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, ...

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3
votes
0answers
145 views

What can we say about a semigroup that's acted on by an ideal of polynomials?

This got no response on MSE, so posting here. Let $S = \{ (a,b) \in \Bbb{Z}^2 : \gcd(a,b) \neq 1 \} \cup (1,1)$. Then $S$ forms a semigroup. The operation being componentwise multiplication. Let ...
10
votes
3answers
662 views

Probability of coprime polynomials

Given positive integer $N$, we choose $m_1, m_2, n_1, n_2$ independently and with equal probabilities from $\{0,1,\ldots,N\}$, and let $f_1 = x^{m_1} + (1+x)^{n_1}$ and $f_2 = x^{m_2} + (1+x)^{n_2}$ ...
2
votes
2answers
163 views

Finding a simpler “local” lower bound for a rational function

I have obtained as the expression for some quantity the following gargantuan formula: $$ \frac{k^8 + 3k^7 + 8k^6 + 3k^5 - 16k^4 - 32k^3 + 63k^2 - 34k + 6}{k^6 + 3k^5 + 6k^4 - 24k^2 + 21k - 5}$$. ...
1
vote
1answer
140 views

Roots of the derivative as symmetric functions of the roots of the polynomial

Let $p(t)=(t^2-a_1^2)\ldots(t^2-a_n^2)$ be an even polynomial with distinct real non-zero roots. Can the roots of its derivative $p'(t)$ be expressed nicely (e.g. as rational symmetric functions) in ...
6
votes
2answers
597 views

“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 ...
1
vote
0answers
138 views

Finite fields: alternating sums of values of polynomials

Notation In what follows let $p$ be a (odd, if needed) prime, $e$ a positive integer, $q = p^e$; $\mathbb{F}_q$ will denote a finite field with $q$ elements whose prime subfield will be denoted as ...
0
votes
0answers
137 views

Image of critical points

Let $K$ be a field of characteristic $0$, $f:K^n\rightarrow K^n$ be an algebraic function, that is, $n$ polynomial functions in $n$ variables. Let $S$ be the set of critical points of $f$. If ...
14
votes
1answer
592 views

When is $f(x_1, \dots, x_n)+c$ an irreducible polynomial for almost all constants $c$?

Let $f\in \mathbb C[x_1, \dots, x_n]$, $n\ge 1$, be a non-constant polynomial. Consider the polynomial $f+t\in \mathbb C[t, x_1,\dots, x_n]$. This is an irreducible polynomial in $\mathbb C(t)[x_1, ...
3
votes
1answer
270 views

Menon’s identity

I also put this question in stackexchange, but remained unanswered. http://math.stackexchange.com/questions/506996/menons-identity Let $G$ be a group of order $n$. Consider an action of $U_n$, the ...
5
votes
1answer
198 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 ...
7
votes
0answers
455 views

Getting a bound via polynomial equations

When studying the existence problem of power residue deference sets, I came across the following system of polynomial equations over $\mathbb{C}$, \begin{cases} ...
2
votes
1answer
339 views

About the practice of Bernstein-Kushnirenko theorem

The following refers to common roots of bivariate polynomial equations and, in particular to the quim's and auniket's comments. The BKK theorem (cf. arXiv:0812.4688. Theorem 5.4) asserts that if we ...
4
votes
2answers
309 views

On $a^4+nb^4 = c^4+nd^4$ and Chebyshev polynomials

In a 1995 paper, Choudhry gave a table of solutions to the quartic Diophantine equation, $a^4+nb^4 = c^4+nd^4\tag{1}$ for $n\leq101$. Seiji Tomita recently extended this to $n<1000$ and solved ...
3
votes
0answers
118 views

Elliptic curves and quasi-self-reciprocal polynomials

I am reading Shoichi Kihara's On the rank of the elliptic curve $y^2=x^3+k, II$ [Proc. Japan Acad. Ser. A Math. Sci. Volume 72, Number 10 (1996), 228-229] which is available here ...
7
votes
2answers
451 views

Dimension of a homogeneous polynomial system

Let $m\geq4$ be an even integer, $V\subset\mathbb{C}^{m-1}$ be the solution set of the following polynomial equations: \begin{cases} ...
5
votes
1answer
351 views

Is an ideal generated by multilinear, irreducible, homogeneous polynomials of different degrees always radical?

I asked this question on math.se and someone even put a bounty on it, yet there was no answer. Hence, I am asking here. Assume $\Bbbk$ to be a field of characteristic zero. Definition. A ...
4
votes
3answers
335 views

Set of primes dividing polynomials and composition

For a non-constant polynomial $A \in \mathbb{Z}[x]$, let $\mathcal{P}(A)$ denote the set of prime numbers $p$ which divide $A(n)$ for some integer $n$. If $\mathcal{P}(A) \subseteq \mathcal{P}(B)$ for ...
4
votes
0answers
106 views

Is the normalised Kauffman bracket more powerful than the Kauffman bracket?

The Kauffman bracket polynomial for a knot diagram $D$ is a Laurent polynomial $\langle D \rangle \in \mathbb{Z}[A, A^{-1}]$. Although it is invariant under Reidemeister moves of type II and III, ...
3
votes
0answers
139 views

Basis of multivariate polynomials with a specified set of roots

I'm studying certain polynomials of 2 complex variables, say x and y. These polynomials have roots at the non-negative integers, that is both $x$ and $y$ have to be $x,y \in \mathbb{N}$ ...
2
votes
4answers
1k views

Solving a System of Quadratic Equations

I have many polynomial equations in many variables which I want to jointly minimize (in a mean square sense, but you could pick a different reasonable measure which favors anything where all ...
3
votes
1answer
132 views

Irreducibility of trinomials over number fields

I wonder if the following is known or, not very difficult to see: Let $K$ be a number field and $\alpha \in K$ be nonzero. Does there necessarily exist a positive integer $n > 1$ such that the ...
21
votes
4answers
1k views

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 ...
4
votes
1answer
394 views

On composition of polynomials

Given two irreducible polynomials $f_{u}(x),f_{r}(x) \in \Bbb Q[x]$, can one find two polynomials or rational functions $h_{u}(x),h_{r}(x) \in \Bbb Q[x]$ or $\Bbb Q(x)$ respectively such ...
7
votes
3answers
375 views

How few terms may appear in a polynomial with given (cyclotomic) roots and nonnegative coefficients?

Given $W \subset \mathbb C$, let $S_W$ be the set of polynomials in $\mathbb R[x]$ that vanish on $W$ and have only nonnegative coefficients. Warm-up question: It's clear that if $W$ contains a ...
1
vote
1answer
322 views

Is there a quick way to find all roots of a real polynomial with multiple variables?

If I am asked to find the roots of a polynomial of one variable, I will use a computer to estimate the eigenvalues of its companion matrix. Now suppose I'm given a real polynomial of multiple ...
1
vote
0answers
215 views

Every antisymmetric (alternating) polynomial is divisible by Vandermonde product

I am looking for a proof of the statement: Every antisymmetric (alternating) polynomial is divisible by Vandermonde product, yielding a symmetric polynomial in the result. The statement is really ...
3
votes
1answer
174 views

Irreducibility of trinomials

I wonder if the following is known or, not very difficult to see: Let $K$ be a number field and $A, B \in \mathcal{O}_K$ be nonzero integers of $K$. Does there necessarily exist a positive integer $n ...
1
vote
1answer
147 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 ...
0
votes
1answer
170 views

Integral points on genus 0 curves related to polynomial identities

Consider the identity $$(x+1)^5+(y+1)^5 = (2x + 6y) ( -x +y )^4 + f(x,y)$$ where $f(x,y)=-x^5 + 2*x^4*y + 12*x^3*y^2 - 28*x^2*y^3 + 22*x*y^4 - 5*y^5 + 5*x^4 + 5*y^4 + 10*x^3 + 10*y^3 + 10*x^2 + ...
7
votes
3answers
471 views

Idempotent polynomials

Let $R$ be a commutative ring with identity and let $f \in R[x]$. There are well known characterizations for $f$ to be a nilpotent element of $R[x]$ or to have a multiplicative inverse in $R[x]$. Is ...
9
votes
3answers
727 views

Why is the gcd so large in an identity related to the $abc$ conjecture?

Consider the identity $$ (x+z)^5+(y-z)^5 = (-3 x + 4 y)^2 (x + y)^3 + (x+y) f(x,y,z) $$ Where $f(x,y,z)=(-8*x^4 + 5*x^3*y + 24*x^2*y^2 - 9*x*y^3 - 15*y^4 + 5*x^3*z - 5*x^2*y*z + 5*x*y^2*z - 5*y^3*z + ...
3
votes
0answers
81 views

Usefulness of polynomial ideals for graphs

Given a graph $G=(V,E)$ on $n$ vertices, one can associate to it the polynomial $f \in k[x_1,\dots,x_n]$ given by $f = \prod_{\{i,j\}\in E}(x_i - x_j)$. Then one can map various graph properties into ...
28
votes
3answers
2k views

Are surjectivity and injectivity of polynomial functions from $\mathbb{Q}^n$ to $\mathbb{Q}$ algorithmically decidable?

Is there an algorithm which, given a polynomial $f \in \mathbb{Q}[x_1, \dots, x_n]$, decides whether the mapping $f: \mathbb{Q}^n \rightarrow \mathbb{Q}$ is surjective, respectively, injective? -- And ...
5
votes
2answers
150 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 ...
7
votes
2answers
194 views

Finding a low-degree polynomial vanishing on half the zeroes of a polynomial system

Let $f(x)$ be a real polynomial of degree $2d$ without real roots. Let the complex roots be $z_1$, $\bar{z_1}$, $z_2$, $\bar{z_2}$, ..., $z_d$, $\bar{z_d}$ with $z_i$ in the upper half plane. Let ...
4
votes
0answers
164 views

Pair of two-variable polynomial equations of high order

I have the following pair of equations to be solved for two variables $\rho$ and $D$ resulting from a certain Maximum Likelihood Estimation for a time series $X_n > 0$, $n=0, \ldots, N+1$ with $N ...
2
votes
1answer
103 views

Degree principles for non-symmetric polynomials

A theorem of Timofte says that a symmetric polynomial inequality of degree $d$ holds on $\mathbb{R}^{n}_{+}$ if and only if it holds for all vectors in $\mathbb{R}^{n}_{+}$ with at most $\max\{\lfloor ...
8
votes
1answer
327 views

certain trigonometric homeomorphisms

Are there any simple characterizations of rational functions $f(x,y)$ with real coefficients such that $\theta\mapsto f(\cos\theta,\sin\theta)$ is a homeomorphism from $\mathbb R\bmod 2\pi$ to ...
4
votes
4answers
568 views

The Icosahedron Equation

$$1728 V^5 + F^3 = E^2 \;.$$ Can anyone point me to a concise, modern derivation and explanation of the significance of the icosahedron equation, more modern and concise than Klein's description in ...
1
vote
1answer
98 views

“Symmetric” Polynomial 4-cocycles

It is an old theorem of Heaton's (based on work of Eilenberg and MacLane), that a polynomial 3-cocycle $f(x,y,z)$ which is "symmetric," in the sense that $f(x,y,z)-f(x,z,y)+f(z,x,y)=0$, is always a ...
10
votes
1answer
398 views

Least prime $p$ such that an irreducible polynomial of degree $n$ has no root modulo $p$?

This question is inspired by an old question of Greg Kuperberg, about how small is the first prime $p$ which makes a given monic polynomial $P$ with integral coefficient have a (simple) root modulo ...
33
votes
2answers
2k views

A question on maps from $\mathbb{Z}/p\mathbb{Z}$ to itself

Let $p\geq 3$ be a prime number, and let $u:\mathbb{Z}/p\mathbb{Z}\to \mathbb{Z}/p\mathbb{Z}$ be a map such that, for all $l\in \mathbb{Z}/p\mathbb{Z}$,$l\neq 0$, the map $k\mapsto u(k+l)-u(k)$ is a ...
3
votes
2answers
339 views

Counting polynomials with same coefficient sum and a given value at a point

Call an univariate polynomial $f(x) = \sum_{i=0}^{n}a_{i}x^{i} \in \Bbb{Z}[x]$ symmetric if $a_{i} = a_{n-i}$ and $a_{0} = a_{n} > 0$. For a given $\sum_{i=0}^{n}a_{i}$ and $a_{i} \geq 0$, how ...
2
votes
1answer
191 views

Algorithm for representing a polynomial as a composition of lower degree polynomials

Let $q$ be a large prime and $e$ an integer such that $GCD(e,q-1)=1$. Let $p(x)$ be a polynomial of degree $e^n$ with coefficients in $\mathbb Z_q$ such that there exists a progression of polynomials ...
2
votes
1answer
281 views

Is there an irreducible integral polynomial in two variables which is reducible for every value of one of the variables?

Is there a polynomial $f(x,y)$ in two variables, with integer coefficients, such that $f$ is irreducible over the complex numbers (i.e., in $\mathbb{C}[x,y]$), but for every integer $n$, the ...
28
votes
1answer
1k views

Zeros of polynomials with real positive coefficients

The following problem arose in collaborative work with Subhro Ghosh: Question: To any polynomial $P_n(z)=\sum_{i=0}^n a_i z^i =a_n \prod_{i=1}^n(z-z_i)$, attach the empirical measure of zeros ...
1
vote
0answers
60 views

Jack symmetric functions and their inner products

I have some questions regarding Jack polynomials. I use the notation of of I.G. Macdonald's book "Symmetric Functions and Hall polynomials". Let $\Lambda$ be the ring of symmetric functions over ...
1
vote
2answers
144 views

Polynomials giving Lower Degree Elements in an Algebraic Number Field

My earlier related question Lower Degree Elements in an Algebraic Number Field has been given a clean answer for the first part. My present question is below: Take a number field ...
9
votes
1answer
350 views

Irreducibility of the trinomial over Q

I'm trying to find an algebraic proof of irreducibility of the polynomial $x^n-x-1$ over rational numbers (or integers, which the same). I've read the Selmer's paper "On the irreducibility of certain ...
1
vote
1answer
332 views

Existence of non-trivial solution to non linear polynomial system

I need to find conditions for the existence of non-trivial solutions to a multivariable polynomial system in two cases: The first case: $f1: a_1x^2+a_2xy+a_3y^2+a_4z^2=0$ $f2: ...