Questions tagged [polynomials]
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.
380
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Distribution of roots of complex polynomials
I generated random quadratic and cubic polynomials with coefficients in $\mathbb{C}$
uniformly distributed in the unit disk $|z| \le 1$. The distribution of the roots of 10000
of these polynomials are ...
28
votes
2
answers
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Are there irreducible polynomials with all zeros on two concentric circles?
This is somewhat similar to this recent question, but extending in a different direction.
Let $f(x)$ be an irreducible polynomial of degree $n$ with integer coefficients. Call such $f$ a bicycle ...
25
votes
4
answers
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Using higher-order Bring radicals to solve arbitrary polynomials
It is well known that there is no general formula for the solution of the quintic. Of course, what this really means is that there is no general formula that only involves addition, subtraction, ...
25
votes
5
answers
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Monic polynomial with integer coefficients with roots on unit circle, not roots of unity?
There are certainly non-monic polynomials of degree 4 with all roots on the unit circle, but no roots are roots of unity; $5 - 6 x^2 + 5 x^4$ for example.
Now, for a monic polynomial of degree $n$, ...
23
votes
1
answer
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Hilbert's Theorem on $L_2$ norm of polynomials in $\mathbb{Z}[X]$ - Explicit construction and a converse?
Consider the set of polynomials with real coefficients as a vector space with the following inner-product: $\langle f, g \rangle = \int_{a}^{b} f(x)g(x) dx$.
Hilbert showed, in a paper from 1894, ...
20
votes
1
answer
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Density of polynomials in $C^k(\overline\Omega)$
Let $\Omega$ be an open and bounded subset of $\mathbb{R}^2$ and let $C^k(\Omega)$, $1\leq k<\infty$, be the space of functions $f$ with continuous derivatives of order $\leq k$ in $\Omega$, ...
19
votes
1
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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 that really so?
As far as I know, it is an open ...
18
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3
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858
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$\prod_k(x\pm k)$ in binomial basis?
Let $x$ be an indeterminate and $n$ a non-negative integer.
Question. The following seems to be true. Is it?
$$x\prod_{k=1}^n(k^2-x^2)=\frac1{4^n}\sum_{m=0}^n\binom{n-x}m\binom{n+x}{n-m}(x+2m-n)^...
18
votes
4
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Expressing power sum symmetric polynomials in terms of lower degree power sums
Is there an explicit formula expressing the power sum symmetric polynomials
$$p_k(x_1,\ldots,x_N)=\sum\nolimits_{i=1}^N x_i^k = x_1^k+\cdots+x_N^k$$
of degree $k$ in $N < k$ variables entirely ...
17
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3
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About the prime divisors of values of polynomials
Let $P$ be a polynomial having integer coefficients (and degree $\geq 3$), and let $\mathscr P_P$ be the set of prime numbers dividing some value $P(n)$ with $n \in \mathbb Z$.
Is it true that $\...
17
votes
1
answer
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Is $x^{n}-x-1$ irreducible?
Is it true that for every $n \in \mathbb{N}$, $x^{n}-x-1$ is irreducible in $\mathbb{Z}[x]$?
The standard irreducibility criteria seem to fail.
17
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6
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What's an example of a transcendental power series?
Let $k$ be a field. What is an explicit power series $f \in k[[t]]$ that is transcendental over $k[t]$?
I am looking for elementary example (so there should be a proof of transcendence that does ...
16
votes
1
answer
552
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Cyclotomic polynomials 2
My naive question may actually lead to something interesting.
Let $\Phi_m(x)$, $\Phi_n(x)$ be cyclotomic polynomials, $m<n$. These polynomials are relatively prime and so there are polynomials $...
16
votes
1
answer
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Reduced resultants and Bezout's identity
Introduction:
Given two univariate and coprime integer polynomials $f(x), g(x)$, we can always write
\begin{equation}
u(x)f(x)+v(x)g(x)=c
\end{equation}
for a unique pair of integer polynomials $u(...
15
votes
1
answer
639
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Is a wild automorphism of $k[x_1,\ldots,x_n]$, $n \geq 3$, necessarily of infinite order?
Let $k[x_1,\ldots,x_n]$ be a polynomial ring over a field $k$ of characteristic zero.
When $n=2$, it is known that every automorphism of $k[x_1,x_2]$ is tame, namely, a finite product of elementary ...
13
votes
3
answers
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Effective algorithm to test positivity
Let $f(x_1,\ldots, x_n)$ be a real polynomial in several variables. Is there an effective algorithm to test whether $f$ is positive (or nonnegative) on the whole of ${\mathbb{R}}^n$?
13
votes
2
answers
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Galois group of a product of polynomials
How can I compute the Galois group of the polynomial $fg\in K[x]$ assuming that I know the Galois groups of $f\in K[x]$ and $g\in K[x]$? Let's suppose for simplicity that the field $K$ is perfect.
13
votes
2
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Irreducible polynomial over number field with roots in every completion?
Let K/Q be a field, probably not a finite extension. Is it possible for a polynomial to be irreducible over K but have a root in every completion of K? What about all but finitely many completions?
...
11
votes
1
answer
330
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Analogue of conic sections for the permutohedra, associahedra, and noncrossing partitions
Slicing cones in various ways with a plane generates conic sections identified geometrically as hyperbolas, parabolas, or ellipses and algebraically, when suitably rotated, as certain rescaled ...
9
votes
3
answers
861
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Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in Z[t]$ be unbounded as n varies?
We've recently seen this question: Can the number of solutions $ab(a+b+1)=n$ for $a,b,n \in \mathbb{Z}$ be unbounded as $n$ varies? It appears initially plausible that the answer is yes, but evidently ...
9
votes
0
answers
673
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Van der Pol's identity for the sum of divisors and a quartic polynomial equation for odd perfect numbers
In Touchard (1953) it is mentioned that the sum of divisors $\sigma(n)$ satisfies the following recurrence relation ($n>1$):
$$n^2(n-1) = \frac{6}{\sigma(n)} \sum_{k=1}^{n-1}(3n^2-10k^2)\sigma(k)\...
8
votes
0
answers
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Is the Collatz conjecture known to be true for interesting unbounded classes of numbers?
The Collatz or the $3n+1$ conjecture is open.
Is there a specific polynomial $f(x)\in\mathbb Z[x]$ whose range is unbounded for which every integer of form $|f(m)|$ at $m\in\mathbb Z$ satisfies $3n+1$...
8
votes
2
answers
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Moment matching: construction of a mixture of Gaussian distribution with lower moments identical to Gaussian
This is a question related to the statistical model behind independent component analysis (ICA).
We assume that $Z \sim N(0,1)$. Our goal is to construct a random variable $X$ that follows a ...
8
votes
3
answers
597
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Is it possible to create a polynomial $p(x)$ with this relation between $p(0)$ and $p(c)$?
Given $b$ and $c$ with $b,c>1$, is it possible to construct a polynomial $p(x)$, whose degree is $n$ for all $c$ and $b$, such that:
$|p|$ is strictly increasing on $[1,c]$
and $|b \cdot p(c)| &...
8
votes
2
answers
304
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Bounds on coefficients of factors of a multivariate polynomial
Given a multivariate polynomial $F(x, y, ..)$ what is the smallest bound B that can be quickly found such that $|G|_{\infty} \le B$ for all factors $G$ of $F$. (I'm using $|G|_{\infty}$ to denote the ...
8
votes
7
answers
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Old question of Serre on discriminants of a sequence of polynomials
Let $P_n(t)$ be polynomials with integer coefficients with $d_n = \deg(P_n(t))$ going to infinity when $n$ goes to infinity
and with nonzero discriminants $disc(P_n(t)) \neq 0$.
Question: Is
$$
\...
8
votes
2
answers
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Expression for the sum of square roots of zeros of a polynomial
Let $f(x)$ be a polynomial of degree $n$ with rational coefficients whose zeroes are nonnegative real numbers: $x_1, \dots, x_n\geq 0$.
General question. Does there exist a simple expression for the ...
8
votes
1
answer
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polarization formula for homogeneous polynomials
given a homogeneous polynomial p of dgree n on $R^d$, there is a unique symmetric n-linear functional $B$ on $(R^d)^n$ such that $p(x)=B(x,..,x)$. The question is: Can we get $B$ by means of a ...
8
votes
2
answers
430
views
Inductive definition of Bernstein polynomials
For $n\in \mathbb{N}$ let $B_n$ be the linear operator taking a function $f$ on the unit interval $I=[0,1]$ to its $n$-th Bernstein polynomial $B_nf$,
$$ B_nf(x):=\sum_{k=0}^n\binom{n}{k} f\Big(\...
7
votes
2
answers
778
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"MultiCatalan numbers"
Could anyone provide a reference for the following (sort of) generalization of Catalan numbers: the multinomial coefficient
$$
\binom{2k_1+3k_2+4k_3+...}{k_1+2k_2+3k_3+...,k_1,k_2,k_3,...}
$$
is ...
7
votes
1
answer
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Why are all these families of polynomials finally log-concave?
This started when I was examining certain families of unimodal polynomials, i.e. $\sum_{k=0}^n a_kx^k$ where $a_0\le a_1\le\cdots \le a_k\ge\cdots \ge a_n$.
(Notation: in the following, the $a_k$ ...
6
votes
1
answer
915
views
Irreducible polynomial $p_{n}(x)=\sum_{k=0}^{n}\frac{x^k}{k!}$ for all positive integers $n$
Let $n$ be a positive integer greater than $1$, and define the polynomial $$p_{n}(x)=\sum_{k=0}^{n}\dfrac{x^k}{k!}$$
Is $p_{n}(x)$ irreducible in $\mathbf{Q}[x]$?
I can show it when $n$ is a ...
6
votes
1
answer
697
views
Find the maximum of $|a_{p}|$, if $a_0+a_1x+\dots+a_nx^n:[-1,1]\mapsto [-1,1]$
Let $n$ be a given positive integer, and let $f(x)=\displaystyle\sum_{k=0}^{n}a_{k}x^k$, where $a_{i}\in \mathbb{R}$, $0 \le i \le n$. If
$$|f(x)|\le 1,\qquad \text{for } ~|x|\le 1,$$
what is the ...
6
votes
2
answers
784
views
Is every square root of an integer a linear combination of cosines of $\pi$-rational angles?
For example, $\sqrt 2 = 2 \cos (\pi/4)$, $\sqrt 3 = 2 \cos(\pi/6)$, and $\sqrt 5 = 4 \cos(\pi/5) + 1$. Is it true that any integer's square root can be expressed as a (rational) linear combinations of ...
6
votes
2
answers
923
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Counterexample to Proposition of Granville related to abc conjecture
Looks like there is counterexample to Proposition related to
abc conjecture. Confusion is likely.
From RATIONAL AND INTEGRAL POINTS ON QUADRATIC TWISTS OF A GIVEN HYPERELLIPTIC CURVE, Andrew ...
6
votes
1
answer
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An equivalence relation on the space of polynomials in one complex variable
Let $P(z)$ be a polynomial with complex variable $z$. We consider the following distribution for the roots of $P(z)=0$: the distribution is a triple $(n_{1},n_{2},n_{3})$
where these integers are ...
6
votes
1
answer
515
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Is every polynomial a factor of a trinomial?
We say that $T(X) \in \mathbb{Q}[X]$ is a trinomial if there exist $A,B,C \in \mathbb{Q}$ such that $T(X) = AX^n + BX^m + C$ for some $n \geq m \in \mathbb{N}$.
Is it true that for each irreducible ...
5
votes
0
answers
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Nullstellensatz with nilpotents and $I=J(V(I))$
Let $R$ be the ring $$\mathbb{R}[t_1,t_2\ldots]/(t_1^2,t_2^2,\ldots)$$
Let $p_1,\ldots p_\ell$ be polynomials in $R[x_1,\ldots,x_n]$ whose constant terms are 0.
Let $f$ be a polynomial which is zero ...
5
votes
1
answer
348
views
(Variation of an old question) Are these functions polynomials?
This is a followup to the question here: How to show that the following function isn't a polynomial over Q?.
As before, let $b_1,b_2,\dots$ be an enumeration of $\mathbb Q$. The question might ...
5
votes
0
answers
515
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Log-concave polynomial is a log-concave function?
A polynomial $\sum\limits_{k=0}^n a_kx^k$ is log-concave if $a_0,\ldots,a_n$ constitute a log-concave sequence. I wonder whether the log-concave polynomial is also a log-concave function with respect ...
5
votes
1
answer
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Ideal generated by two univariate, coprime, integer polynomials
Let $f(x)$, $g(x)$ be two univariate, coprime, integer polynomials and let $I=\big(f(x),g(x)\big)$ the ideal of $\mathbb{Z}[x]$ generated by $f, g$. Let $I \cap \mathbb{Z}$, that is, the elements of $\...
4
votes
2
answers
332
views
algorithm for finding radical expressions of all conjugates of an arbitrary algebraic number expressed in radicals
By an algebraic number expressed in radicals, I mean one that is an element of a set $S$ characterized as follows:
$\mathbb{Z}\subset S$.
For any $a,b\in S$, $a+b,a·b\in S$.
For $a,b\in S$ with $b\...
4
votes
1
answer
239
views
A discrete operator begets even/odd polynomials
Given a function $f(x)$ define the forward shift operator by $Ef(x)=f(x+1)$ and the discrete derivative $\delta f(x)=(E-1)f(x)=f(x+1)-f(x)$.
Given a partition $\lambda=(\lambda_1,\lambda_2,\dots,\...
3
votes
0
answers
261
views
Guises of the refined Eulerian numbers, generated by tangent vectors (OEIS A145271)
The Eulerian numbers (OEIS A008292, not to be confused with the Euler numbers) pop up in numerous scenarios in combinatorics and advanced analysis, one as the components of the h-vectors of the ...
3
votes
1
answer
249
views
Is the restriction of a graded automorphism of a polynomial ring to a polynomial subring linearizeable?
Let $k$ be a field and let $A=k[x_1,\dots,x_n]$ be a polynomial algebra over $k$, and let $B\subset A$ be a graded subalgebra that is itself a polynomial ring, i.e. $B=k[f_1,\dots,f_m]$ for some ...
3
votes
0
answers
199
views
On an exact expression for the squares of the distances of the critical points to a given zero of a polynomial
Let $p(z) = \prod_{j=1}^{l+1} (z - z_j)^{M_j}$ be a complex polynomial of degree $n$, where the $z_j$ are distinct for $1, \ldots, l+1$. The first $l$ entries in the list $\{z'_1, \ldots, z'_{n-1} \}$ ...
2
votes
1
answer
593
views
Half spaces free of roots of a given polynomial
I thank Loic Teyssier and Emil Jerabek who helped me to revise the two previous version
This question is motivated by the following fact in complex variable:(I learned this fact from the book of ...
2
votes
0
answers
185
views
Infinite products from the fake Laver tables-Now with no set theory
We say that a sequence of algebras $(\{1,\dots,2^{n}\},*_{n})_{n\in\omega}$ is an inverse system of fake Laver tables if for $x\in\{1,\dots,2^{n}\}$, we have
$2^{n}*_{n}x=x$,
$x*_{n}1=x+1\mod 2^{n}$,...
2
votes
0
answers
94
views
Cryptography signature scheme based on hardness of finding points on varieties?
Related to this question Complexity of finding solutions of trapdoored polynomial.
I am trying to build signature scheme based on hardness
of finding points on varieties.
Let $K$ be field and $M=K[x_1,...
2
votes
2
answers
2k
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A bounded polynomial having bounded coefficients: several variables
Consider the multivariate case for the question "Approximation theory reference for a bounded polynomial having bounded coefficients" (Approximation theory reference for a bounded polynomial having ...