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.
2,555
questions
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2
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Kernel of evaluation map into field of quotients
Let $R$ be an integral domain and for $a \in R$ denote by $\text{eval}_a: R[X] \to R$ evaluation at $a$. It's well-known (and easy to see) that
$$\ker(\text{eval}_a)=(X-a).$$
The next more ...
- 1,999
6
votes
2
answers
783
views
Roots of this equation in x
I am examining the roots of the equation in $x$, $\sum_{q=0}^{2k-1} (-1)^{q} {2k+1 \choose q+1} x^{2k-q} m^{q}+r=0$ where $m$ and $r$ are positive integers.
I want to know whether the roots of this ...
6
votes
1
answer
878
views
What are prime number values of the trinomial $q(n) = n^2 + n + 41$? Assuming $n$ is a positive integer
Are there infinitely many integer values $n$ such that $q(n)$ is a prime number?
Numerical evidence points to a yes answer.
This is similar to Landau's 4th problem from 1912.
(The conjecture that ...
- 187
6
votes
1
answer
308
views
$p_n(x,y)=\sum_{i=0}^{n-1}x^{n-1-i}y^{i}$ is always an integer
Does anyone know if the following problem has ever been studied?
Let $a$ and $b$ be two real numbers and consider the polynomial: $$p_n(x,y)=\sum_{i=0}^{n-1}x^{n-1-i}y^{i}$$
where $n$ is a ...
- 2,929
6
votes
1
answer
301
views
Which criteria for "uniformly splitting" polynomials?
Let $P(x)$ be an irreducible monic polynomial of degree $\ge4$ with integer coefficients. We all know that over a finite field $\mathbb F_p$, $P$ will often split, and I am interested in polynomials ...
- 13.2k
6
votes
1
answer
810
views
Algebraic closure of a polynomial ring
What could be conditions on $k\in\mathbb{C}[x,y,z]$ that would ensure that any polynomial $f\in\mathbb{C}[x,y,z]$ that is algebraically dependent of $k$ is indeed a polynomial in $k$, ie $f\in\mathbb{...
- 63
6
votes
1
answer
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explicit formula for the j-invariant of binary quartic form
A binary quartic form
$aX^4+bX^3Y+cX^2Y^2+dXY^3+Y^4$
decomposes as a product of linear factors $Y-t_jX$, $j=1,...,4$.
I would like to have an explicit formula for symmetrization of the crossratio ...
- 105
6
votes
1
answer
860
views
Fundamental Theorem of Algebra, via algebra
I know there are already a couple of questions on this on the site, but I haven't seen an answer to this particular form...
We know, from the Fundamental Theorem of Algebra, that the complex ...
- 1,687
6
votes
1
answer
308
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A "known" Pythagorean identity in algebra?
Some will recognize this as similar to a question I asked before, but
I want to ask it without the trigonometry.
Let $e_k$ be the $k$th-degree elementary symmetric polynomial in
$x_1,x_2,x_3,\ldots$. ...
- 11.9k
6
votes
2
answers
609
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Can this system of equations about Newton's formula have concrete result?
Try to solve this system of equations:
$$
S_1=x_1+\dots+x_n=a;\\
S_2=x_1^2+\dots+x_n^2=a;\\
{}\cdots\\
S_n=x_1^n+\dots+x_n^n=a;
$$
And find the value of $S_{n+1}=x_1^{n+1}+\dots+x_n^{n+1},a\in\mathbb{...
- 75
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votes
3
answers
342
views
Second order recurrence relation for third order polynomial root
Consider this recurrence relation:
$$
\begin{eqnarray*}
f_0&=&1\\
f_n&=&
\sum_{m=0}^{n-1} \frac{\left(\frac{m+3}{2}\right)_{m-1}}{\left(\frac{m+2}{2}\right)_m} f_{n-m-1} f_m\ \ \ \...
- 237
6
votes
1
answer
265
views
If $f,g \in D[x,y]$ are algebraically dependent over $D$, then $f,g \in D[h]$ for some $h\in D[x,y]$?
This question asks: If $f,g \in k[x,y]$ are two algebraically dependent polynomials over an arbitrary field $k$, is it true that there exists a polynomial $h \in k[x,y]$ such that $f,g \in k[h]$,
...
- 2,783
6
votes
1
answer
645
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On the distribution of roots modulo primes of an integral polynomial
For motivation and related questions, see below.
Rough sketch of the question.
View $\bigsqcup_{p \text{ prime}} (\mathbb{Z}/p\mathbb{Z})$ as a ‘subset’ of the unit circle, via $a\pmod{p} \mapsto e^{...
- 83
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1
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461
views
On property of monic polynomial with integer coefficients
For a monic polynomial with integer coefficients $f$ where $\partial f = 2$, we have
$$
\textrm{inf}(f(x)) > 0 \implies
\textrm{inf}(f(x)) \geq \frac{3}{4} .
$$
Could we generalize this (for ...
- 1,531
6
votes
2
answers
572
views
What is the set of possible values of the degree of the sum of two algebraic numbers with fixed degrees?
This question is related to Degree of sum of algebraic numbers and algebraic numbers of degree 3 and 6, whose sum has degree 12.
In this last question I asked a very special case of the following ...
- 3,565
6
votes
1
answer
329
views
Check irreducibility of an explicit polynomial, without computer
I have a polynomial of degree 8 in 6 variables given explicitly by
$$ (\sqrt{1+(x_1+x_2+x_3)^2+(y_1+y_2+y_3)^2}+\sqrt{1+x_1^2+y_1^2}+\sqrt{1+x_2^2+y_2^2}+\sqrt{1+x_3^2+y_3^2})\times\text{the other ...
- 5,129
6
votes
1
answer
382
views
Calculate Ramanujan's class invariant by using modular equation of degree $5$
Let $$K(k):=\int_{0}^{\frac{\pi}{2}}\frac{d\phi}{\sqrt{1-k^2\sin^2\phi}}=\frac{\pi}{2}{ _2F_1\bigg(\frac{1}{2},\frac{1}{2},1;k^2 \bigg)}$$
where $0<k<1$.
Let $K, K′, L$ and $L′$ denote the ...
- 493
6
votes
1
answer
516
views
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 ...
- 11.2k
6
votes
2
answers
701
views
Can we decompose a polynomial into difference of convex polynomials?
Given a multivariate polynomial $p(x_1, ..., x_n)$ on $\mathbb{R}^n$, can we always decompose it into the difference of two convex polynomials? i.e., is there a pair of convex polynomials $f$ and $g$, ...
- 81
6
votes
1
answer
367
views
Polynomiality of functions over residue rings
Suppose $\mathbb{Z}/m \mathbb{Z}$ is a residue ring for some $m \in \mathbb{N}$. If $m=p$ is a prime number then every function $f:\mathbb{Z}/p \mathbb{Z} \rightarrow \mathbb{Z}/p \mathbb{Z}$ is a ...
- 411
6
votes
2
answers
307
views
Collision polynomials
Consider $P_n(x)$ polynomials defined through the recurrence relations
$$P_n(x)=2(1-x)P_{n-1}(x)-(1+x)^2P_{n-2}(x),$$ with $P_0(x)=1$ and $P_1(x)=1-3x$.
In fact, the explicit solution of these ...
- 16.4k
6
votes
3
answers
2k
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Multivariate Bernstein polynomials for approximation of derivatives.
If I have a $C^\infty$ function $f: [0,1]^n \to \mathbb{R}$ then its Bernstein polynomials
$$
B_m(x) = \sum_{k_1,\dots,k_n=0}^m f\left(\frac{k_1}{m}, \dots, \frac{k_m}{m}\right)
\prod_{i=1}^n \binom{m}...
- 413
6
votes
1
answer
1k
views
Low degree polynomial approximation for the entropy function
Let $X$ be a discrete random variable with possible values
$\{x_1,\ldots,x_n\}$, and let $p$ denote the probability mass function of
$X$. In addition, denote $p_i=p(x_i)$.
The entropy of $X$ is ...
user17253
6
votes
1
answer
2k
views
Polynomials are dense in weighted $L^2$ space
Hi,
It seems to be a common knowledge that the polynomials $x^n$ are dense in $L^2$ spaces with various probability weights, such as the gamma distribution weight $x^{\alpha-1}e^{-x}/\Gamma(\alpha)\;...
- 1,745
6
votes
2
answers
554
views
Distribution of zeros of a polynomial mod. a prime
Let $\mathbb P$ be the set of prime numbers.
Is there a non constant polynomial $f \in \mathbb Z[X]$ such that the set $$I_f := \{ \textstyle\frac{z}{p} : z \in \mathbb Z, p \in \mathbb P, p \mid f(...
- 11.9k
6
votes
1
answer
188
views
What is the smallest $\mathbb{Z}[x]$ multiple of $(x-1)^n$ in the coefficient vector $\ell_1$ sense?
Define a norm $\lVert p \rVert_1$ for $p\in \mathbb{Z}[x]$ as the sum of absolute values of the coefficients of $p$, as expressed in the ordinary monomial basis. What is the smallest norm of a ...
- 163
6
votes
1
answer
400
views
Rectangles in rectangles and $(b^2-a^2)^2\le (ax-by)^2+(bx-ay)^2$
When does an $a\times b$ rectangle fit inside an $x\times y$ rectangle? I have an algebraic condition which I can diagram geometrically, and I'd like a good geometric argument.
Assume $0<a<b$, $...
user44143
6
votes
1
answer
2k
views
$f,g \in \mathbb{Z}[x,y]$ satisfying: $\operatorname{Jac}(f,g)=0$ and $f,g \notin \mathbb{Z}[h]$ for every $h \in \mathbb{Z}[x,y]$?
Is it possible to find $f,g \in \mathbb{Z}[x,y]$ (with $\deg(f),\deg(g) \geq 1$) such that the following two conditions are satisfied:
(1) $\operatorname{Jac}(f,g)=f_xg_y-f_yg_x = 0$.
(2) ...
- 2,783
6
votes
1
answer
246
views
Knots indistinguishable by HOMFLY
Is there any list (incomplete of course) of knots, that have similar HOMFLY polynomials? I am mainly interested in torus knots.
- 63
6
votes
1
answer
1k
views
The resultant of an arbitrary polynomial and a cyclotomic polynomial
This is a natural generalization of this question.
Let $f$ be a monic irreducible polynomial over $\mathbb Z$. Let $S_f$ be the set of natural numbers $n$ such that one of the three equivalent ...
- 137k
6
votes
2
answers
726
views
bound for zeros of a polynomial with bounded integer coefficients
Let $f$ be a monic polynomial with bounded integer coefficients and such that all zeros are (in absolute value) greater than $1$. How close can the zeros of $f$ reach $1$ (in absolute value)?
More ...
- 567
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votes
2
answers
2k
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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 ...
- 1,775
6
votes
1
answer
483
views
A numerical matrix of power sum polynomials
Let $p_i=x_1^i+x_2^i+\cdots+x_m^i=\sum_{k=1}^mx_k^i$ be the power sum polynomials. Then, the determinant of the $m\times m$ Hankel matrix $M_m=(p_{i+j-2})$, for $1\leq i,j\leq m$, has a neat ...
- 41.8k
6
votes
1
answer
374
views
Values of the determinants $\det[(j-k)^m+\delta_{jk}]_{1\le j,k\le n}\ (m=1,2,3,\ldots)$
For positive integers $m$ and $n$, let $D_m(n)$ denote the determinant $\det[(j-k)^m+\delta_{jk}]_{1\le j,k\le n}$, where the Kronecker delta $\delta_{jk}$ is $1$ or $0$ according as $j=k$ or not.
...
- 14.5k
6
votes
1
answer
352
views
Does solving polynomial equations commute with tropicalization? (particularly for the field of Puiseux series)
The field of Puiseux series over an algebraically closed field of characteristic zero is also an algebraically closed field, and furthermore it has a valuation so that our Puiseux series can be ...
- 1,823
6
votes
1
answer
358
views
Negative coefficient in an almost cyclotomic polynomial
Let $a,b,c,d$ be four prime numbers. We set the polynomial :
$$P(X)=\frac{(1-X^{abc})(1-X^{abd})(1-X^{acd})(1-X^{bcd})(1-X^a)(1-X^b)(1-X^c)(1-X^d)}{(1-X)^2(1-X^{ab})(1-X^{ac})(1-X^{ad})(1-X^{bc})(1-X^{...
- 75
6
votes
1
answer
623
views
Are roots of transcendental elements transcendental?
This looks extremely easy, but then again it's late at night...
Let $k$ be a commutative ring with unity. An element $a$ of a $k$-algebra $A$ is said to be transcendental over $k$ if and only if ...
- 33.2k
6
votes
3
answers
827
views
Error in Polynomial Root Finding Algorithm with Synthetic Division
I have written a program which finds the roots of polynomial using Newton's Method. After finding the first root to within a tolerance (note that this also finds complex roots), I use synthetic ...
- 163
6
votes
1
answer
1k
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Specializations of Schur functions at consecutive integers
Given a partition λ = (λ1, λ2, ..., λn) denote with sλ the associated Schur function.
There exists a nice product formula for the principal specializations:
sλ...
- 1,372
6
votes
1
answer
190
views
Can the Chebyshev polynomials be constructed from the extremal property?
It is a known fact that the Chebyshev polynomials, defined as $T_n := \cos(n \arccos x)$, have the following extremal property:
Theorem: Of all monic degree $n$ polynomials, $\frac1{2^{n-1}} T_n$ has ...
- 63
6
votes
1
answer
258
views
The combinatorics of the Nullstellensatz for the variety of nilpotent matrices
Let $H_n$ denote the set of $n \times n$ nilpotent matrices with complex entries. The set $H_n$ may be regarded as an algebraic variety. Indeed, consider the polynomial ring $\mathbb{C}[A_{i,j} : 1 \...
- 445
6
votes
1
answer
303
views
Irreducibility of a polynomial when the sum of its coefficients is prime
I came up with the following proposition, but don't know how to prove it.
I used Maple to see that it holds when $ a + b + c + d <300 $.
Let $a,b,c$ and $d$ be non-negative integers such that $d\...
- 61
6
votes
1
answer
747
views
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 ...
- 312
6
votes
2
answers
1k
views
Finding the inertia group
Set $h(x) = x^5+x^4+x^3+x^2+x-1$, let $L$ be the splitting field of $h$ over $\mathbb{Q}$, and let $p$ be a prime of $L$ lying over $2$.
What is the isomorphism class of the inertia group $I_p$, ...
- 11.2k
6
votes
1
answer
529
views
An S-lemma for polynomials of degree 4 in three variables
Might the following be true:
Let $p,q\in\mathbb{R}[x,y,z]$ be homogeneous polynomials, with $\deg(p)\leq 4$ and $\deg(q)= 2$. Suppose $q(x,y,z)>0$ for some $x,y,z\in\mathbb{R}$. Then the ...
- 211
6
votes
1
answer
308
views
Is the minimal polynomial of an algebraic formal Laurent series always separable?
Let $f(x)\in K((x))$ be an algebraic formal Laurent series and let $P(x,y)\in K(x)[y]$ be its minimal polynomial. Is $P(x,y)$
always separable? An example of non separable polynomial comes
from ...
- 365
6
votes
1
answer
608
views
Maximum of a B-spline
Given $p+2$ nondecreasing (and not all identical) knots $t_0, \ldots, t_{p+1}$ on the real line, consider the normalized B-spline of degree $p$ defined over these knots.
We know that the B-spline is ...
- 278
6
votes
1
answer
638
views
Probability of a set of random vectors over finite field being a spanning set
Suppose I have a set of random vectors $f(a_1, \ldots, a_\ell) := (v_1, \ldots, v_m) \subset F_p^n$, $m \ge n$, given by a matrix valued polynomial function $f$, where the $a_i$'s are independent, ...
- 4,354
6
votes
1
answer
488
views
Polynomials with prescribed points to match prescribed bounds
Consider real polynomials on the interval $I=[-1,1]$. It is easy
to see that the smallest degree for a non-negative polynomial
with given zeros $x_1,\dots,x_s\in I^\circ$ is $n=2s$ (e.g.
$P(x) = \...
- 12.3k
6
votes
2
answers
2k
views
Examples of nice families of irreducible polynomials over Z
Hi,
i search for irreducible polynomials over Z which have variable coefficients you can "choose".
Since I found nearly nothing in books or the internet i hope you can help me.
Here 3 examples:
Let g ...
- 891