f} > 1$ excludes $x-1$). So suppose $f$ is non-cyclotomic and irreducible, of degree $d$, an …
- 13.8k
4
votes
Old question of Serre on discriminants of a sequence of polynomials
(Another reference is Prasolov's book Polynomials, which reproduces the same calculation). … However, one may wish to restrict to reciprocal polynomials; for Lehmer's problem this would be sufficient. …
- 13.8k
11
votes
2
answers
1k
views
Can there be a power basis for a totally real field of high degree?
A number field $K$ is said to have a power basis if there is an $\alpha \in K$ such that the full ring of integers $O_K$ is the $\mathbb{Z}$-linear span of $1,\alpha,\alpha^2,\ldots,\alpha^{\deg{K}-1} …
- 13.8k
29
votes
Accepted
Is $x^{n}-x-1$ irreducible?
To this I may add Prasolov's monograph Polynomials as a (probably) more accessible reference; there, you will find a complete treatment of Ljunggren's result.
Added. … Osada's paper to which he refers is The Galois groups of the polynomials $X^n + aX^l+b$ (J. Number Theory 25, pp. 230-238, 1987), although the result already appears in an earlier paper by E. …
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