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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-2
votes
0answers
22 views

Find the number of connected components in pseudospectra [on hold]

Suppose: $B_i \in \mathbb{C}^{n \times n}$, $0<w_i\in \mathbb{R}$ $(i = 0,1,2,\ldots,m)$ ${\rm P}(x) ={\rm{B}_m} x ^m + \cdots + B_1 x + B_0$ is a matrix polynomial, and $x $ is a complex ...
5
votes
0answers
99 views

Has anyone seen these binary trees (Catalan-type related to the Gegenbauer polynomials and Motzkin paths)?

The OEIS entry A121448 enumerates binary trees with $n$ edges and $k$ vertices with outdegree 1. Has anyone seen these trees? The o.g.f. for this entry, $G(x,t)$, is essentially a discriminant ...
1
vote
0answers
58 views

Boundary of pseudospectra

Suppose: $B_i \in \mathbb{C}^{n \times n}$, $0<w_i\in \mathbb{R}$ $(i = 0,1,2,\ldots,m)$ ${\rm P}(x) ={\rm{B}_m} x ^m + \cdots + B_1 x + B_0$ is a matrix polynomial, and $x $ is a complex ...
0
votes
0answers
54 views

“Dimension” of ideals in $F_q[x]/\langle x^n-1\rangle$? [closed]

I'm very much confused by algebra. Hoping to get a bit more comfortable I tried to compute different things and see what happens... Let $F_q$ be the finite field with $q$ elements and ...
-1
votes
0answers
153 views

Where I can find a Carlitz's paper? [closed]

I am looking for a PDF of the paper: Carlitz, Lewis, Mills, Straus - Polynomials over finite fields with minimal value sets. Only I can find about it is this data: Mathematika / Volume 8 / Issue ...
1
vote
1answer
157 views

Methods for searching for prime generating polynomials

I am currently using a non-systematic, pseudo-random method for finding prime-generating polynomials, based on the Bateman-Horn method for finding likely candidates, and then narrowing down. I have ...
0
votes
1answer
79 views

approximation of products of polynomials

I am wondering whether the following can be proved: Suppose $p(z)$ and $q(z)$ are polynomials of degree $n$ with real coefficients and leading coefficient 1. Moreover, they have quite different ...
1
vote
1answer
107 views

When is a homogeneous polynomial an inner function on the unit torus?

What is the necessary and sufficient condition(s) for a homogeneous polynomial on torus(bidisk) to be an inner function? More precisely I want to know when absolute value of a homogeneous polynomial ...
5
votes
2answers
104 views

Moment matching: construction of a mixture of Gaussian distribution with lower moments identical to Gaussian

This is a question related to the statistical model behind independent component analysis (ICA). We assume that $Z \sim N(0,1)$. Our goal is to construct a random variable $X$ that follows a ...
0
votes
0answers
33 views

finite orbits of transformations of the rational function field

Let $K : = k(x_1, x_2, \cdots, x_t)$ be the rational function field and consider the transformation $\tau$ of $K$ defined by $u \mapsto \frac{\alpha(u) + b}{a}$, where $a \ne 0 , b \in k$ and $\alpha$ ...
0
votes
1answer
95 views

fixed points of an affine polynomial automorphism

Let $K:- k[x_1, x_2, \cdots, x_n]$ be the polynomial ring over a field $k$. Let $a_i, b_i \in K$ where $a_i \ne 0$. Consider the automorphism $\alpha$ of $K$ defined by $x_i \mapsto a_ix_i + b_i$. ...
0
votes
0answers
48 views

Is there any way to approximate the largest root of a polynomial? [duplicate]

I know Newton's method will always give a root, but is there a modified version of the method that will always give the largest root?
1
vote
1answer
107 views

Is there a limit definition for the roots of a polynomial with arbitrary degree? [closed]

I know there's no general formula for all the roots of a polynomial with a degree greater than 4, but is there some sort of limit (or other) definition to calculate the roots (particularly the largest ...
2
votes
0answers
110 views

Polynomials with some roots whose product is 1

I asked this question in this post but have not got a full answer. So I post it again on MO. Consider the complex coefficient polynomial equation \begin{eqnarray} ...
4
votes
1answer
174 views

Polynomial factoring over finite fields

What is known in general about the complexity of factoring polynomials over finite fields? For instance given $\Bbb F_q$ where $q=p^n$ and total degree $d$ polynomial in $m$ variables what can we say ...
30
votes
3answers
937 views

Polynomials with the same values set on the unit circle

Assume that $P(z)$, $Q(z)$ are complex polynomials such that $P(S)=Q(S)$, where $S=\{z\colon |z|=1\}$ (equality is understood in the sense of sets, but I do not know the answer even for multisets). ...
1
vote
1answer
89 views

Ternary cyclotomic polynomials with $n=15r$

Let $n=15r$ where $r>5$ is an odd prime number. If $r\!\!\! \mod 15 \equiv w$ then is it true that $\Phi_{n}(x)$ is not flat whenever $2<w<13$ ? In other words, are the flat ones necessarily ...
2
votes
0answers
92 views

Log-concave polynomial is a log-concave function?

A polynomial $\sum\limits_{k=0}^n a_kx^k$ is log-concave if $a_0,\ldots,a_n$ constitute a log-concave sequence. I wonder whether the log-concave polynomial is also a log-concave function with respect ...
0
votes
0answers
14 views

Complexity of computing the multivariate Tutte polynomial of clique where each edge have distinct label

The multivariate Tutte polynomial $Z_G(q,v)$ is generalization of the Tutte polynomial and each edge is labelled by variable $v_e$. $Z_G(q,v)$ is linear in $v_i$. Let $G$ be a clique where each edge ...
1
vote
0answers
19 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
134 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, ...
2
votes
0answers
138 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
200 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 ...
4
votes
0answers
150 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
votes
1answer
113 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
votes
0answers
41 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
votes
1answer
67 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
342 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
86 views

system of complex number equations

Let $a_1,a_2,a_3,a_4\in \mathbb{C}$ be distinct such that $$a_1^3+a_2^3+a_3^3+a_4^3=0$$ $$(1+|a_1|^2)a_1^2+(1+|a_2|^2)a_2^2+(1+|a_3|^2)a_3^2+(1+|a_4|^2)a_4^2=0$$ ...
5
votes
1answer
199 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 ...
1
vote
0answers
54 views

Octahedron and System of trigonometric equations

Could somebody help me to prove the following? $$\sum_{k=1}^6 \cos(2 \theta_k) (\cos(2\phi_k)-1))=0$$ $$\sum_{k=1}^6 \sin(2 \theta_k) (\cos(2\phi_k)-1))=0$$ $$\sum_{k=1}^6 \cos (\phi_k)=0$$ ...
-3
votes
1answer
195 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
144 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
vote
1answer
163 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 ...
5
votes
0answers
360 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 ...
7
votes
2answers
309 views

regular polygon and constant potential function

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
102 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
votes
0answers
40 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 ...
0
votes
0answers
65 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 ...
2
votes
0answers
99 views

Zeros of polynomials modulo 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 a root ...
4
votes
0answers
76 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
77 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
271 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
73 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
95 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
481 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 ...
4
votes
0answers
155 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 ...
8
votes
0answers
174 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 + ...
18
votes
1answer
489 views

Why can the general quintic be transformed to $w^5-5\beta w^3+10\beta^2w-\beta^2 = 0$?

The quintic can be transformed to the one-parameter Brioschi quintic, $$w^5-10\alpha w^3+45\alpha^2w-\alpha^2 = 0\tag1$$ 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 ...