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-...

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0answers
20 views

Complexity of computing the Tutte polynomial of multigraph when the Tutte polynomial of the underlying simple graph is known

Let $G$ be multigraph with $l$ loops and $m$ multiple edges and $G'$ be the underlying simple graph (loops and multiple edges removed). Assume the Tutte polynomial of $G'$ is given. Q1 What is ...
5
votes
2answers
153 views

Generalized cycle index polynomial for the symmetric group

The answer to a particular calculation in quantum information theory gives me the following expression: Given $M$ specific elements of the symmetric group $S_n$, define the polynomial $$Z_n(\pi_1, \...
4
votes
0answers
164 views

Polynomials representing locally constant functions

Let $K$ be a finite field with $p$ elements. (a) Let $f\in K\lbrack x\rbrack$ be such that (i) $\deg(f)<p$ and (ii) $f(2x) = f(x)$ for $\geq (1-\epsilon) p$ values of $x$ in $K$. What can we say ...
2
votes
1answer
213 views

On the divisibility of a certain power sum

Does $1^n + 2^n + \cdots + m^n$ divide $(1+2+ \cdots +m)^n$ for any even integers $m, n\geq 2$ ?. For $n\leq 4$, the solution easily follows from the relevant identities. For $n\geq 6$, i suspect ...
5
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0answers
162 views

Is a wild automorphism of $k[x_1,\ldots,x_n]$, $n \geq 3$, necessarily of infinite order?

Let $k[x_1,\ldots,x_n]$ be a polynomial ring over a field $k$ of characteristic zero. When $n=2$, it is known that every automorphism of $k[x_1,x_2]$ is tame, namely, a finite product of elementary ...
0
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1answer
120 views

Standard rational functions from matrices

In linear algebra we get introduced to standard polynomials that are associated to matrices such as characteristic polynomials and determinants. What are some of the standard rational functions that ...
0
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0answers
45 views

Approximation by polynomials with coefficients sum

In this paper Erdos p.1176 remarked that if the coefficients of $f(z):\sum_{v=0}^{n}a_{v}z^v$ are all real,then $\sum_{v=0}^{n}|a_{v}|$ is maximal for $f(z)=\pm T_{n}(z)$,where $T_{n}(z)$ is Chebyshev ...
2
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1answer
75 views

Distribution of values of quadratic polynomials over a finite field

Let $S$ denote the set of squares in the finite field ${\mathbb F}_p$. Now let $f(x) \in {\mathbb F}_p[x]$ denote a degree $2$ polynomial, which is not a square of a linear polynomial. If we consider ...
0
votes
2answers
474 views

Solving a system of equations using Gröbner basis

In Sage (or any other package) when using Gröbner basis to solve a system of equations (some of which are non-linear equations) does computing the Gröbner basis for the ideal ID generated by the ...
2
votes
0answers
130 views

system of complex number equations

Let $a_1,a_2,a_3,a_4\in \mathbb{C}$ be distinct such that $$a_1^3+a_2^3+a_3^3+a_4^3=0$$ $$(1+|a_1|^2)a_1^2+(1+|a_2|^2)a_2^2+(1+|a_3|^2)a_3^2+(1+|a_4|^2)a_4^2=0$$ $$(1+2|a_1|^2+2|a_1|^4)a_1+(1+2|a_2|^...
5
votes
1answer
218 views

Estimating size of greatest prime divisor of a sequence of integers

Consider the numbers of the form: $$A_n = \prod_{\pm}\left(\pm 1\pm \sqrt{2} \pm \cdots \pm \sqrt{n}\right)$$ where, the product in taken oven all $2^n$ terms with variations in sign. We know such ...
2
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0answers
77 views

Octahedron and System of trigonometric equations

Could somebody help me to prove the following? $$\sum_{k=1}^6 \cos(2 \theta_k) (\cos(2\phi_k)-1))=0$$ $$\sum_{k=1}^6 \sin(2 \theta_k) (\cos(2\phi_k)-1))=0$$ $$\sum_{k=1}^6 \cos (\phi_k)=0$$ $$\sum_{k=...
-3
votes
1answer
209 views

Does differentiation widen, or narrow, the class of functions?

Let $\cal F^k$ be a set of functions, each of class $C^k$, i.e., both, for every function in $\cal F^k$: $k^{\textrm{th}}$ derivatives exist, and are continuous. Let $D(\cal F^k)$ be the set of ...
3
votes
1answer
155 views

Hermite-Kakeya Theorem for entire functions

In a question asked by Bobby Ocean, the following theorem is cited: Hermite-Kakeya Theorem(for polynomials) - Given two real-valued polynomials, $f$ and $g$, then $f(x)+g(x) r$ has only real zeros ...
1
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1answer
176 views

Reduced resultant of monic polynomials

Let $f(x)$ and $g(x)$ be coprime monic polynomials in $\mathbf{Z}[X]$ of positive degrees $m$ and $n$ respectively. It seems that in this case their reduced resultant can be obtained from the ...
8
votes
1answer
572 views

Sort-of Converse of Kolmogorov Zero-One Theorem

Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. The Kolmogorov Zero-One Theorem states that Suppose we have independent random variables $X_1, X_2, ...$. Then $\forall \ A \in \...
13
votes
2answers
459 views

regular polygon question

Let $a_1,a_2,\ldots,a_n$ be distinct points on the complex plane $\mathbb{C}$ and $L$ be a circle in $\mathbb{C}$ such that $$f(z):=\sum_{i=1}^n|z-a_i|^{2n-2}$$ is constant on $L.$ Could somebody ...
2
votes
2answers
106 views

Quick tests to differentiate eigenvalues

Given two real square symmetric matrices $A$ and $B$ are there any quick tests to make sure at least one of their eigenvalues differ without computing the eigenvalues and likely more robust or looking ...
3
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0answers
50 views

numerical stability of root identification via Newton-Raphson iteration of Stieltjes residue sums

I have asked several questions on math.SE in order to compute numerically the poles of high-degree Padé approximations for $e^{-x}$, because a computation directly from the polynomial ...
2
votes
1answer
240 views

On a (possible?) equivalence of Bunyakovsky conjecture

Dirichlet's Theorem on primes in arithmetical progressions is equivalent to the following statement: for all integer numbers $a,b$, where $\gcd(a,b)=1$, there exists at least one prime of the form $an+...
4
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0answers
187 views

Zeros of polynomials modulo a non-prime

Suppose I have a set $S$ and I want to find a polynomial $p$ such that $p(s) = 0 \mod n$ if $s \in S$, and that it is non-zero modulo $n$ otherwise. In the literature such an $S$ is sometimes called ...
4
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0answers
84 views

symmetric systems of polynomial equations

Suppose I have a polynomial $p(x_1,...,x_N)$ in $N$ complex variables, and I wish to solve $p(x_{\pi(1)},...,x_{\pi(N)})=0$ for all permutations $\pi \in S_N$. Clearly this is overdetermined for ...
2
votes
1answer
92 views

Pointwise convergence of polynomials to a function on a compact set K that is 1 on some disc D and zero outside D

Motivation of my question: Let $A$ be a bounded selfadjoint operator with spectral measure $E$ and $x$ a vector. Then it is easily seen that the closed linear span of all $A^nx$ ($n\in\mathbb N$) ...
3
votes
0answers
277 views

A relation between a ring with its polynomial ring

Let $\{f_i(x)\}_{i\in I}$ be a subset of $R[x]$ where $R[x]$ is the polynomial ring of $R$(a commutative ring with identity). If the ring $R/\langle f_i^2(n)-f_i(n)\rangle_{i\in I, n\in A}$, for every ...
0
votes
2answers
105 views

Number of polynomial terms for certain degree and certain number of variables

I would like to calculate the maximum number of polynomial terms given a certain number of variables and a certain degree. eg. given that the number of variables is 2 and the degree is 3, the maximum ...
5
votes
2answers
99 views

(Re)construction of a polygon from all inter-vertex distances

For a $n$-polygon in $\mathbb{R}^3$ the set of distances between all pairs of vertices is given. (How) is it possible to reconstruct the geometric structure of the polygone? Symbolically: For a set ...
0
votes
2answers
495 views

What is wrong with this counterexample to the Weak Bunyakovsky's conjecture and reformulation of Bunyakovsky's conjecture?

From HYPOTHESIS H AND AN IMPOSSIBILITY THEOREM OF RAM MURTY. On p. 13 BUNYAKOVSKY’S CONJECTURE ( WEAK FORM ). Let $f$ be a polynomial with integer coefficients and positive leading coefficients ...
5
votes
0answers
162 views

Plane real curves such that their intersections with lines are hyperbolic

Let $R$ be an (irreducible) plane real algebraic curve (without isolated points). Consider Zarissky closure of $R$ in ${\mathbb C}P^1$ (as real variety). Suppose that $\lambda\in R \Rightarrow\...
10
votes
0answers
192 views

Precise relationship between “finite” Fourier analysis and Galois theory in solving the cubic?

Suppose we want to solve$$x^3 - ax^2 + bx - c = 0.$$We know a priori that this can be factored as $(x - r_0)(x - r_1)(x - r_2)$; by Vieta's formulas, we know$$a = r_0 + r_1 + r_2,\quad b = r_0r_1 + ...
20
votes
1answer
884 views

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
votes
1answer
1k views

Is Lehmer's polynomial solvable?

The degree 10 polynomial $$\displaystyle x^{10} + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1$$ given by D.H. Lehmer in 1933 has the property that its largest real root, $\beta = 1.176280 \cdots$ is ...
2
votes
2answers
380 views

Polynomials which always assume perfect power values

Let $f(x)$ be a non-constant polynomial with integer coefficients. It is a well-known result that if $f(n)$ is a square for all integers $n$, then $f$ must in fact be the square of a polynomial (see, ...
2
votes
0answers
120 views

Integral representation of the complex homogeneous polynomial $z_1\cdots z_n$

Consider the transform (see e.g., (5.1) in this paper): \begin{equation*} \Lambda_\mu(q)(z) := \int_{\Delta_n} q(\zeta)\,\Re\left(\frac{1+\langle\zeta,z\rangle}{1-\langle\zeta,z\rangle}\right)d\mu(\...
6
votes
2answers
276 views

non commutative polynomial which is zero for all matrix evaluation

I want to work on $K$ an algebraic closed (commutative) field of characteristic zero (even if it seems to be more general). We can define the free K-algebra of polynomials in non commutative ...
1
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0answers
111 views

Algebraic independence criterion

Is there any criterion for checking algebraic independence of a set of polynomials in $n$ variables in terms of the leading monomials with respect to some monomial order ? The Jacobian criterion is ...
24
votes
4answers
847 views

Deep/precise relationship between two approaches to FLT for polynomials, $n = 3$

David Speyer commented the following here. I saw Brian Conrad give an excellent one hour talk to undergraduates where he proved that there do not exist nonconstant, relatively prime, polynomials $...
8
votes
0answers
160 views

$k[x_1, \dots, x_n]$ is free iff $\mathbb{C}[x_1, \dots, x_n]^G \otimes \text{Harm}(\mathbb{R}^n, G) \to k[x_1, \dots, x_n]$ isomorphism?

For any subgroup $G \subset \text{GL}_n(\mathbb{R})$ the set $\mathbb{C}[x_1, \dots, x_n]^G$, of $G$-invariant polynomials, is a graded subalgebra of $\mathbb{C}[x_1, \dots, x_n]$, resp. the set $\...
3
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0answers
52 views

$G$-harmonic polynomials, dimension of $\text{Harm}(\mathbb{R}^n, S_n)$? [closed]

Definition. Let $\text{Harm}(\mathbb{R}^n, G)$ be the space of $G$-harmonic polynomials on $\mathbb{R}^n$. My question is, what is the dimension of $\text{Harm}(\mathbb{R}^n, S_n)$?
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0answers
76 views

$\mathbb{C}[x_1, \dots, x_n]$ is a free $\mathbb{C}[x_1, \dots, x_n]^{S_n}$-module with certain generators [duplicate]

Let the symmetric group $S_n$ act on $\mathbb{R}^n$ by permutation of coordinates. This makes $S_n$ a subgroup of $\text{GL}_n(\mathbb{R}$ and the algebra $\mathbb{C}[x_1, \dots, x_n]^{S_n}$ is the ...
12
votes
2answers
568 views

No nonconstant coprime polynomials $a(t)$, $b(t)$, $c(t) \in \mathbb{C}[t]$ where $a(t)^3 + b(t)^3 = c(t)^3$

See David Speyer's answer here. I saw Brian Conrad give an excellent one hour talk to undergraduates where he proved that there do not exist nonconstant, relatively prime, polynomials $a(t)$, $b(t)...
1
vote
1answer
142 views

Connection between the Chebyshev polynomials and the Faber polynomials

From a comment on this question: @draks, there is a connection between the Chebyshev polynomials and the Faber polynomials (a.k.a. Shur polynomials), which 'invert" the cyclic partition ...
2
votes
0answers
55 views

How to approximate higher-degree multivariate polynomial in space of lower-degree multivariate polynomials with some constraints?

For a polynomial $P_{1}(x)$, $x\in {\mathbb R}^n$ with a higher-degree, how to find a lower-degree polynomial $P_{2}(x)$ with determined structure or bounded degree to approximate it with the ...
0
votes
2answers
307 views

Is there a way to find out how many distinct roots a polynomial has? [closed]

Let say we have an arbitrary polynomial over the reals, and we do not know whether it is separable or not. Is there some algorithm to find the number of roots it has in the complex number?
11
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1answer
435 views

$\pi e$ and an unfamiliar polynomial

Ever since my exposure to this integral involving $\pi e$, I've conjectured and set about evaluating the possible nature of the following integral $$\int_0^1 x^m \sin(\pi x) x^x (1-x)^{1-x} \ dx, \...
29
votes
2answers
978 views

Polynomial $g:\mathbb R^n \rightarrow\mathbb R^n$ with no critical point may have no root

Version 1 (solved): If $g$ : $\mathbb R^n \rightarrow \mathbb R^n$ is a polynomial, $Dg(x)$ is non-degenerate for every $x$, then there exists $x$, such that $g(x)=0$. Version 2: If $f$ : $\mathbb R^n ...
3
votes
0answers
69 views

Is there a natural covariant of sextic polynomials with the following coefficients?

Let $$\displaystyle f(x) = a_6 x^6 + a_5 x^5 + a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0 $$ be an irreducible sextic polynomial with integer coefficients. Write $\theta_1, \cdots, \theta_6$ for the ...
4
votes
1answer
259 views

Is the Veronese variety “enough” to describe all the $SL(V)$-orbits in $\mathbb{P}(\textrm{Sym}^dV)$?

I apologise in advance if the question will look ridicolous to experienced eyes: in this case a good reference will be enough to clarify my doubts. Let $V$ be a complex vector space of dimension $n$, ...
3
votes
1answer
292 views

If $F(x,y)$ is a polynomial which is not a square, then how often is the specialization $F(x,a)$ a square?

Suppose $F(x,y)$ is a polynomial in two variables over a field $K$, and $F(x,y)$ is not a square. When is $F(x,a)$ a square for $a\in K$? I would guess that Hilbert's Irreducibility Theorem might help ...
6
votes
2answers
390 views

Seeking an explanation for a peculiar factorization

Recently during my work, I encountered the following family of sextic polynomials $$\displaystyle x^6 - 3x^5 + cx^4 + (5 - 2c)x^3 + cx^2 - 3x + 1.$$ By plugging in various values of $c$, I noticed ...
3
votes
2answers
155 views

Rings all of whose torsion modules are cyclic

Let us call a (possibly non-commutative) ring $R$ "very good" if every finitely generated torsion left $R$-module is cyclic. Here is an example of such a ring: Let $k=\mathbb{C}((t))$ and let $R=k[\...