We changed our privacy policy. Read more.

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.

4,810 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
94
votes
0answers
2k views

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 $K$. ...
22
votes
2answers
1k views

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 ...
20
votes
0answers
572 views

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 $f$...
18
votes
0answers
364 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 $|r_1(P)| \leq |r_2(P)|\leq\dots\leq |r_d(P)|$. ...
18
votes
1answer
271 views

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 ...
16
votes
0answers
357 views

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

Is $z=e^{\frac{1}{\log(x)}}$ a solution to a polynomial? $x\in\Bbb Q~\setminus\{0,1\}.$ Wolfram is telling me $z$ is algebraic but I'm not sure that I believe this. I believe it would follow from ...
15
votes
0answers
329 views

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)\...
15
votes
1answer
581 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-...
14
votes
0answers
612 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), \...
14
votes
0answers
445 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 $n+1$...
14
votes
0answers
378 views

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 ...
14
votes
0answers
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,...
13
votes
0answers
242 views

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....
13
votes
0answers
939 views

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. ...
12
votes
0answers
138 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}...
11
votes
0answers
117 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\...
11
votes
0answers
329 views

The Heegner Polynomials

What is special about $x^3- 6 x^2 + 4 x -2$? The 24th power of the real root - 24 is curiously close to two other numbers, one being the Ramanujan constant. There are more of these polynomials ...
10
votes
0answers
172 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 ...
10
votes
0answers
272 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 $K\left[x_1,x_2,...,x_{n} \right]$, ...
10
votes
0answers
222 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: ...
10
votes
0answers
992 views

Intuition about the Bernstein polynomials proof of the Weierstrass approximation theorem

The Weierstrass approximation theorem can be stated as follows: Let $f\in C([a,b])$, then there exists a sequence $(p_n)_{n\in \mathbb{N}}$ of polynomials in $[a,b]$ such that $(p_n)$ converges ...
10
votes
0answers
240 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 ...
9
votes
0answers
190 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 $k$-...
9
votes
0answers
176 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 ...
9
votes
0answers
139 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) \...
9
votes
0answers
176 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 ...
9
votes
0answers
348 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 ...
9
votes
0answers
740 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 (...
9
votes
1answer
252 views

Find the least possible value of $n$ such that there exist $P(x), Q(x) \in \mathbb{Z}[x]$

Find the least possible value of $n, n \geq 2015$ such that there exists polynomial $P(x)$ with degree $n$, integer coefficients, the coefficient of the term $x^n$ is positive and polynomial $Q(x)$ ...
9
votes
0answers
1k views

What is a fewnomial?

I came across the theory of "fewnomials" (by Khovanskii), which (I guess) are related to polynomials. However, I was surprised that there is no single question on stackexchange concerning fewnomials, ...
9
votes
0answers
349 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 $...
9
votes
1answer
254 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 ...
9
votes
0answers
337 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 ...
8
votes
0answers
152 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 ...
8
votes
0answers
118 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 ...
8
votes
0answers
108 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,...
8
votes
0answers
130 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,...
8
votes
0answers
217 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 $\Frac(R)$ ...
8
votes
0answers
374 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 ...
8
votes
0answers
209 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\...
8
votes
1answer
898 views

Proof of Cohn's Irreducibility Criterion

I was looking for an elementary (or involving introductory level abstract algebra/analysis) proof of Cohn's Irreduciblity Criterion: If $$ a_0, a_1, \dots, a_n \in \Bbb{Z} $$ and $$ 0 \le ...
8
votes
0answers
907 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 ...
7
votes
0answers
249 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 ...
7
votes
0answers
149 views

Polynomials with integer coefficients that can be iterated for infinitely many times and the result is always a prime number

Denote $f^{(n)}(x)=\underbrace{f(f(f(f(\cdots f}_{n \;\text{times}}(x)))))$. Does there exists a polynomial function $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ in $\mathbb{Z}[x]$ such that $\forall ...
7
votes
0answers
124 views

How to prove that all the roots $x_0$ of $p_n(x) = \frac{x}{f(n)} \sum\limits_{k=1}^n a_k p_{n-k}(x)$ satisfy $- \frac{4d}{a_1^2} f(n) < x_0 \leq 0$?

This question is an extension of Why are the roots of this recursive defined polynomial bound by the roots of the discriminant of the characteristic polynomial? As I discovered numerically, I can ...
7
votes
0answers
178 views

Examples of polynomials $f(x) = x^d + c, c \in \mathbb{Q}$ with rational periodic points of period $N \geq 4$

Let $f(x) = x^d + c$ be a polynomial where $ c \in \mathbb{Q}, d \in \mathbb{N}, d \geq 2 $. For $ n \in \mathbb{N}$, the $n^{\text{th}}$ iterate of $f$ is defined recursively by $$ f^n (x) = \left\...
7
votes
0answers
54 views

Let $\mathbb F$ be a field, find a necessary and sufficient condition on $\mathbb F$ such that the only semi-polynomial maps are the polynomials.

Let $\mathbb F$ be a field. A map $f : \mathbb F^2 \rightarrow \mathbb F$ is semi-polynomial if for every fixed $y$ the map $ x \rightarrow f(x,y)$ is a polynomial and for every fixed $x$ the map $y ...
7
votes
0answers
140 views

Number of distinct roots between complex roots among three polynomials

I want to prove that any three relatively prime polynomials $A, B, C \in \mathbb C[X]$ verifying $A+B+C=0$ have at least $1+\max(\deg A, \deg B, \deg C)$ distinct roots in total among each other. I ...
7
votes
0answers
522 views

$a$ and $b$ are solutions of $ \frac{1}{x^{2} - 10x-29} + \frac{1}{x^{2} - 10x-45} - \frac{2}{x^{2} - 10x-69} = 0 $, $a+b=?$

$a$ and $b$ are solutions of $$ \frac{1}{x^{2} - 10x-29} + \frac{1}{x^{2} - 10x-45} - \frac{2}{x^{2} - 10x-69} = 0 $$ What is $a+b=?$ $$ $$ Are there better approaches than the one below? Solution: ...
7
votes
0answers
110 views

$P(x,y) = n$ iff $n$ is NOT a perfect square

Does there exist a two variable polynomial $P(X,Y)$ with integer coefficients such that a positive integer $n$ is a perfect square iff there do not exists a tuple $(X,Y)$ of positive integers such ...

1
2 3 4 5
97