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
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Are $\mathbb C$ , $\mathbb C[X]$ definable in $\mathbb C[[X]]$?
Let $L$ be a first-order language and $M$ be an $L$-structure. Let $D \subseteq M^n$ . Let us say $D$ is definable in $M$ if for some finite set (possibly empty) $A=\{a_1,...,a_m\} \subseteq M$ and ...
user111524
21
votes
1
answer
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Primes represented by two-variable quadratic polynomials
I'm looking over a paper, "Primes represented by quadratic polynomials in two variables" [1] which attempts to characterize the density of the primes in two-variable quadratic polynomials. ...
- 8,994
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1
answer
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Why can the general quintic be transformed to $v^5-5\beta v^3+10\beta^2v-\beta^2 = 0$?
The quintic can be transformed to the one-parameter Brioschi quintic,
$$u^5-10\alpha u^3+45\alpha^2u-\alpha^2 = 0\tag1$$
This form is well-known for its connection to the symmetries of the ...
- 12.1k
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votes
1
answer
719
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What is the best probabilistic estimate from below for a random polynomial on an arc?
I'm currently involved in a small (but quite time consuming) project where we are trying to get some decent bound for the number $N(P)$ of real zeroes of a random polynomial $P(x)=\sum_{k=0}^n\xi_k x^...
- 59.8k
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13
answers
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Finding all roots of a polynomial
Is it possible, for an arbitrary polynomial in one variable with integer coefficients, to determine the roots of the polynomial in the Complex Field to arbitrary accuracy? When I was looking into this,...
- 333
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votes
5
answers
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How can you find an integer coefficient polynomial knowing its values only at a few points (but requiring the coefficients be small)?
Example: How can you guess a polynomial $p$ if you know that $p(2) = 11$? It is simple: just write 11 in binary format: 1011 and it gives the coefficients: $p(x) = x^3+x+1$. Well, of course, this ...
20
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6
answers
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What are the properties of this polynomial sequence?
Consider following polynomial sequence.
$$\begin{cases}a_{-1}=0,~a_0=1, \\a_{n+1}=x \cdot a_n \pm a_{n-1}\end{cases}$$
Here $+$ or $-$ is taken in such a way that all coefficients in $a_n$ do not ...
- 1,141
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votes
1
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Minimum value of $|p(1)|^2+|p(2)|^2 +...+ |p(n+3)|^2$ over all monic polynomials $p$
Let $n$ be a positive integer. Determine the smallest possible value of $|p(1)|^2+|p(2)|^2 +...+ |p(n+3)|^2$ over all monic polynomials $p$ of degree $n$.
This question was proposed (problem A.611)
...
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1
answer
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Irreducibility of the polynomial $x^n+5x+3$ over $\mathbb{Q}$
This question is first asked by me on MSE, but I haven't recieve a nice answer yet.
I would like to determine whether the polynomial $p(x)=x^n+5x+3$ is irreducible over $\mathbb{Q}$ when $n\ge 2$. ...
- 315
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2
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Maximal Ideals in the ring k[x1,...,xn ]
Hi. From one of the forms of Hilbert's Nullstellensatz we know that all the maximal ideals in a polynomial ring $k[x_1, \dots, x_n]$ where $k$ is an algebraically closed field, are of the form $(x_1 - ...
20
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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 ...
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2
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Multiple roots of polynomials with coefficients $\pm 1$
Question P. Can a polynomial $P(x)=\sum_{n=0}^ma_nx^n$ with coefficients $a_n\in\{-1,1\}$ (and $P(1)=0$) have a multiple root in the interval $(\tfrac12,1)$?
Also I am interested in a similar question ...
- 40.9k
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Symmetric polynomial from graphs
Let $g$ be a directed, connected multigraph, on $n$ vertices, without loops.
Define
$$P_g(x_1,\dots,x_n) := Sym\left[ \prod_{(i,j) \in g} (x_i-x_j) \right]$$
where $(i,j)$ is the directed edge ...
- 15.6k
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Refinement of mean value conjecture for complex polynomials?
I was playing around with Smale's Mean Value Conjecture and found a curious formulation of it which would be stronger (and which may simply be false). It seems to hold for `generic' random polynomials ...
20
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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 $\Omega$, ...
- 3,781
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0
answers
658
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Polynomials with roots in convex position
Let $\mathcal P_n$ denote the set of all monic polynomials of degree $n$ with real or complex coefficients such that $P\in\mathcal P_n$ if for all $k\in\lbrace 0,1,\dots,n-2\rbrace$ the $n-k$ roots of ...
- 17.5k
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4
answers
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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 ...
- 4,909
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votes
3
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783
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What is the shortest polynomial divisible by $(x-1)(y-1)(x^2y-1)$
I am interested in polynomials with few terms ("short polynomials", "fewnomials") in ideals. A simple to state question is
Given an ideal $I\subset k[x_1,\dots,x_n]$, what is the shortest polynomial ...
- 1,961
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3
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How to show that the following function isn't a polynomial over Q?
Enumerate the rationals as $b_1,b_2,\dots$ and define the (set) function:
$$f(x) = (x-b_1)^2 + (x-b_1)^2(x-b_2)^2 + \dots.$$
At any particular $x$, only finitely many terms are non zero so this is ...
- 7,646
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2
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Is there an explicit expression for Chebyshev polynomials modulo $x^r-1$?
This is an immediate successor of Chebyshev polynomials of the first kind and primality testing and does not have any other motivation - although original motivation seems to be huge since a positive ...
- 17.5k
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3
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What is the story behind the Chebyshev polynomials?
Is there anything reliable known about who actually discovered the Chebyshev polynomials and what the motivation and circumstances were?
The reason why I am interested in knowing, is that I needed a ...
- 12.7k
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1
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Are the following identities well known?
$$
x \cdot y = \frac{1}{2 \cdot 2 !} \left( (x + y)^2 - (x - y)^2 \right)
$$
$$
\begin{eqnarray}
x \cdot y \cdot z &=& \frac{1}{2^2 \cdot 3 !} ((x + y + z)^3 - (x + y - z)^3 \nonumber \\
&-...
- 201
19
votes
2
answers
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Constants for Rolle's Theorem applied to polynomials
Rolle's Theorem states that $f(1/2)=f(-1/2)+f'(x)$ has a root in the open real
interval $(-1/2,1/2)$ if $f$ is continuous and differentiable. How large can the absolute value of such a root
$\xi$
be ...
- 17.5k
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votes
3
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Cyclotomic polynomials: $\Phi_n(p)$ is like $p^{\phi(n)}$ for big enough $p$, right?
Apologies in advance if this turns out to be simple. So far I haven't found a proof or a reference.
Although I like $p$ to be a prime, I can ask the following for positive integers $n$ and $p$, ...
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2
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Which algebraic relations are possible between algebraic conjugates?
For which non-constant rational functions $f(x)$ in $\mathbb{Q}(x)$ is there $\alpha$, algebraic over $\mathbb{Q}$, such that $\alpha$ and $f(\alpha) \neq \alpha$ are algebraic conjugates? More ...
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2
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Why is a matrix pencil called a pencil?
I'm trying to understand the historical context behind the word pencil in matrix pencils, or pencil of curves so on.
I am aware that even Gantmacher 1959 has this terminology however I don't know ...
- 295
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votes
2
answers
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Conditions for smooth dependence of the eigenvalues and eigenvectors of a matrix on a set of parameters
Let $A\in\mathcal M_n$ be an $n\times n$ real [symmetric] matrix which depends smoothly on a [finite] set of parameters, $A=A(\xi_1,\ldots,\xi_k)$. We can view it as a smooth function $A:\mathbb R^k\...
- 4,304
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votes
1
answer
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Counting real zeros of a polynomial
I recently came across a criteria to count the number of real zeros of a polynomial $P(x)$ with real coefficients. Unfortunately I cannot find the reference! The criteria is the following: Form the ...
- 1,021
19
votes
2
answers
922
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Zeros of MacLaurin polynomials for the exponential function
Asked but never answered at MSE.
Let $\exp_n(z)$ denote the nth degree Taylor polynomial of $e^z$ :
$\exp_n(z) = 1 + z + z^2/2! + ... + z^n/n! \;$ .
The zeros of $\exp_n(z)$ were studied by ...
- 1,401
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votes
2
answers
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How can I distinguish a genuine solution of polynomial equations from a numerical near miss?
Cross-posted from MSE, where this question was asked over a year ago with no answers.
Suppose I have a large system of polynomial equations in a large number of real-valued variables.
\begin{align}
...
- 1,292
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votes
1
answer
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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 ...
- 24.2k
19
votes
1
answer
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"Local" Gauss-Lucas theorem?
The Gauss-Lucas theorem relates the location of zeros of a polynomial to the location of zeros of its derivative:
Suppose $f(z)\in \mathbb{C}[z]$ is a non-constant polynomial with roots $\alpha_1,\...
- 2,125
19
votes
0
answers
520
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univariate integer version of Hilbert's 17th problem
Let $f(x)$ be a polynomial of degree $d$ with integer coefficients such that $f(x)\geqslant 0$ for all real $x$. Is it necessarily true that there exists an integer $N(d)$ such that $N(d)\cdot f$ is a ...
- 103k
18
votes
5
answers
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Is $x^p-x+1$ always irreducible in $\mathbb F_p[x]$?
It seems that for any prime number $p$ and for any non-zero element $a$ in the finite field $\mathbb F_p$, the polynomial $x^p-x+a$ is irreducible over $\mathbb F_p$. (It is of course obvious that ...
- 17.5k
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7
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Proofs of the Chevalley-Warning Theorem
A well known proof of the Chevally-Warning Theorem is the one listed on wikipedia: http://en.wikipedia.org/wiki/Chevalley%E2%80%93Warning_theorem
Are there any other proofs of this, or ...
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3
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Strengthening Ax-Grothendieck
The question was cross-posted from Math.SE: https://math.stackexchange.com/questions/4566017/strengthening-ax-grothendieck
The question is simple. The Ax-Grothendieck theorem says a polynomial map $p\...
- 1,897
18
votes
1
answer
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Distinct integer roots for a degree 7+ polynomial and its derivative
Question: Is there a polynomial $f \in \mathbb{Z}[x]$ with $\deg(f) \geq 7$ such that
all roots of $f$ are distinct integers; and
all roots of $f'$ are distinct integers?
Background:
I asked a ...
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18
votes
4
answers
8k
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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$$
of degree $k$ in $N < k$ variables entirely ...
- 183
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votes
2
answers
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Can Schwartz-Zippel be formulated for commutative rings instead of fields?
The polynomials which occur in the Schwartz-Zippel lemma could be defined for any commutative ring, yet the lemma is restricted to fields. This makes it inapplicable for $(1+x^n)=1+x^n(\operatorname{...
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3
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$\prod_k(x\pm k)$ in binomial basis?
Let $x$ be an indeterminate and $n$ a non-negative integer.
Question. The following seems to be true. Is it?
$$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)^...
- 41.8k
18
votes
2
answers
1k
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Polynomials with many zeros of absolute value 1
Let $S$ be a finite subset of the positive integers. Define $N_S(x) =
1-(1-x)\sum_{j\in S}x^j$. Assume that $N_S(x)$ is symmetric, i.e.,
$x^dN_S(1/x)=N_S(x)$, where $d=\deg N_S(x)$. It seems that $N_S(...
- 49.5k
18
votes
1
answer
691
views
Is the p-adic density of the image of a polynomial always rational?
This question was previously posted here on MSE.
Let $P(x)$ be a polynomial with integer coefficients, and let $p$ be a prime number. For $n\in\mathbb N$, let $I_n$ be the number of integers $i\in\{1,\...
- 537
18
votes
1
answer
468
views
Iterated antiderivatives of polynomials having many real roots
Question For which polynomials $p_n:\mathbb{R} \rightarrow \mathbb{R}$ having $n$ distinct real roots can we find an infinite sequence of polynomials
$$ p_n, p_{n+1}, p_{n+2} , p_{n+3}, \dots, $$
such ...
17
votes
6
answers
3k
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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 ...
- 3,254
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votes
4
answers
1k
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A mixing property for finite fields of characteristic $2$
In connection with this MO post, here is a question somewhat implicitly contained in a joint paper of S. Kopparty, S. Saraf, M. Sudan, and myself.
Let ${\mathbb F}$ be a finite field, and suppose ...
- 22.8k
17
votes
2
answers
833
views
Non-negative polynomials on $[0,1]$ with small integral
Let $P_n$ be the set of degree $n$ polynomials that pass through $(0,1)$ and $(1,1)$ and are non-negative on the interval $[0,1]$ (but may be negative elsewhere).
Let $a_n = \min_{p\in P_n} \int_0^1 ...
- 779
17
votes
4
answers
976
views
Finite interpolation by a nondecreasing polynomial
Let $x_1 < x_2 < \ldots < x_n$ and $y_1 < y_2 < \ldots < y_n$ be two sequences
of $n$ real numbers. It is well known that there are polynomials that "interpolate"
in that $f(x_i)...
- 3,565
17
votes
3
answers
2k
views
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 $\...
- 3,826
17
votes
2
answers
1k
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$P(x)=P(y)$ has infinitely many integer solutions
Determine all polynomials $P(x)$ with integer coefficients such that $P(x)=P(y)$ has infinitely many integer solutions in integer $x$ and $y$ with $x \neq y$.
Choose $P(x)=a_n(x-k)^{2n}+a_{n-1}(x-k)^{...
- 501
17
votes
2
answers
1k
views
Images of polynomials
Let $f,g \in \mathbb{Q}[x]$ be polynomials such that $\{f(a) : a \in \mathbb{Q}\} \subseteq \{g(a) : a \in \mathbb{Q} \}$. Must there be some $h \in \mathbb{Q}[x]$ such that $f(x) = g(h(x))$ for all $...
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