Questions tagged [polynomials]

For both basic and advanced questions on polynomials in any number of variables, including, but not limited to solving for roots, factoring, and checking for irreducibility.

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If polynomials are almost surjective over a field, is the field algebraically closed?

Let $K$ be a field. Say that polynomials are almost surjective over $K$ if for any nonconstant polynomial $f(x)\in K[x]$, the image of the map $f:K\to K$ contains all but finitely many points of $K$. ...
Eric Wofsey's user avatar
39 votes
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Does there exist a polynomial $P(x,y)$ which detects all non-squares?

Problem. Does there exist a two-variable polynomial $P(x, y)$ with integer coefficients such that a positive integer $n$ is not a perfect square if and only if there is a pair $(x, y)$ of positive ...
Prism's user avatar
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31 votes
2 answers
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How to find area of a polygon built on the roots of a given polynomial?

How to find the area of a (maximum area convex) polygon, built on the roots of a given polynomial in the complex plane? For example, consider the equation: $$2x^5+3x^3-x+1=0$$ It has one real and four ...
Yuriy S's user avatar
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25 votes
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When is a polynomial contained in the ideal generated by its partial derivatives?

Let $R = k[x_1,\dots,x_n]$ be a multivariate polynomial ring over a field $k$ of characteristic zero, and let $f\in R$. Is there an easy-to-test necessary and sufficient condition on $f$ such that $f$...
Ben Blum-Smith's user avatar
23 votes
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716 views

Is $z=e^{\frac{1}{\log(x)}}$ a solution to an algebraic equation?

Is $z=e^{\frac{1}{\log(x)}}$ with $x\in\Bbb Q~\cap(0,1)$ a solution to an algebraic equation? Wolfram is telling me $z$ is algebraic but I'm not sure that I believe this. I believe it would follow ...
John Zimmerman's user avatar
20 votes
0 answers
410 views

A Polynomial Formed from the Roots of Another Polynomial ad infinitum

Let $P(x)$ be a monic polynomial of degree $d$ with complex coefficients. Let $r_1(P),r_2(P),\dots, r_d(P)$ denote the set of roots, ordered so that $|r_1(P)| \leq |r_2(P)|\leq\dots\leq |r_d(P)|$. ...
Simon Segert's user avatar
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19 votes
1 answer
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Existence of rational sequence such that a polynomial is split over $\Bbb{Q}$

Does there exist a sequence $(a_n)_{n\in \Bbb{N}}$ of rationals such that for all $n\in \Bbb{N}$, $a_n\neq 0$ and the polynomial $a_0+a_1X+\cdots+a_nX^n$ is split over $\Bbb{Q}$? I was asked this ...
user avatar
19 votes
0 answers
1k views

Gauss-Lucas Theorem (roots of derivatives)

Gauss-Lucas Theorem states: "Let f be a polynomial and $f'$ the derivative of $f$. Then the theorem states that the $n-1$ roots of $f'$ all lie within the convex hull of the $n$ roots $\alpha_1,\ldots,...
wieschoo's user avatar
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18 votes
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662 views

Bounding a polynomial from below

Let $\sigma >0$ be fixed. For even $k \in \mathbb{N} \cup \{0\}$, we consider the polynomial \begin{equation} \varphi_k(x) = \sum_{j=0}^{k} (-1)^j {k \choose j} b_j \, x^{2j} \quad x \in (-1,1), \...
char's user avatar
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18 votes
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525 views

Increasing derivatives of recursively defined polynomials

Consider recursively defined polynomials $f_0(x) = x$ and $f_{n+1}(x) = f_n(x) - f_n'(x) x (1-x)$. These polynomials have some special properties, for example $f_n(0) = 0$, $f_n(1) = 1$, and all $n+1$...
TomH's user avatar
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18 votes
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Polynomial in the components of the curvature tensor

Consider a closed Riemannian manifold $(M,g)$ of dimension n and let $K(t,x,y)$ be its heat kernel. Then it is known that the heat kernel has an asymptotic expansion as $t\downarrow 0$: $$K(t,x,x)\...
asd's user avatar
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17 votes
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Polynomials with same image on the rationals?

I am struggling with this problem: If $P_1$ and $P_2$ are two polynomials such that $P_1(\mathbb{Q})=P_2(\mathbb{Q})$, show that $P_1(x)=P_2(ax+b)$ for some constants $a,b$. Here is what I have done....
Chris Sanders's user avatar
16 votes
1 answer
707 views

Is $\bigl(X(X-a)(X-b)\bigr)^{2^n} +1$ an irreducible polynomial over $\mathbb{Q}[X]$?

Let $a, b \in \mathbb{Q}$, with $a\neq b$ and $ab\neq 0$, and $n$ a positive integer. Is the polynomial $\bigl(X(X-a)(X-b)\bigr)^{2^n} +1$ irreducible over $\mathbb{Q}[X]$? I know that $\bigl(X(X-...
user84673's user avatar
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15 votes
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How to prove this polynomial inequality?

How can we prove the following? If $\frac{dP_{n}}{dz}|_{z=z_{0}}=0$ then $|P_{n}(z_{0})|<2$ for all $n>1$, where $P_{n}(z)\equiv P_{n-1}^{2}+z$ and $P_{1}\equiv z$ $z$ is in the complex plane. ...
Jerry Guern's user avatar
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14 votes
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289 views

Gauss Sums of Cubic Characters

Let $p$ be a prime number, $p\equiv 1 \mod 6$, and $g$ a primitive root modulo $p$, i.e. a generator of the group $\mathbb F_{p} ^ {\times} \cong C_{p-1}$. The element $g^3 \in \mathbb F_{p} ^ {\times}...
Nanhui Lee's user avatar
13 votes
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316 views

On the properties of sum-of-squares polynomials

Definition 1. If a multivariate polynomial $f$ can be written as a finite sum of squared polynomials, i.e., $f(x)=\sum_{i = 1}^n g_i^2(x)$, then $f$ is SOS. Definition 2. If an $n$-variate polynomial ...
khashayar's user avatar
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13 votes
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How close can polynomials be in magnitude?

If $p$ and $q$ are polynomials and $|p(t)|=|q(t)|$ on the interval $[0,1]$, then in fact $p(t)=\omega q(t)$ for some constant phase $\omega$. I am curious about quantitative strengthenings of this ...
felipeh's user avatar
  • 3,790
12 votes
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How to prove this $p^{j-\left\lfloor\frac{k}{p}\right\rfloor}\mid c_{j}$

Let $p$ be a prime number and $g\in \mathbb{Z}[x]$. Let $$\binom{x}{k}=\dfrac{x(x-1)(x-2)\cdots(x-k+1)}{k!} \in \mathbb{Q}[x]$$ for every $k \geq 0$. Fix an integer $k$. Write the integer-valued ...
network o's user avatar
  • 459
11 votes
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160 views

The fair soup division and approximating numbers

Recently The Vee confessed: Literally every time I'm serving some soup I'm thinking of this little mathematical problem I devised. Solving the problem, I introduced the following notion. A number $q\...
Alex Ravsky's user avatar
11 votes
0 answers
898 views

The probability that a random (real) cubic has three real roots

We can formalize the notion of the probability that a randomly selected quadratic real polynomial has real roots as follows: Suppose $R > 0$, and suppose the random variables $a, b, c$ are (...
Travis Willse's user avatar
11 votes
1 answer
3k views

What does the degree of a matrix minimal polynomial encode?

Let $\mathsf{F}$ be any field. Let $A$ be an $n \times n$ matrix over $\mathsf{F},$ whose rank is $r \le n.$ Let $\mu \in \mathsf{F}[x]$ be the minimal polynomial of $A.$ What does $\deg(\mu)$ tell ...
user avatar
10 votes
0 answers
197 views

Is there a notion of polynomial ring in "one half variable"?

Let $C$ be the category of commutative rings. Is there a functor $F :C \to C$ such that $F(F(R)) \cong R[X]$ for every commutative ring $R$ ? (Here, we may assume those isomorphisms to be natural ...
Watson's user avatar
  • 23.8k
10 votes
0 answers
329 views

Expanding a product of linear combinations with coefficients $1$ and $-1$

For any odd natural number $n$, denote $t \equiv \frac{n-1}{2}$. Let $K$ be a field such that $\operatorname{char} K \neq 2$. Working over the polynomial ring $K\left[x_1,x_2,...,x_{n} \right]$, ...
PalmTopTigerMO's user avatar
10 votes
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169 views

Irreducibility of q-factorial plus 1

Is it true that $[n]_q! + 1$ is an irreducible polynomial over $\mathbb{Z}$ for all positive integers $n$ ? I checked that this is true for $n$ up to $20$. Here $[n]_q! := 1 (1 + q) (1 + q + q^2) \...
Penchez's user avatar
  • 201
10 votes
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266 views

$\gcd(p_{n-1}, \ n^5 - n^3 + n^2 - n + 1) = 1$ where $p_n = n$th prime.

How can I prove in general that, for all $n\geq 2$: $$ \gcd(p_{n-1}, \ n^5 - n^3 + n^2 - n + 1) = 1 $$ Seems to always be true: ...
Daniel Donnelly's user avatar
10 votes
0 answers
249 views

When does a polynomial fixing a subring imply its coefficients are in that subring?

Let $S$ be a subring of $R$. If $p$ is a polynomial with coefficients in $S$, then $p$ fixes $S$ (as a function, that is, $p(s)\in S$ for all $s\in S$). A converse statement is: If $p$ is a ...
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9 votes
0 answers
192 views

Prove or disprove that the power of positive term polynomial will be eventually single peak

This is a question that a classmate asked me three years ago. Let $P(x)=\sum_{i=0}^n a_ix^i$ be a polynomial such that each $a_i>0$. Prove or disprove that there exists a positive integer $r$ such ...
JetfiRex's user avatar
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9 votes
0 answers
219 views

When does a polynomial over a field define an injective function?

In the specific case of algebraically closed fields, the only injective polynomials are the linear ones. Finite fields are also fairly easy to describe since they effectively reduce down to looking at ...
Merosity's user avatar
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9 votes
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312 views

Ideal of $k[x,y]$ invariant under an involution

Let $k$ be a field of characteristic zero. The two-dimensional case: Let $p,q \in k[x,y]$, $I=\langle p,q \rangle$ a proper ideal of $k[x,y]$ and $\delta$ an involution on $k[x,y]$, namely, a $k$-...
user237522's user avatar
  • 6,467
9 votes
0 answers
221 views

Generalized series for $\pi$ - What is the polynomial?

The Madhava-Leibniz series for $\pi$ is $$4\sum_{n=0}^\infty\frac{(-1)^n}{2n+1}$$ When we regroup odd and even terms into individual new terms, $b_n=a_{2n}+a_{2n+1}$, we have the other well known ...
Jam's user avatar
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9 votes
0 answers
193 views

When the element-wise product of two ideals produces an ideal

Consider the ring $R=\mathbb C[X,Y]$. For every two ideals $I,J$ of $R$, define $I*J:=\{ij : i\in I, j\in J\}$. Now definitely, $I*J=J*I$ always holds. If $I$ is principal, then actually $I*J$ is an ...
user102248's user avatar
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9 votes
0 answers
306 views

Looking for references about a graphical representation of the set of roots of polynomials depending on a parameter

Answering, some time ago, to this question : Change in eigenvalues on changing one entry of a matrix, I had the idea of a graphical representation of roots of polynomial equations $$P(x,a)=P_{a}(x) \...
Jean Marie's user avatar
9 votes
0 answers
405 views

A Ramanujan-type trigonometric identity

At the end of the following article: http://www.ijpam.eu/contents/2013-85-1/15/15.pdf It is asserted that the russian mathematician, Sergey Markelov, in private communication, told them that he ...
user avatar
9 votes
0 answers
441 views

Why do these two integrals use roots of reciprocal polynomials?

There is the nice integral by V. Reshetnikov, $$\int_0^1\frac{dx}{\sqrt[3]x\ \sqrt[6]{1-x}\ \sqrt{1-x\,\alpha^2}}=\frac{1}{N}\,\frac{2\pi}{\sqrt{3}\;\alpha}\tag1$$ also discussed in this post. By ...
Tito Piezas III's user avatar
9 votes
1 answer
399 views

Elementary proof of irreducibility criterion

From ``Problems from the Book'' by Andreescu and Dospinescu, the following irreducibility criterion is presented: Let $f$ be a monic polynomial with integer coefficients and let $p$ be a prime. If $...
WeierstrassSauce's user avatar
9 votes
1 answer
450 views

If $F(a_1,\ldots,a_k)=0$ whenever $a_1,\ldots,a_k$ are integers such that $f(x)=x^k-a_1x^{k-1}-\cdots-a_k$ is irreducible, then $F\equiv0$

I'm trying to understand a proof of the following theorem (from section II of Hall's paper An Isomorphism Between Linear Recurring Sequences and Algebraic Rings): If $F(a_1, \ldots, a_k)$ is a ...
modnar's user avatar
  • 981
9 votes
0 answers
347 views

On the continuation of a polynomial

This exrcise is from the first section of Marden: Exercise 12. Let the interior of a piecewise regular curve $C$ contain the origin $\cal O$ and be star-shaped with respect to $\cal O$. If the ...
Antonio Vargas's user avatar
8 votes
0 answers
195 views

Prove that $\Phi_{420}(69) > \Phi_{69}(420)$

Let $\Phi_n(x)$ denote the nth cyclotomic polynomial. Prove that $\Phi_{420}(69) > \Phi_{69}(420)$. Observe that if $\phi$ denotes the Euler-phi function, \begin{align} \phi(420) &= (2^2-2) \...
user3472's user avatar
  • 1,195
8 votes
0 answers
172 views

Quantify the similarity between a polynomial roots and the roots of its derivatives

On $\mathbb{C}[X]$, many theorems and conjectures deal with relations between a polynomial roots and the roots of its derivatives. When looking at a graph, the derivative roots distribution somewhat ...
Jean-Armand Moroni's user avatar
8 votes
0 answers
122 views

Are circle-tangent polynomials always either odd or even?

By saying circle-tangent polynomial I mean a polynomial of degree $N>2$ that lies mostly inside a unit circle except for the two tails exiting the circle and going to infinity and is tangent to the ...
Shieru Asakoto's user avatar
8 votes
0 answers
72 views

To get all open sets in $\mathbb{R}^n$ with a unique polynomial

At the beginning of the book Tame topology and o-minimal structures, by Lou van den Dries, it said that all the open sets of $\mathbb{R}^n$ can be obtained from a single polynomial equation. More ...
QuinnLesquimau's user avatar
8 votes
0 answers
314 views

Is theory of equations a dead field today?

Is theory of equations a dead field today? By theory of equations I mean, specially, the study of polynomials and solving algebraic equations through radicals. There seem to be very few journals on ...
Mr. N's user avatar
  • 516
8 votes
0 answers
171 views

Does a sequence of $d$-SOS polynomials converge to a polynomial that is also $d$-SOS?

Let $\mathbb{R}[X]_{\leq 2d}$ denote the real vector space of polynomials of degree at most $2d$ in the coordinate ring $\mathbb{R}[X]$ of variety $X$. Definition: A polynomial $f$ is $d$-SOS if ...
Tio Miserias's user avatar
8 votes
0 answers
218 views

Sufficient conditions for $\mathbb{Q} (\alpha , \beta )=\mathbb{Q} (\alpha +c\beta ) \forall c\in \mathbb{Q}^*$

What are some sufficient conditions for $\mathbb{Q} (\alpha , \beta )=\mathbb{Q} (\alpha +c\beta ) \forall c\in \mathbb{Q}^*$ with $\alpha , \beta $ algebraic over $\mathbb{Q}$? We know that, for ...
Lucio Tanzini's user avatar
8 votes
0 answers
133 views

When do polynomial equations come from complexification?

If $f(z) \in \mathbb{C}[z]$ is a polynomial of degree $d$, then it has $d$ complex zeros. Writing the complexification $$f(x+iy)=u(x,y)+iv(x,y)$$ we observe that the real polynomial system $u(x,y)=v(x,...
Taylor's user avatar
  • 161
8 votes
0 answers
147 views

If prime $p=a_n10^n+a_{n-1}10^{n-1}+\ldots+a_110+a_0$ then $f(x)=a_nx^n+\ldots+a_0$ is irreducible in $\mathbb{Z}[x]$

I have been trying to solve this problem on my own for four days now, and I cannot figure out how to prove it: If we express a prime $p$ in base $10$ as $$p= a_m10^m+a_{m-1}10^{m-1}+\ldots +a_110+a_0,...
Ana's user avatar
  • 267
8 votes
0 answers
252 views

Domains such that $R[X] \cong S[X]$ but $\mathrm{Frac}(R)[X] \not \cong \mathrm{Frac}(S)[X]$

Is it possible to find two integral domains $R,S$ such that $R[X] \cong S[X]$ but $\mathrm{Frac}(R)[X] \not \cong \mathrm{Frac}(S)[X]$ ? $\renewcommand{\Frac}{\mathrm{Frac}}$ Here $\Frac(R)$ ...
Watson's user avatar
  • 23.8k
8 votes
0 answers
231 views

Fixed point of a polynomial mapping - what's the relation between the two views

Let $\sigma : \Bbb{C}^3 \to \Bbb{C}^3$ be a polynomial mapping. Let $P:= \Bbb{C}[x,y,z]$ denote the space of polynomial in 3 variables. Then $\sigma$ induces a (linear) mapping $\tilde{\sigma} : P\...
Teddy's user avatar
  • 2,396
8 votes
3 answers
1k views

Minimum number of terms resulting from the product of two polynomials with a given number of terms

Given two integers ($n$, $m$), what is the smallest number of terms that could result from the product of two polynomials with $n$ and $m$ non-zero terms respectively? That is, what is the smallest ...
thesquaregroot's user avatar
8 votes
0 answers
1k views

Runge's phenomen: interpolation error using Chebyshev nodes oscillates

We're trying to approximate the Runge function $f(x) = \dfrac{1}{1+25x^2}$ using Chebyshev nodes. When calculating the interpolation error, using different degrees ranging from 0 to 50, we get the ...
gieldops's user avatar
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